Finite Group Databasebeta

← #10778232

Subgroups — #10778232

Order 768.

592 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×#3076C211yesmaximal, characteristic
2#83348C211yesmaximal, characteristic, complemented
3#82534C211yesmaximal, characteristic
4#82519C211yesmaximal, characteristic
5#71832C211yesmaximal, characteristic, complemented
6#71151C211yesmaximal, characteristic, complemented
7#71148C211yesmaximal, characteristic, complemented
8#71139C211yesmaximal, characteristic, complemented
9#71136C211yesmaximal, characteristic, complemented
10#71118C211yesmaximal, characteristic, complemented
11#71117C211yesmaximal, characteristic, complemented
12#71106C211yesmaximal, characteristic, complemented
13#71105C211yesmaximal, characteristic, complemented
14#70439C211yesmaximal, characteristic, complemented
15#70420C211yesmaximal, characteristic, complemented
16#19185—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#479—21no—
21C3×#478—21no—
22C3×#477C2211yescharacteristic
23C3×#476C2211yescharacteristic
24C3×#475C2211yescharacteristic
25C3×#474C2211yescharacteristic
26C3×#427—21no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#416—21no—
29#5239—21no—
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5080C2211yescharacteristic
33#5079C2211yescharacteristic
34#5052—21no—
35#5052—21no—
36#5031C2211yescharacteristic
37#5030—21no—
38#5030—21no—
39C4×#216—21no—
40Q8×C3⋊C8C2211yescharacteristic
41Q8×C3⋊C8C2211yescharacteristic
42#5020C2211yescharacteristic
43#5019C2211yescharacteristic
44C4×#214—21no—
45#5012—21no—
46#5012—21no—
47#5012C2211yescharacteristic
48#5012C2211yescharacteristic
49#5007C2211yescharacteristic
50#4996—21no—
51#4926—21no—
52#4926C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3185—31no—
69#3107—31no—
70#3076D311yescharacteristic
71#2971—31no—
72#1703—31no—
73#1699—31no—
74#1691—31no—
75#1689—31no—
76#1684—31no—
77#1680—31no—
78#1672—31no—
79#1670—31no—
80#1565—31no—
81#1554—31no—
82#1552—31no—
83C2×C3×#41D411yescharacteristic
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×C4×Q8D411yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4D411yescharacteristic
97C3×C4×D4C2311yescharacteristic
98C3×#117—21no—
99C3×#117—21no—
100C3×#117D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#216—21no—
104C2×#216—21no—
105C2×#216—21no—
106C2×#214—21no—
107#926—21no—
108#924—21no—
109#924—21no—
110C2×#208—21no—
111C2×#208—21no—
112C2×#208D411yescharacteristic
113C2×#208D411yescharacteristic
114#870—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C3×#107—21no—
120C3×#107—21no—
121C3×#106—21no—
122C3×#106—21no—
123C3×C4⋊C8—21no—
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#103—21no—
127C3×#103—21no—
128C3×#102—21no—
129C3×#102—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#832—21no—
134#830—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#808—21no—
143#808—21no—
144#808—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803D411yescharacteristic
154#803D411yescharacteristic
155C4×C3⋊C8—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533C2×D311yescharacteristic
162C4×#41C2×D311yescharacteristic
163#498—31no—
164#494—31no—
165#487—31no—
166#487—31no—
167#485—61no—
168#479—21no—
169#478—21no—
170#477C2×D311yescharacteristic
171#476C2×D311yescharacteristic
172#475—31no—
173#475C2×D311yescharacteristic
174#474—31no—
175#474C2×D311yescharacteristic
176#470—61no—
177#470—61no—
178#462—31no—
179#461—31no—
180#452—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#441—61no—
184#441—61no—
185C4×Q16—61no—
186C4×SD16—61no—
187#436—31no—
188#436—31no—
189#436—61no—
190#436—61no—
191#432—31no—
192#432—61no—
193#431—31no—
194#427—21no—
195C2×C4⋊C8C2×D311yescharacteristic
196#416—21no—
197C4×M16—31no—
198C4×M16—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#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C3×#41—22no—
212C3×#41—41no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4#4111yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—42no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
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×C4⋊C4C2×D411yescharacteristic
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—41no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—41no—
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42#4111yescharacteristic
250C3×C42C2411yescharacteristic
251#217—41no—
252#216—42no—
253#216—42no—
254#216—42no—
255#214—42no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262C4×Dic3—41no—
263#208—22no—
264#208—41no—
265#208—42no—
266#208—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×#41#6711yescharacteristic
276C2×Q16—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—61no—
283C2×M16—61no—
284C22×C8—21no—
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—61no—
298C4×Q8—61no—
299C4×Q8#6711yescharacteristic
300C4×Q8C22×D311yescharacteristic
301C4×D4—21no—
302C4×D4—21no—
303C4×D4#6711yescharacteristic
304C4×D4C22×D311yescharacteristic
305#117—21no—
306#117—21no—
307#117—61no—
308#117#6711yescharacteristic
309C2×C42—21no—
310C2×C42C22×D311yescharacteristic
311#107—21no—
312#107—21no—
313#107—61no—
314#106—21no—
315#106—21no—
316#106—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—21no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—21no—
339#102—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—31no—
345#97—31no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8—61no—
352C23×C3—21no—
353C23×C3#12111yescharacteristic
354C3×Q8—21no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—81no—
358C3×Q8#13712yes—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—81no—
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4#12111yescharacteristic
378C2×C3×C4#12111yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×Dic3—41no—
383C2×Dic3—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8#13512yes—
398#41—22no—
399#41—41no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8C2×#6711yescharacteristic
405C2×D4—21no—
406C2×D4C2×#6711yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4D3×D411yescharacteristic
411C22×C4#23711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413Q16—122no—
414Q16—122no—
415Q16—122no—
416SD16—122no—
417M16—62no—
418M16—122no—
419M16—121no—
420M16—122no—
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—61no—
431C2×C8—61no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4C2×#6711yescharacteristic
447C4⋊C4C2×#6711yescharacteristic
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—41no—
452C22⋊C4—121no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—41no—
457C42—121no—
458C42D3×D411yescharacteristic
459C42#23711yescharacteristic
460C42C2×#6711yescharacteristic
461C42C23×D311yescharacteristic
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×#12111yescharacteristic
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4C2×#13511yescharacteristic
479C3×C4C2×#13711yescharacteristic
480C3×C4#57511yescharacteristic
481C3×C4#57511yescharacteristic
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23#93911yescharacteristic
486Q8—21no—
487Q8—41no—
488Q8—41no—
489Q8—81no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8#94212yes—
495D4—21no—
496D4—21no—
497D4—42no—
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—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4#93911yescharacteristic
515C2×C4#93911yescharacteristic
516C2×C4#93911yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×#23711yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—32no—
521C8—41no—
522C8—41no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—121no—
527C8—121no—
528C8—121no—
529C8—121no—
530C2×C3—41no—
531C2×C3—81no—
532C2×C3#303511yescharacteristic
533C2×C3#305611yescharacteristic
534C2×C3#306911yescharacteristic
535C22—21no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22C2×#93911yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—241no—
552C4—241no—
553C4D3×#13511yescharacteristic
554C4#576811yescharacteristic
555C4C2×#94211yescharacteristic
556C4#582211yescharacteristic
557C3#1918511yescharacteristic, complemented, Sylow
558C2—41no—
559C2—81no—
560C2#8273211yescharacteristic
561C2#8390011yescharacteristic
562C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3076C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenter#3076D3
hypocenterC3#19185
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
2. #83348C2semidirect
5. #71832C2semidirect
6. #71151C2semidirect
7. #71148C2semidirect
8. #71139C2semidirect
9. #71136C2semidirect
10. #71118C2semidirect
11. #71117C2semidirect
12. #71106C2semidirect
13. #71105C2semidirect
14. #70439C2semidirect
15. #70420C2semidirect
557. C3#19185semidirect

Maximal subgroups

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

subgroupindex
C3×#30762
#833482
#825342
#825192
#718322
#711512
#711482
#711392
#711362
#711182
#711172
#711062
#711052
#704392
#704202
#191853