Finite Group Databasebeta

← #10778176

Subgroups — #10778176

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
1C3×#2970C211yesmaximal, characteristic
2#82534C211yesmaximal, characteristic
3#82507C211yesmaximal, characteristic
4#78540C211yesmaximal, characteristic, complemented
5#70967C211yesmaximal, characteristic, complemented
6#70966C211yesmaximal, characteristic, complemented
7#70963C211yesmaximal, characteristic, complemented
8#70962C211yesmaximal, characteristic, complemented
9#70855C211yesmaximal, characteristic, complemented
10#70854C211yesmaximal, characteristic, complemented
11#70851C211yesmaximal, characteristic, complemented
12#70850C211yesmaximal, characteristic, complemented
13#70821C211yesmaximal, characteristic, complemented
14#70820C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19181—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5470C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×#446C2211yescharacteristic
22C3×#435—21no—
23C3×#435—21no—
24C3×C8×D4C2211yescharacteristic
25C3×C8×D4C2211yescharacteristic
26C3×#432—21no—
27C3×#432C2211yescharacteristic
28C3×C4×M16C2211yescharacteristic
29C2×C3×#97—21no—
30#5239—21no—
31#5233—21no—
32#5137—21no—
33#5137—21no—
34#5031C2211yescharacteristic
35C4×#216—21no—
36#5022—21no—
37#5021C2211yescharacteristic
38#5020C2211yescharacteristic
39#5020C2211yescharacteristic
40#5019C2211yescharacteristic
41C4×#214—21no—
42#5009—21no—
43#5008C2211yescharacteristic
44#5007C2211yescharacteristic
45#5007C2211yescharacteristic
46#4998—21no—
47#4996—21no—
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4716—21no—
51#4716—21no—
52#4699C2211yescharacteristic
53#4681C2211yescharacteristic
54#4681C2211yescharacteristic
55#4680C2211yescharacteristic
56#4486C2211yescharacteristic
57#4486C2211yescharacteristic
58#4482C2211yescharacteristic
59#4482C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3185—31no—
69#3107—31no—
70#3076—31no—
71#2970D311yescharacteristic
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#1552—31no—
83C2×C3×#41D411yescharacteristic
84C2×C3×M16D411yescharacteristic
85C2×C3×M16D411yescharacteristic
86C22×C3×C8—21no—
87C22×C3×C8—21no—
88C3×C4×Q8D411yescharacteristic
89C3×C4×Q8C2311yescharacteristic
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4D411yescharacteristic
93C3×C4×D4C2311yescharacteristic
94C3×#117—21no—
95C3×#117D411yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#216—21no—
99C2×#214—21no—
100#926—21no—
101#926—21no—
102#924—21no—
103#924—21no—
104#924—21no—
105#924—21no—
106C22×C3⋊C8—21no—
107C2×Dic12—21no—
108C2×Dic12—21no—
109#870—21no—
110#870—21no—
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6—21no—
116C4×Dic6—21no—
117C4×Dic6—21no—
118C4×Dic6—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C22⋊C8—21no—
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×#97—21no—
127C3×#97—21no—
128C3×#97—21no—
129C3×#97D411yescharacteristic
130C3×#97D411yescharacteristic
131C3×C4×C8C2311yescharacteristic
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#835—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#830—21no—
140#817—21no—
141#817—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#810—21no—
147#810—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160#533—31no—
161#533—31no—
162C4×#41C2×D311yescharacteristic
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#446C2×D311yescharacteristic
182#446C2×D311yescharacteristic
183#441—61no—
184#441—61no—
185C4×Q16—61no—
186C4×SD16—61no—
187#435—21no—
188#435—21no—
189C8×D4C2×D311yescharacteristic
190C8×D4C2×D311yescharacteristic
191#432—21no—
192#432C2×D311yescharacteristic
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16C2×D311yescharacteristic
198C2×#97—21no—
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#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C3×#41—22no—
212C3×#41—41no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C3×M16—22no—
227C3×M16—41no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—42no—
231C2×C3×C8—42no—
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42C2×D411yescharacteristic
247C3×C42C2×D411yescharacteristic
248C3×C42#4111yescharacteristic
249C3×C42C2411yescharacteristic
250#217—41no—
251#217—41no—
252#216—42no—
253#214—42no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262Dic3⋊C4—41no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265Dic3⋊C4—41no—
266C4×Dic3—41no—
267C4×Dic3—41no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—42no—
273Dic12—42no—
274Dic12—42no—
275C2×#41D1211yescharacteristic
276C2×Q16—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16#6711yescharacteristic
281C2×M16#6711yescharacteristic
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—61no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289#125—61no—
290#125—61no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C4×Q8—61no—
296C4×Q8—61no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8D1211yescharacteristic
300C4×Q8C22×D311yescharacteristic
301C4×D4—21no—
302C4×D4—21no—
303C4×D4D1211yescharacteristic
304C4×D4C22×D311yescharacteristic
305#117—21no—
306#117—61no—
307#117—61no—
308#117D1211yescharacteristic
309C2×C42—21no—
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—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
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—21no—
342C22⋊C8—21no—
343C22⋊C8—61no—
344#97—21no—
345#97—21no—
346#97—21no—
347#97#6711yescharacteristic
348#97#6711yescharacteristic
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C4×C8C22×D311yescharacteristic
352C23×C3—21no—
353C23×C3#12111yescharacteristic
354C3×Q8—21no—
355C3×Q8—21no—
356C3×D4—21no—
357C3×D4—42no—
358C3×D4#13712yes—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—81no—
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8#13512yes—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—22no—
399#41—41no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8C2×D1211yescharacteristic
405C2×D4—21no—
406C2×D4C2×D1211yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4C2×D1211yescharacteristic
411C22×C4#23511yescharacteristic
412C22×C4C2×#6711yescharacteristic
413Q16—122no—
414Q16—122no—
415Q16—122no—
416SD16—122no—
417M16—22no—
418M16—41no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—122no—
428C2×C8C2×#6711yescharacteristic
429C2×C8C2×#6711yescharacteristic
430C2×C8C2×#6711yescharacteristic
431C2×C8C2×#6711yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4C2×D1211yescharacteristic
447C4⋊C4C2×D1211yescharacteristic
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—121no—
457C42—121no—
458C42C2×D1211yescharacteristic
459C42#23511yescharacteristic
460C42C2×#6711yescharacteristic
461C42C23×D311yescharacteristic
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×#12111yescharacteristic
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4C2×#13511yescharacteristic
477C3×C4C2×#13711yescharacteristic
478C3×C4#57511yescharacteristic
479C3×C4#57511yescharacteristic
480Dic3—81no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23#93011yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—241no—
495D4—21no—
496D4—42no—
497D4#90912yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—81no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4#93011yescharacteristic
515C2×C4#93011yescharacteristic
516C2×C4#93011yescharacteristic
517C2×C4C22×D1211yescharacteristic
518C2×C4C2×#23511yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—121no—
526C8—121no—
527C8—121no—
528C8—121no—
529C8#95012yes—
530C2×C3—41no—
531C2×C3—42no—
532C2×C3#303511yescharacteristic
533C2×C3#305611yescharacteristic
534C2×C3#306911yescharacteristic
535C22—21no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22C2×#93011yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—241no—
550C4—241no—
551C4—241no—
552C4—241no—
553C4C2×#90911yescharacteristic
554C4#574811yescharacteristic
555C4C2×#95011yescharacteristic
556C4#582211yescharacteristic
557C3#1918111yescharacteristic, complemented, Sylow
558C2—41no—
559C2—42no—
560C2#8252511yescharacteristic
561C2#8377611yescharacteristic
562C2#8381611yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2970C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenter#2970D3
hypocenterC3#19181
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2970

Complements

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

NH ≅ G/Ntype
4. #78540C2semidirect
5. #70967C2semidirect
6. #70966C2semidirect
7. #70963C2semidirect
8. #70962C2semidirect
9. #70855C2semidirect
10. #70854C2semidirect
11. #70851C2semidirect
12. #70850C2semidirect
13. #70821C2semidirect
14. #70820C2semidirect
15. #69850C2semidirect
557. C3#19181semidirect

Maximal subgroups

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

subgroupindex
C3×#29702
#825342
#825072
#785402
#709672
#709662
#709632
#709622
#708552
#708542
#708512
#708502
#708212
#708202
#698502
#191813