Finite Group Databasebeta

← #10847347

Subgroups — #10847347

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3077C211yesmaximal, characteristic
2#83342C211yesmaximal, characteristic, complemented
3#83330C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic
5C4×#833C211yesmaximal, characteristic
6#70671C211yesmaximal, characteristic, complemented
7#70670C211yesmaximal, characteristic, complemented
8#70669C211yesmaximal, characteristic, complemented
9#70668C211yesmaximal, characteristic, complemented
10#70525C211yesmaximal, characteristic, complemented
11#70520C211yesmaximal, characteristic, complemented
12#70318C211yesmaximal, characteristic
13#70318C211yesmaximal, characteristic
14#70317C211yesmaximal, characteristic
15#70317C211yesmaximal, characteristic
16#25493—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#870C2211yescharacteristic
20C3×#478—21no—
21C3×#478—21no—
22C3×#476—21no—
23C3×#474—21no—
24C3×#428C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C3×#418C2211yescharacteristic
28C3×#418C2211yescharacteristic
29C2×#833C2211yescharacteristic
30C2×#833C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#822C2211yescharacteristic
34C2×#822C2211yescharacteristic
35#5081—21no—
36#5081—21no—
37#5065—21no—
38#5065—21no—
39#5024—21no—
40#5024—21no—
41#5024—21no—
42Q8×C3⋊C8—21no—
43#5009—21no—
44#5009—21no—
45#5009—21no—
46#5009—21no—
47#4999C2211yescharacteristic
48#4999C2211yescharacteristic
49#4997C2211yescharacteristic
50#4997C2211yescharacteristic
51#4995C2211yescharacteristic
52#4995C2211yescharacteristic
53#4995C2211yescharacteristic
54#4995C2211yescharacteristic
55C2×#808C2211yescharacteristic
56C2×#807C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58C4×#208C2211yescharacteristic
59C4×#208C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3166—31no—
69#3077D311yescharacteristic
70C2×#478—31no—
71#2962—31no—
72#2090—31no—
73#2089—31no—
74#2063—31no—
75#2063—31no—
76#1998—31no—
77#1952—31no—
78#1951—31no—
79#1950—31no—
80#1931—31no—
81#1931—31no—
82C4×#100—31no—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C3×C4⋊Q8D411yescharacteristic
88C3×C4⋊Q8D411yescharacteristic
89C3×#125D411yescharacteristic
90C3×#125D411yescharacteristic
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×#117C2311yescharacteristic
100C3×#117C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C42C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#870—21no—
115#870—21no—
116#870—21no—
117#870—21no—
118C3×#107—21no—
119C3×#107—21no—
120C3×#106—21no—
121C3×#106—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8—21no—
124C3×#102—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128#833—22no—
129#833Q814yescomplemented
130#832—21no—
131#832—21no—
132#832—21no—
133#832—21no—
134#822—22no—
135#822—22no—
136#808—21no—
137#808—21no—
138#808—21no—
139#808—21no—
140#808—21no—
141#808—21no—
142#807—21no—
143#807—21no—
144#807—21no—
145#807—21no—
146#807—21no—
147#807—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)—21no—
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)—21no—
152C4⋊(C3⋊C8)D411yescharacteristic
153C4⋊(C3⋊C8)D411yescharacteristic
154C4⋊(C3⋊C8)D411yescharacteristic
155C4⋊(C3⋊C8)D411yescharacteristic
156#803—21no—
157#803—21no—
158#803—21no—
159#803—21no—
160C4×C3⋊C8—21no—
161C4×C3⋊C8—21no—
162C4×C3⋊C8—21no—
163C4×C3⋊C8—21no—
164#533C2×D311yescharacteristic
165C2×#125—31no—
166C2×C4×D4C2×D311yescharacteristic
167#501—61no—
168#501—61no—
169#499—61no—
170#499—61no—
171#478—21no—
172#478—21no—
173#478—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—21no—
178#474—21no—
179#446—61no—
180#446—61no—
181#446—61no—
182C8×Q8—61no—
183C2×#108—31no—
184C2×#108—31no—
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428C2×D311yescharacteristic
190#428C2×D311yescharacteristic
191C2×#107—31no—
192C2×#106—31no—
193#424—31no—
194#424—31no—
195#423C2×D311yescharacteristic
196C2×C4⋊C8—31no—
197#418C2×D311yescharacteristic
198#418C2×D311yescharacteristic
199C2×#102—31no—
200C2×#102—31no—
201C2×#100—31no—
202C2×#100—31no—
203#405—31no—
204#405—31no—
205C4×M16—31no—
206C4×M16—31no—
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#341—31no—
215C24×C3—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4C2×Q812yes—
224C22×C3×C4—21no—
225C22×C3×C4—41no—
226C22×C3×C4—41no—
227C22×C3×C4—82no—
228C22×C3×C4#4111yescharacteristic
229C22×C3×C4#4111yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—41no—
254C3×C22⋊C4—41no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—41no—
266Dic3⋊C4—41no—
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
269#208—21no—
270#208—21no—
271#208—21no—
272#208—21no—
273#208—21no—
274#208C2×Q812yes—
275#208C2×Q812yes—
276#208#4112yes—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D411yescharacteristic
286C2×C3⋊C8C2×D411yescharacteristic
287C2×C3⋊C8C2×D411yescharacteristic
288C2×C3⋊C8C2×D411yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—21no—
291C2×M16—31no—
292C2×M16—31no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16C22×D311yescharacteristic
296C2×M16C22×D311yescharacteristic
297C22×C8—31no—
298C22×C8—31no—
299C4⋊Q8#6711yescharacteristic
300C4⋊Q8#6711yescharacteristic
301#125—61no—
302#125—61no—
303#125—61no—
304#125—61no—
305#125#6711yescharacteristic
306#125#6711yescharacteristic
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315#117C22×D311yescharacteristic
316#117C22×D311yescharacteristic
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D311yescharacteristic
324#108—62no—
325#108—62no—
326#107—21no—
327#107—21no—
328#107—61no—
329#107—61no—
330#107—61no—
331#107—61no—
332#107—61no—
333#107—61no—
334#106—21no—
335#106—21no—
336#106—61no—
337#106—61no—
338#106—61no—
339#106—61no—
340#106—61no—
341#106—61no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—31no—
345C4⋊C8—31no—
346C4⋊C8—31no—
347C4⋊C8—31no—
348C4⋊C8—61no—
349C4⋊C8—61no—
350C4⋊C8—61no—
351C4⋊C8—61no—
352#102—21no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—61no—
357#102—61no—
358#102—61no—
359#102—61no—
360#100—34no—
361#100—62no—
362#97—61no—
363#97—61no—
364#97—61no—
365#97—61no—
366C4×C8—61no—
367C4×C8—61no—
368C4×C8—61no—
369C4×C8—61no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—81no—
374C23×C3—81no—
375C23×C3C2×#4111yescharacteristic
376C3×Q8—42no—
377C3×Q8—81no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—82no—
390C2×C3×C4—82no—
391C2×C3×C4—81no—
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4#12111yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×Q811yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—81no—
405C2×Dic3—81no—
406C3×C8—41no—
407C3×C8—41no—
408C3⋊C8—21no—
409C3⋊C8—21no—
410C3⋊C8—21no—
411C3⋊C8—21no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C24—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4D3×Q812yes—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—82no—
431C22×C4#23711yescharacteristic
432C22×C4#23711yescharacteristic
433C22×C4C23×D311yescharacteristic
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—32no—
439M16—32no—
440M16—32no—
441M16—61no—
442M16—61no—
443M16—61no—
444M16—61no—
445M16—61no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—31no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—121no—
473C4⋊C4—121no—
474C4⋊C4—121no—
475C4⋊C4—121no—
476C4⋊C4C2×#6711yescharacteristic
477C4⋊C4C2×#6711yescharacteristic
478C4⋊C4C2×#6711yescharacteristic
479C4⋊C4C2×#6711yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—21no—
485C42—21no—
486C42D3×D411yescharacteristic
487C42D3×D411yescharacteristic
488C42C2×#6711yescharacteristic
489C42C2×#6711yescharacteristic
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—42no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3C2×#12111yescharacteristic
498C22×C3#53311yescharacteristic
499C22×C3#54111yescharacteristic
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4#54711yescharacteristic
510C3×C4#54811yescharacteristic
511C3×C4D4×Q811yescharacteristic
512C3×C4D4×Q811yescharacteristic
513Dic3—81no—
514Dic3—81no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—81no—
519C23—81no—
520C23C2×#23711yescharacteristic
521Q8—42no—
522Q8—81no—
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—241no—
540C2×C4—241no—
541C2×C4#93911yescharacteristic
542C2×C4#93911yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×Q811yescharacteristic
547C2×C4#100711yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4#101011yescharacteristic
550C2×C4C22×#6711yescharacteristic
551C8—41no—
552C8—41no—
553C8—62no—
554C8—62no—
555C8—62no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—61no—
560C8—121no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—82no—
564C2×C3#242911yescharacteristic
565C2×C3#306711yescharacteristic
566C2×C3#306811yescharacteristic
567C22—21no—
568C22—21no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—81no—
573C22—81no—
574C22#562211yescharacteristic
575C22#562511yescharacteristic
576C22C2×#93911yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—241no—
587C4—241no—
588C4#563111yescharacteristic
589C4#563211yescharacteristic
590C4Q8×#6711yescharacteristic
591C4#583111yescharacteristic
592C3#2549311yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—82no—
596C2#8275811yescharacteristic
597C2#8394311yescharacteristic
598C2#8400111yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3077C2
socleC22×C3C2×#121
prime radicalC22×C3×D4C23
hypercenter#3077D3
hypocenterC3#25493
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #83342C2semidirect
3. #83330C2semidirect
6. #70671C2semidirect
7. #70670C2semidirect
8. #70669C2semidirect
9. #70668C2semidirect
10. #70525C2semidirect
11. #70520C2semidirect
129. #833Q8semidirect
592. C3#25493semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833422
#833302
C2×#50092
C4×#8332
#706712
#706702
#706692
#706682
#705252
#705202
#703182
#703172
#254933