Finite Group Databasebeta

← #29695396

Subgroups — #29695396

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840411C211yesmaximal, characteristic
2#21454551C212yesmaximal, complemented
3C2×#16839746C211yesmaximal, characteristic, complemented
4C2×#16839743C211yesmaximal, characteristic, complemented
5#21446661C212yesmaximal, complemented
6#21446285C212yesmaximal, complemented
7#21446284C212yesmaximal, complemented
8#21445809C211yesmaximal, characteristic, complemented
9#21445744C211yesmaximal, characteristic, complemented
10#100×Dic61C211yesmaximal, characteristic, complemented
11C61×#1785C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×#11758194C2212yes—
14C2×M16×D61C2212yes—
15#16840411—22no—
16#16840411—22no—
17#16840411—22no—
18#16840411—22no—
19C2×#14519775C2211yescharacteristic
20C2×#14519760C412yes—
21C2×#14519758C412yes—
22C2×#14519756C2211yescharacteristic
23C2×#14519753C2211yescharacteristic
24C2×#14519749C412yes—
25C2×D4×Dic61C2211yescharacteristic
26C2×C4×C22⋊D61C2211yescharacteristic
27C22×#11758196C412yes—
28#16839837—22no—
29#16839836—22no—
30#16839830—22no—
31#16839825—22no—
32#16839785C2212yes—
33#16839746—24no—
34#16839743—24no—
35C2×#14519696C2212yes—
36C2×#14519695C2212yes—
37C2×#14519695C2212yes—
38C2×#14519694C2211yescharacteristic
39C2×#14519694C2211yescharacteristic
40#16839725C2212yes—
41M16×Dic61C2212yes—
42#16839530C2212yes—
43#16839530C2212yes—
44C22×#41×C61C2211yescharacteristic
45C2×#101×C61C2212yes—
46C2×#100×C61C2211yescharacteristic
47C2×#100×C61C2211yescharacteristic
48C61×#407C2212yes—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—22no—
52C2×#11758224—22no—
53C2×#11758224—42no—
54C22×Dic122C2311yescharacteristic
55C22×C22⋊D61C2×C412yes—
56C22×C22⋊D61C2×C411yescharacteristic
57C23×Dic61C2×C412yes—
58C23×Dic61C2×C412yes—
59C22×C4×D61C2311yescharacteristic
60C2×#11758194—22no—
61C2×#11758194—24no—
62C2×#11758194—42no—
63C2×#11758194C2312yes—
64C2×#11758194C2312yes—
65C2×#11758189—22no—
66M16×D61—24no—
67M16×D61—42no—
68C22×#92418—22no—
69C2×C8×D61—22no—
70#14519775—22no—
71#14519775—22no—
72#14519760—24no—
73#14519760—24no—
74#14519758—24no—
75#14519758—24no—
76#14519756—22no—
77#14519756—22no—
78#14519753—22no—
79#14519753—22no—
80#14519749—24no—
81#14519749—24no—
82D4×Dic61—22no—
83D4×Dic61—22no—
84C4×C22⋊D61D412yes—
85C4×C22⋊D61D412yes—
86C4×C22⋊D61D412yes—
87C4×C22⋊D61D412yes—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—22no—
90C2×#11758196—22no—
91C2×#11758196—22no—
92C2×#11758196—22no—
93C2×#11758196C2×C412yes—
94C2×#11758196C2×C412yes—
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×#11758196C2×C411yescharacteristic
98C2×C22⋊Dic61C2×C412yes—
99C2×C22⋊Dic61C2×C412yes—
100C2×C22⋊Dic61C2×C411yescharacteristic
101C2×#11758198C2×C412yes—
102C2×#11758198C2×C411yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104#14519706—22no—
105#14519706—22no—
106#14519696—24no—
107#14519695—24no—
108#14519695—42no—
109#14519694—22no—
110#14519694D414yescomplemented
111#14519689—22no—
112#14519689—22no—
113#14519687—22no—
114C8×Dic61—22no—
115C2×#41×C61—22no—
116C2×#41×C61—22no—
117C2×#41×C61—22no—
118C2×#41×C61—22no—
119C2×#41×C61—42no—
120C22×Q8×C61C2311yescharacteristic
121C22×D4×C61C2×C412yes—
122C22×D4×C61C2311yescharacteristic
123C23×C4×C61C2×C412yes—
124C23×C4×C61C2311yescharacteristic
125C2×M16×C61C2312yes—
126C2×M16×C61C2312yes—
127#101×C61—42no—
128#100×C61—22no—
129#100×C61—22no—
130C22⋊C8×C61—22no—
131C22⋊C8×C61—22no—
132C24×C61C22×C411yescharacteristic
133C24×C61C22×C412yes—
134C23×D61C22×C411yescharacteristic
135#41×C61—44no—
136#41×C61—44no—
137#41×C61—44no—
138#41×C61—44no—
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—42no—
148C2×D4×C61C2×D412yes—
149C2×D4×C61#4112yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—24no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—42no—
157C22×C4×C61C22×C412yes—
158C22×C4×C61C22×C411yescharacteristic
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—24no—
163C2×C22⋊D61—24no—
164C2×C22⋊D61—42no—
165C2×C22⋊D61—42no—
166C2×C22⋊D61C22⋊C412yes—
167C2×C22⋊D61C22⋊C412yes—
168C22×Dic61—24no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—42no—
174C22×Dic61—44no—
175C22×Dic61C22×C412yes—
176C22×Dic61C22×C411yescharacteristic
177C22×Dic61C22×C411yescharacteristic
178C22×Dic61C22×C412yes—
179C22×Dic61C22×C411yescharacteristic
180#11758224—44no—
181#11758224—44no—
182#11758224—44no—
183#11758224—44no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×C4×D61—22no—
187C2×C4×D61C2×D412yes—
188C2×C4×D61C2×D412yes—
189C2×Dic122—21no—
190C2×Dic122—22no—
191C2×Dic122—21no—
192C2×Dic122—22no—
193M16×C61—22no—
194M16×C61—22no—
195M16×C61—22no—
196M16×C61—22no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C22⋊Dic61—24no—
200C22⋊Dic61—24no—
201C22⋊Dic61C22⋊C412yes—
202C22⋊Dic61C22⋊C412yes—
203#11758198—24no—
204#11758198C22⋊C412yes—
205#11758198C22⋊C412yes—
206C4⋊Dic61—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—24no—
214#11758196C22⋊C411yescharacteristic
215#11758196C22⋊C411yescharacteristic
216#11758196C22⋊C411yescharacteristic
217#11758196C22⋊C411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220#11758194—22no—
221#11758194—22no—
222#11758194—22no—
223#11758194—44no—
224#11758194C2×D414yes—
225C2×#92418—22no—
226C2×#92418—22no—
227C2×#92418—44no—
228#11758189—44no—
229C8×D61—44no—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—42no—
237C23×C61—42no—
238C23×C61C2×C22⋊C412yes—
239C23×C61C4×D414yes—
240C23×C61C23×C411yescharacteristic
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61C2×C22⋊C412yes—
246Q8×C61—42no—
247Q8×C61—42no—
248D4×C61—24no—
249D4×C61—24no—
250D4×C61—48no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—44no—
261C2×C4×C61—44no—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D412yes—
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)12yes—
268C22⋊D61—24no—
269C22⋊D61—42no—
270C22⋊D61—48no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—44no—
280C2×Dic61—44no—
281C2×Dic61—44no—
282C2×Dic61—44no—
283C2×Dic61—42no—
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C2×C22⋊C412yes—
286C2×Dic61C2×C22⋊C411yescharacteristic
287C2×Dic61C2×C22⋊C411yescharacteristic
288C2×Dic61C2×C22⋊C411yescharacteristic
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293Dic122—42no—
294Dic122—42no—
295C8×C61—42no—
296C8×C61—42no—
297#92418—24no—
298#92418—42no—
299#22411—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—24no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—44no—
307C22×C61—44no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61C22×C22⋊C411yescharacteristic
313C22×C61C2×C4×D411yescharacteristic
314C22×C61#51811yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—81no—
319C2×D61—81no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61D4212yes—
324C4×C61#54612yes—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—22no—
329Dic61—42no—
330Dic61—42no—
331Dic61#41312yescomplemented
332C2×#547—611no—
333D4×M16—612no—
334C2×#413—611no—
335C2×#413—611no—
336#1966—612no—
337#1901—612no—
338#1901—612no—
339#1800—611no—
340#1785—611no—
341#1785D6111yescharacteristic, complemented
342C4×#100—611no—
343C2×C61—21no—
344C2×C61—21no—
345C2×C61—42no—
346C2×C61—44no—
347C2×C61D4×C22⋊C411yescharacteristic
348C2×C61C2×#41311yescharacteristic
349C2×C61#288911yescharacteristic
350D61—42no—
351C22×#41—611no—
352C22×#41C2×D6111yescharacteristic
353C22×M16—612no—
354C22×M16—612no—
355#547—1222no—
356#547—1222no—
357#547—1222no—
358#547—1222no—
359C2×C4⋊Q8—611no—
360C2×#123—612no—
361C2×#123—612no—
362C2×C22⋊Q8—611no—
363C2×C22⋊Q8—611no—
364C2×#121—612no—
365C2×C4×D4—611no—
366C2×C4×D4—611no—
367C22×C4⋊C4—612no—
368#436—1222no—
369#435—1222no—
370#435—1222no—
371C8×D4—1222no—
372#423—612no—
373#413—1224no—
374#413—1224no—
375C2×#101—612no—
376C2×#101—612no—
377C2×#101—612no—
378C2×#101C2×D6112yes—
379C2×#100—611no—
380C2×#100—611no—
381C2×#100C2×D6111yescharacteristic
382C2×#100C2×D6111yescharacteristic
383#407—612no—
384#407C2×D6112yes—
385C4×M16—612no—
386#343—612no—
387#343—612no—
388C61#2241111yescharacteristic, complemented, Sylow
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—22no—
392C2×#41—22no—
393C2×#41—42no—
394C2×#41—1222no—
395C2×#41—1222no—
396C2×#41—1222no—
397C2×#41—1222no—
398C2×#41—2442no—
399C22×Q8—611no—
400C22×Q8C22×D6111yescharacteristic
401C22×D4—611no—
402C22×D4—612no—
403C22×D4C4×D6112yes—
404C22×D4C22×D6111yescharacteristic
405C23×C4—611no—
406C23×C4—612no—
407C23×C4—612no—
408C23×C4C4×D6112yes—
409C23×C4C22×D6111yescharacteristic
410C2×M16—612no—
411C2×M16—612no—
412C2×M16—1222no—
413C2×M16—1222no—
414C2×M16—1224no—
415C2×M16—1224no—
416C2×M16—2442no—
417C2×M16—2442no—
418C2×M16C22×D6112yes—
419C2×M16C22×D6112yes—
420C22×C8—1222no—
421C22×C8—1222no—
422C4⋊Q8—1222no—
423C4⋊Q8—1222no—
424#123—1224no—
425#123—1224no—
426#123—1224no—
427#123—1224no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430C22⋊Q8—1222no—
431C22⋊Q8—1222no—
432#121—1224no—
433#121—1224no—
434C4×D4—612no—
435C4×D4—612no—
436C4×D4—612no—
437C4×D4—612no—
438C4×D4—1222no—
439C4×D4—1222no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1222no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—612no—
453C2×C22⋊C4—612no—
454C2×C22⋊C4—612no—
455C2×C42—611no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458#101—42no—
459#101—1224no—
460#101—1224no—
461#101—2442no—
462#100—22no—
463#100—22no—
464#100—614no—
465#100—1222no—
466C22⋊C8—22no—
467C22⋊C8—22no—
468C22⋊C8—1222no—
469C22⋊C8—1222no—
470#97—1222no—
471C4×C8—1222no—
472C24—611no—
473C24C2×C4×D6111yescharacteristic
474C24C2×C4×D6112yes—
475#41—44no—
476#41—44no—
477#41—44no—
478#41—44no—
479#41—2444no—
480#41—2444no—
481#41—2444no—
482#41—2444no—
483C2×Q8—21no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—22no—
487C2×Q8—1221no—
488C2×Q8—1222no—
489C2×Q8—1221no—
490C2×Q8—1222no—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—42no—
496C2×D4—612no—
497C2×D4—612no—
498C2×D4—1224no—
499C2×D4—1224no—
500C2×D4—2442no—
501C2×D4—2442no—
502C2×D4D4×D6112yes—
503C2×D4#1175822412yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—42no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1222no—
525C22×C4—1224no—
526C22×C4—2442no—
527C22×C4—2444no—
528C22×C4C2×C4×D6112yes—
529C22×C4C2×C4×D6111yescharacteristic
530C22×C4D4×D6112yes—
531C22×C4D4×D6112yes—
532C22×C4C23×D6111yescharacteristic
533M16—22no—
534M16—22no—
535M16—22no—
536M16—22no—
537M16—614no—
538M16—1222no—
539M16—1222no—
540M16—1222no—
541M16—2444no—
542M16—2444no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—2444no—
548C2×C8—2444no—
549C4⋊C4—611no—
550C4⋊C4—611no—
551C4⋊C4—611no—
552C4⋊C4—611no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1224no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C22⋊C4—612no—
562C22⋊C4—612no—
563C22⋊C4—612no—
564C22⋊C4—612no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C42—612no—
569C42—612no—
570C23—22no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—612no—
579C23—1221no—
580C23—1221no—
581C23—2441no—
582C23—2441no—
583C23C22⋊C4×D6112yes—
584C23#1451973714yes—
585C23C22×C4×D6111yescharacteristic
586Q8—42no—
587Q8—42no—
588Q8—2442no—
589Q8—2442no—
590D4—24no—
591D4—24no—
592D4—48no—
593D4—1224no—
594D4—2442no—
595D4—2448no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—611no—
608C2×C4—612no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—2444no—
624C2×C4—2441no—
625C2×C4—2444no—
626C2×C4—2444no—
627C2×C4—2444no—
628C2×C4—2442no—
629C2×C4C22⋊C4×D6111yescharacteristic
630C2×C4C22⋊C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×D4×D6112yes—
633C2×C4C2×#1175822411yescharacteristic
634C2×C4#1451982212yes—
635C8—42no—
636C8—42no—
637C8—1224no—
638C8—2442no—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—1222no—
652C22—1222no—
653C22—1222no—
654C22—4881no—
655C22—4881no—
656C22C2×C22⋊C4×D6111yescharacteristic
657C22C2×#1451973711yescharacteristic
658C22#1684023511yescharacteristic
659C4—21no—
660C4—21no—
661C4—42no—
662C4—612no—
663C4—1221no—
664C4—1221no—
665C4—1221no—
666C4—1222no—
667C4—2442no—
668C4—2442no—
669C4#1684039412yes—
670C4#1684039912yes—
671C2—21no—
672C2—21no—
673C2—42no—
674C2—44no—
675C2—2442no—
676C2#2144863511yescharacteristic
677C2#413×D6111yescharacteristic
678C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1785C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840411C2
cosocleC22×C4×C61C24
hypercenter#1785D61
hypocenterC61#22411
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #21454551C2semidirect
3. C2×#16839746C2semidirect
4. C2×#16839743C2semidirect
5. #21446661C2semidirect
6. #21446285C2semidirect
7. #21446284C2semidirect
8. #21445809C2semidirect
9. #21445744C2semidirect
10. #100×Dic61C2semidirect
11. C61×#1785C2semidirect
110. #14519694D4semidirect
331. Dic61#413semidirect
341. #1785D61semidirect
388. C61#22411semidirect

Maximal subgroups

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

subgroupindex
C2×#168404112
#214545512
C2×#168397462
C2×#168397432
#214466612
#214462852
#214462842
#214458092
#214457442
#100×Dic612
C61×#17852
#2241161