Finite Group Databasebeta

← #29723945

Subgroups — #29723945

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839917C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3C22⋊C4×Dic122C211yesmaximal, characteristic
4#21447358C211yesmaximal, characteristic, complemented
5#21447250C211yesmaximal, characteristic
6#21447141C211yesmaximal, characteristic, complemented
7#21447019C211yesmaximal, characteristic
8#21446494C211yesmaximal, characteristic, complemented
9#21446427C211yesmaximal, characteristic
10#21446163C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446072C211yesmaximal, characteristic, complemented
13#21445821C211yesmaximal, characteristic, complemented
14#21445753C211yesmaximal, characteristic
15C61×#2423C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16839917—24no—
22C4×#11758202—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×C22⋊#92418C2211yescharacteristic
35C2×C22⋊#92418C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839622C2211yescharacteristic
42#16839620—21no—
43C22⋊C4×Dic61—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C22⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#399—21no—
53C61×#398—21no—
54C61×#391C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#388—21no—
57C61×#387—21no—
58C61×#384C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C4×C22⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—22no—
65C22×Dic122—22no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202—22no—
74C2×#11758202—22no—
75C2×#11758202—41no—
76C2×#11758202D414yes—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80C22×#92418C2311yescharacteristic
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—41no—
84C4×Dic122—41no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92#14519710—22no—
93#14519710—41no—
94C4⋊#92418—41no—
95#14519701—22no—
96#14519701—41no—
97#14519700—22no—
98#14519700—22no—
99#14519700—22no—
100#14519700—22no—
101#14519700—22no—
102#14519700—41no—
103C22⋊#92418—22no—
104C22⋊#92418—41no—
105C4×#92418—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×Q8×C61C2311yescharacteristic
112C23×C4×C61C2311yescharacteristic
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—41no—
115C4×Q8×C61—22no—
116C4×Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#95×C61—21no—
133#95×C61—21no—
134#95×C61—21no—
135#95×C61—21no—
136#95×C61D411yescharacteristic
137#95×C61D411yescharacteristic
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—24no—
176C2×Dic122—21no—
177C2×Dic122—44no—
178C2×Dic122—44no—
179C2×Dic122—44no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—44no—
183C2×Dic122C2×D412yes—
184C2×Dic122#4112yes—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61—42no—
195C4⋊C4×C61—42no—
196C4⋊C4×C61—81no—
197C4⋊C4×C61—81no—
198C22⋊C4×C61—41no—
199C22⋊C4×C61—42no—
200C22⋊C4×C61—81no—
201C22⋊C4×C61Q1614yescomplemented
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61—21no—
206C42×C61—42no—
207C22⋊Dic61—42no—
208#11758202—28no—
209#11758202—44no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×D412yes—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61C2×Q1612yes—
237C23×C61C22×D411yescharacteristic
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61C2×Q1612yes—
272C2×C4×C61C2×Q1612yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61H(5,𝔽2)11yescharacteristic
278C2×C4×C61H(5,𝔽2)11yescharacteristic
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—48no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122—48no—
298Dic122—41no—
299Dic122—41no—
300Dic122#12314yes—
301#92418—22no—
302#92418—41no—
303#92418—41no—
304#92418—41no—
305#24906—611nomaximal, Sylow
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61#52511yescharacteristic
314C22×C61D4211yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61C22×Q1611yescharacteristic
317C22×C61#57511yescharacteristic
318C22×C61#57711yescharacteristic
319C22×C61#57811yescharacteristic
320C4×C61—22no—
321C4×C61—22no—
322C4×C61—41no—
323C4×C61—42no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61—81no—
329C4×C61C2×#12311yescharacteristic
330C4×C61#53811yescharacteristic
331C4×C61#54611yescharacteristic
332C4×C61#55111yescharacteristic
333Dic61—24no—
334Dic61—42no—
335Dic61—81no—
336Dic61—81no—
337C2×#451—611no—
338C2×C4×Q16—611no—
339#2423D6111yescharacteristic
340Q8×C22⋊C4—611no—
341#2095—611no—
342#2076—611no—
343#2056—611no—
344#2036—611no—
345#1933—611no—
346#1923—611no—
347#1878—611no—
348#1878—611no—
349#1863—611no—
350#1803—611no—
351#1787—611no—
352C2×C61—42no—
353C2×C61#241811yescharacteristic
354C2×C61#310411yescharacteristic
355C2×C61#312111yescharacteristic
356C2×C61#327111yescharacteristic
357C2×C61D4×Q1611yescharacteristic
358C2×C61#329911yescharacteristic
359C2×C61#332711yescharacteristic
360C23×Q8—611no—
361C22×Q16—611no—
362C22×Q16—611no—
363C2×C22⋊Q8C2×D6111yescharacteristic
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8C2×D6111yescharacteristic
367#451—1224no—
368C4×Q16—1224no—
369C2×#107—611no—
370C2×#107—611no—
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×C22⋊C8—611no—
381C2×C22⋊C8—611no—
382C2×C4×C8—611no—
383#399—21no—
384#398—21no—
385#391—611no—
386#391—611no—
387#391—1221no—
388#391—1221no—
389#391C2×D6111yescharacteristic
390#389C2×D6111yescharacteristic
391#388—21no—
392#387—21no—
393#384C2×D6111yescharacteristic
394#380—611no—
395#380—1221no—
396#380C2×D6111yescharacteristic
397C4×C22⋊C4—1221no—
398C4×C22⋊C4C2×D6111yescharacteristic
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#336—611no—
404#336—611no—
405C61#2490611yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—611no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—1222no—
412C22×Q8—1221no—
413C22×Q8—2441no—
414C22×Q8—2441no—
415C22×Q8C22×D6111yescharacteristic
416C23×C4—1221no—
417C23×C4C22×D6111yescharacteristic
418C2×Q16—614no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×Q16—1222no—
423C2×Q16—2441no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—611no—
428C22⋊Q8—22no—
429C22⋊Q8—41no—
430C4×Q8—22no—
431C4×Q8—41no—
432C4×Q8—1222no—
433C4×Q8—1222no—
434C4×Q8—2441no—
435C4×Q8—2441no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4C22⋊D6111yescharacteristic
452C2×C22⋊C4C22⋊D6111yescharacteristic
453C2×C42—21no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C2×C42C22×D6111yescharacteristic
458#107—1222no—
459#107—2441no—
460C4⋊C8—2441no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467#103—2441no—
468#103—2441no—
469C22⋊C8—1222no—
470C22⋊C8—2441no—
471C4×C8—1222no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95C22⋊D6111yescharacteristic
482#95C22⋊D6111yescharacteristic
483C24C2×C22⋊D6111yescharacteristic
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—612no—
490C2×Q8—612no—
491C2×Q8—612no—
492C2×Q8—612no—
493C2×Q8—1224no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—2442no—
503C2×Q8—2442no—
504C2×Q8—2444no—
505C2×Q8—2444no—
506C2×Q8—2444no—
507C2×Q8—2444no—
508C2×Q8D4×D6112yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—1221no—
520C22×C4—1224no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C2×C22⋊D6111yescharacteristic
528C22×C4C2×C22⋊D6111yescharacteristic
529C22×C4C2×C22⋊D6111yescharacteristic
530C22×C4C2×C22⋊D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532Q16—1228no—
533Q16—2444no—
534C2×C8—612no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—81no—
556C4⋊C4—81no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C22⋊C4—41no—
564C22⋊C4—42no—
565C22⋊C4—81no—
566C22⋊C4—2442no—
567C22⋊C4#1175820214yescomplemented
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—42no—
573C42—2442no—
574C42—2442no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23C2×#1175820212yes—
582C23C22×C22⋊D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—614no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8—1222no—
592Q8—1222no—
593Q8—2448no—
594Q8—2441no—
595Q8—2441no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—2448no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4C2×#1175820212yes—
639C2×C4C2×#1175820212yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#1451982211yescharacteristic
645C2×C4#1451982311yescharacteristic
646C8—1222no—
647C8—2441no—
648C8—2441no—
649C8—2441no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22#1684026911yescharacteristic
658C22D4×C22⋊D6111yescharacteristic
659C22#1684040011yescharacteristic
660C22C22×#1175820211yescharacteristic
661C22#1684056011yescharacteristic
662C22#1684057311yescharacteristic
663C22#1684057711yescharacteristic
664C4—22no—
665C4—22no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—1224no—
674C4—4881no—
675C4—4881no—
676C4—2442no—
677C4#123×D6111yescharacteristic
678C4#1684034811yescharacteristic
679C4#1684040411yescharacteristic
680C4#1684044011yescharacteristic
681C2—42no—
682C2#2144927811yescharacteristic
683C2#2145585411yescharacteristic
684C2#2145604211yescharacteristic
685C2#2145740011yescharacteristic
686C2D4×#1175820211yescharacteristic
687C2#2145768911yescharacteristic
688C2#2145808111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2423C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2423D61
hypocenterC61#24906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2423

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
4. #21447358C2semidirect
6. #21447141C2semidirect
8. #21446494C2semidirect
10. #21446163C2semidirect
11. #21446160C2semidirect
12. #21446072C2semidirect
13. #21445821C2semidirect
201. C22⋊C4×C61Q16semidirect
405. C61#24906semidirect
567. C22⋊C4#11758202semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399172
C2×C4×#117582022
C22⋊C4×Dic1222
#214473582
#214472502
#214471412
#214470192
#214464942
#214464272
#214461632
#214461602
#214460722
#214458212
#214457532
C61×#24232
#2490661