Finite Group Databasebeta

← #29779577

Subgroups — #29779577

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454169C211yesmaximal, characteristic, complemented
2#21448307C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447226C211yesmaximal, characteristic, complemented
5#21446959C211yesmaximal, characteristic, complemented
6#21446954C211yesmaximal, characteristic, complemented
7#21446517C211yesmaximal, characteristic, complemented
8#21446514C211yesmaximal, characteristic, complemented
9#21446510C211yesmaximal, characteristic, complemented
10#21446507C211yesmaximal, characteristic, complemented
11#21445914C211yesmaximal, characteristic, complemented
12#21445760C211yesmaximal, characteristic
13#21445757C211yesmaximal, characteristic
14#21445583C211yesmaximal, characteristic, complemented
15C61×#1825C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519754—21no—
19C22×C4×Dic61C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839666—21no—
34#16839663C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839659—21no—
38#16839658C2211yescharacteristic
39#16839639—21no—
40#16839639—21no—
41#16839631C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839620—21no—
44C4×#11758198—21no—
45C4×#11758198—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C2×#11758222—22no—
60C2×#11758222—22no—
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—41no—
64C2×#11758222—41no—
65C22×Dic122—21no—
66C22×Dic122C2311yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C23×Dic61D411yescharacteristic
70C23×Dic61D411yescharacteristic
71C22×C4×D61—21no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519776—41no—
75#14519776—41no—
76#14519754—44no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—41no—
90C2×C4×Dic61—41no—
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519702—22no—
96#14519702—41no—
97#14519700—22no—
98#14519700—41no—
99#14519699—22no—
100#14519699—41no—
101#14519697—22no—
102#14519697—41no—
103C22⋊#92418—22no—
104#14519687—41no—
105#14519687—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—22no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C22⋊C8×C61—22no—
126#95×C61—21no—
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—41no—
150C22×Dic61—41no—
151C22×Dic61—41no—
152C22×Dic61—41no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157#11758222—44no—
158#11758222—44no—
159#11758222—44no—
160#11758222—44no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244#4112yes—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122#4112yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊C4×C61—42no—
194C22⋊C4×C61—42no—
195C42×C61—42no—
196C22⋊Dic61—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—41no—
215C4×Dic61—41no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61#12412yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61#12412yes—
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C22⋊D61—48no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—44no—
275C2×Dic61—44no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61AGL(1,ℤ/8)12yes—
285C2×Dic61#13712yes—
286D244—22no—
287D244—22no—
288D244—41no—
289D244—41no—
290C4×D61—48no—
291Dic122—22no—
292Dic122—22no—
293Dic122—41no—
294Dic122—41no—
295Dic122—84no—
296C8×C61—41no—
297C8×C61—41no—
298#92418—41no—
299#92418—41no—
300#29113—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61#52511yescharacteristic
309C22×C61C2×#12411yescharacteristic
310C22×C61#53411yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61C2×#13711yescharacteristic
313C22×C61#57511yescharacteristic
314C22×C61#57511yescharacteristic
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—22no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61#55111yescharacteristic
328C4×C61#55111yescharacteristic
329C4×C61#55111yescharacteristic
330C4×C61#55111yescharacteristic
331Dic61—22no—
332Dic61—22no—
333Dic61—42no—
334Dic61—41no—
335Dic61—41no—
336Dic61—81no—
337Dic61—81no—
338#2857—611no—
339#2297—611no—
340#2072—611no—
341#2071—611no—
342#2024—611no—
343#2023—611no—
344#1936—611no—
345#1936—611no—
346#1935—611no—
347#1935—611no—
348#1825—611no—
349#1825D6111yescharacteristic, complemented
350#1788—611no—
351#1788—611no—
352#1749—611no—
353C2×C61—42no—
354C2×C61#256111yescharacteristic
355C2×C61#311511yescharacteristic
356C2×C61#311611yescharacteristic
357C2×C61#313511yescharacteristic
358C2×C61#314111yescharacteristic
359C2×C61#319411yescharacteristic
360C2×C61#319611yescharacteristic
361D61—82no—
362C22×#41—611no—
363C2×C4⋊Q8—611no—
364C2×C22⋊Q8—1221no—
365C2×#117C2×D6111yescharacteristic
366C22×C42—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×C22⋊C8—611no—
376C2×C22⋊C8C2×D6111yescharacteristic
377C2×#97—611no—
378C2×#97—611no—
379#391—611no—
380#391—611no—
381#391—1221no—
382#390—611no—
383#390—611no—
384#389—611no—
385#389—611no—
386#389—1221no—
387#386—1221no—
388#386—1221no—
389#384—611no—
390#380—611no—
391#380—1221no—
392#380C2×D6111yescharacteristic
393C4×C22⋊C4—1221no—
394C4×C22⋊C4—1221no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400#340C2×D6111yescharacteristic
401#340C2×D6111yescharacteristic
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405C61#2911311yescharacteristic, complemented, Sylow
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—1222no—
410C2×#41—2441no—
411C2×#41—2441no—
412C22×Q8—611no—
413C22×Q8—1221no—
414C22×D4—611no—
415C22×D4—1221no—
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—1221no—
419C23×C4C22×D6111yescharacteristic
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C4⋊Q8—2441no—
425C4⋊Q8—2441no—
426C22⋊Q8—2444no—
427#117—22no—
428#117—22no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—21no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—2441no—
453C2×C42—2441no—
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#103—2441no—
458#102—1222no—
459#102—1222no—
460#102—2441no—
461#102—2441no—
462C22⋊C8—22no—
463C22⋊C8—1222no—
464#97—2441no—
465#97—2441no—
466#95—21no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—1221no—
475C24D4×D6111yescharacteristic
476#41—2444no—
477#41—2444no—
478#41—2444no—
479#41—2444no—
480C2×Q8—612no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×D4—612no—
490C2×D4—1224no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—2442no—
497C2×D4—2442no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—2442no—
517C22×C4—2441no—
518C22×C4—2441no—
519C22×C4—2441no—
520C22×C4—2441no—
521C22×C4—2442no—
522C22×C4#1175822212yes—
523C22×C4D4×D6111yescharacteristic
524C22×C4#1175822412yes—
525C22×C4C23×D6111yescharacteristic
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—42no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C42—2441no—
568C42—2441no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—1222no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23#1451976312yes—
583C23C2×D4×D6111yescharacteristic
584Q8—1222no—
585Q8—1222no—
586Q8—4884no—
587Q8—2441no—
588Q8—2441no—
589D4—1222no—
590D4—1222no—
591D4—2448no—
592D4—2441no—
593D4—2441no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—2448no—
626C2×C4—2444no—
627C2×C4—2444no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4#1451976312yes—
637C2×C4#41×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822411yescharacteristic
640C2×C4C2×#1175822211yescharacteristic
641C2×C4#1451982311yescharacteristic
642C2×C4#1451982311yescharacteristic
643C8—41no—
644C8—41no—
645C8—2441no—
646C8—2441no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—4881no—
658C22—2442no—
659C22—2442no—
660C22—2442no—
661C22#1684027011yescharacteristic
662C22C2×#1451976311yescharacteristic
663C22#1684032311yescharacteristic
664C22AGL(1,ℤ/8)×D6111yescharacteristic
665C22#137×D6111yescharacteristic
666C22#1684055911yescharacteristic
667C22#1684055911yescharacteristic
668C4—22no—
669C4—81no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—1222no—
674C4—1222no—
675C4—4881no—
676C4—4881no—
677C4—2441no—
678C4—2442no—
679C4—2441no—
680C4#1684043511yescharacteristic
681C4#1684043511yescharacteristic
682C4#1684043611yescharacteristic
683C4#1684043611yescharacteristic
684C2—42no—
685C2—4882no—
686C2#2145097611yescharacteristic
687C2#2145596511yescharacteristic
688C2#2145597611yescharacteristic
689C2#2145616211yescharacteristic
690C2#2145621011yescharacteristic
691C2#2145663711yescharacteristic
692C2#2145665311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1825D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
1. #21454169C2semidirect
3. #21447231C2semidirect
4. #21447226C2semidirect
5. #21446959C2semidirect
6. #21446954C2semidirect
7. #21446517C2semidirect
8. #21446514C2semidirect
9. #21446510C2semidirect
10. #21446507C2semidirect
11. #21445914C2semidirect
14. #21445583C2semidirect
15. C61×#1825C2semidirect
349. #1825D61semidirect
405. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214541692
#214483072
#214472312
#214472262
#214469592
#214469542
#214465172
#214465142
#214465102
#214465072
#214459142
#214457602
#214457572
#214455832
C61×#18252
#2911361