Finite Group Databasebeta

← #10778182

Subgroups — #10778182

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83352C211yesmaximal, characteristic
2#83348C211yesmaximal, characteristic, complemented
3#79328C211yesmaximal, characteristic
4#79025C211yesmaximal, characteristic
5C3×#1552C211yesmaximal, characteristic, complemented
6#70821C211yesmaximal, characteristic, complemented
7#70420C211yesmaximal, characteristic, complemented
8#68479C211yesmaximal, characteristic, complemented
9#68477C211yesmaximal, characteristic, complemented
10#68474C211yesmaximal, characteristic, complemented
11#68473C211yesmaximal, characteristic, complemented
12#68447C211yesmaximal, characteristic, complemented
13#68445C211yesmaximal, characteristic, complemented
14#68442C211yesmaximal, characteristic, complemented
15#68441C211yesmaximal, characteristic, complemented
16#19181—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C4×#235C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22#5113—21no—
23#5106—21no—
24#5094C2211yescharacteristic
25#5093C2211yescharacteristic
26#5092C2211yescharacteristic
27#5090C2211yescharacteristic
28#5087C2211yescharacteristic
29#5083C2211yescharacteristic
30#5080C2211yescharacteristic
31#5079C2211yescharacteristic
32#5052—21no—
33#5052—21no—
34#5047—21no—
35#5045—21no—
36#5030—21no—
37#5030—21no—
38C4×#216—21no—
39C4×#215—21no—
40#4925C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4851C2211yescharacteristic
43#4850C2211yescharacteristic
44#4847—21no—
45#4846C2211yescharacteristic
46#4842C2211yescharacteristic
47#4834C2211yescharacteristic
48#4834C2211yescharacteristic
49#4831C2211yescharacteristic
50#4831C2211yescharacteristic
51#4829—21no—
52#4728—21no—
53#4728—21no—
54#4705—21no—
55D3×#97—21no—
56C3×#332C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3185—31no—
69#3107—31no—
70#3076—31no—
71#2970—31no—
72#1704—31no—
73#1700—31no—
74#1692—31no—
75#1690—31no—
76#1683—31no—
77#1679—31no—
78#1671—31no—
79#1669—31no—
80#1565—31no—
81#1554—31no—
82#1552D311yescharacteristic, complemented
83C2×#235D411yescharacteristic
84C22×C3×C8—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#125—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×#117—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#216—21no—
95C2×#216—21no—
96C2×#216—21no—
97C2×#215—21no—
98C4×#67—21no—
99C4×#67—21no—
100#926—21no—
101#926D411yescharacteristic
102#924—21no—
103#924—21no—
104C2×#208D411yescharacteristic
105C2×#208D411yescharacteristic
106C2×D3×C8—21no—
107C2×D3×C8—21no—
108C4×D12D411yescharacteristic
109C4×D12C2311yescharacteristic
110#872—21no—
111C42×D3—21no—
112#870—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6—21no—
116C4×Dic6—21no—
117C4×Dic6D411yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×#97—21no—
126#835—21no—
127#835—21no—
128#835—21no—
129#835—21no—
130#830—21no—
131#821—21no—
132#821—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#818—21no—
136#818—21no—
137#817—21no—
138#817—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145#810—21no—
146#810—21no—
147#810—21no—
148#809—21no—
149#808—21no—
150#808—21no—
151#808—21no—
152#807—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803D411yescharacteristic
156#803D411yescharacteristic
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533C2×D311yescharacteristic
162C4×#41—31no—
163#498—31no—
164#494—31no—
165#487—31no—
166#487—31no—
167#485—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#476—31no—
172#475—31no—
173#475—31no—
174#474—31no—
175#474—31no—
176#470—61no—
177#470—61no—
178#462—31no—
179#461—31no—
180#452—61no—
181#446—31no—
182#446—31no—
183#441—61no—
184#441—61no—
185C4×Q16—61no—
186C4×SD16—61no—
187#435—61no—
188#435—61no—
189C8×D4—31no—
190C8×D4—31no—
191#432—61no—
192#432C2×D311yescharacteristic
193#431—31no—
194#427—61no—
195C2×C4⋊C8C2×D311yescharacteristic
196#416—61no—
197C4×M16—31no—
198C2×#97—61no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332C2×D311yescharacteristic
208#332C2×D311yescharacteristic
209#332C2×D311yescharacteristic
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C2×#67—21no—
217#235—22no—
218#235—41no—
219C2×D12C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—42no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C22⋊C4—41no—
238C3×C42—41no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4111yescharacteristic
242C3×C42C2411yescharacteristic
243#217—21no—
244#217—41no—
245#216—42no—
246#216—42no—
247#216—42no—
248#215—42no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—41no—
265#208—22no—
266#208—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273D3×C8—42no—
274D3×C8—42no—
275C2×#41—31no—
276C2×Q16—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16—31no—
281C2×M16—31no—
282C22×C8—21no—
283C22×C8—61no—
284C22×C8—61no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#125—21no—
290#125—61no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—31no—
298C4×Q8—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305#117—21no—
306#117—31no—
307#117—61no—
308#117—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—61no—
312#107—61no—
313#107—61no—
314#106—61no—
315#106—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—21no—
345#97—31no—
346#97—31no—
347#97—61no—
348#97—61no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C23×C3#12111yescharacteristic
353C22×D3—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4#12111yescharacteristic
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371#67—42no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377D12—21no—
378D12#13712yes—
379C4×D3—42no—
380C4×D3—41no—
381C4×D3—41no—
382C4×D3—81no—
383Dic6—21no—
384Dic6—21no—
385Dic6—41no—
386Dic6—41no—
387Dic6—81no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8#13512yes—
398#41—62no—
399#41—121no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—31no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×D4—31no—
406C2×D4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4#23511yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413Q16—122no—
414Q16—122no—
415Q16—122no—
416SD16—122no—
417M16—62no—
418M16—121no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—42no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—122no—
431C2×C8—122no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—31no—
437C4⋊C4—31no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—121no—
447C4⋊C4—121no—
448C22⋊C4—41no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C42—41no—
454C42—61no—
455C42—61no—
456C42—61no—
457C42—121no—
458C42#23511yescharacteristic
459C42D3×D411yescharacteristic
460C42D3×D411yescharacteristic
461C42C23×D311yescharacteristic
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3C2×#12111yescharacteristic
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4C2×#13511yescharacteristic
476C3×C4C2×#13711yescharacteristic
477C3×C4#57511yescharacteristic
478C3×C4#57511yescharacteristic
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482Dic3—81no—
483Dic3—81no—
484C23—61no—
485C23#88411yescharacteristic
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—61no—
491Q8—61no—
492Q8—121no—
493Q8—121no—
494Q8—241no—
495D4—32no—
496D4—61no—
497D4—122no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—241no—
514C2×C4#88411yescharacteristic
515C2×C4#88411yescharacteristic
516C2×C4#88411yescharacteristic
517C2×C4C2×#23511yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C8—32no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—61no—
526C8—61no—
527C8—61no—
528C8—121no—
529C8—121no—
530C2×C3—41no—
531C2×C3#303511yescharacteristic
532C2×C3#305611yescharacteristic
533C2×C3#306911yescharacteristic
534D3—42no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—62no—
539C22—121no—
540C22—121no—
541C22C2×#88411yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—81no—
547C4—81no—
548C4—121no—
549C4—121no—
550C4—121no—
551C4—241no—
552C4—241no—
553C4D3×#13511yescharacteristic
554C4#576811yescharacteristic
555C4#577311yescharacteristic
556C4D3×#13711yescharacteristic
557C3#1918111yescharacteristic, complemented, Sylow
558C2—41no—
559C2—122no—
560C2#7871211yescharacteristic
561C2#7886111yescharacteristic
562C2#7888911yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1552C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#1552D3
hypocenterC3#19181
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
2. #83348C2semidirect
5. C3×#1552C2semidirect
6. #70821C2semidirect
7. #70420C2semidirect
8. #68479C2semidirect
9. #68477C2semidirect
10. #68474C2semidirect
11. #68473C2semidirect
12. #68447C2semidirect
13. #68445C2semidirect
14. #68442C2semidirect
15. #68441C2semidirect
82. #1552D3semidirect
557. C3#19181semidirect

Maximal subgroups

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

subgroupindex
#833522
#833482
#793282
#790252
C3×#15522
#708212
#704202
#684792
#684772
#684742
#684732
#684472
#684452
#684422
#684412
#191813