Finite Group Databasebeta

← #10784417

Subgroups — #10784417

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79313C211yesmaximal, characteristic, complemented
2#79306C211yesmaximal, characteristic
3#79273C211yesmaximal, characteristic, complemented
4#79272C211yesmaximal, characteristic, complemented
5#79270C211yesmaximal, characteristic
6#79268C211yesmaximal, characteristic
7C3×#1454C211yesmaximal, characteristic, complemented
8#68099C211yesmaximal, characteristic, complemented
9#68099C211yesmaximal, characteristic, complemented
10#68098C211yesmaximal, characteristic, complemented
11#68097C211yesmaximal, characteristic, complemented
12#68096C211yesmaximal, characteristic, complemented
13#68093C211yesmaximal, characteristic, complemented
14#68065C211yesmaximal, characteristic, complemented
15#68061C211yesmaximal, characteristic, complemented
16#19744—31nomaximal, Sylow
17#5477C2211yescharacteristic
18#5470C2211yescharacteristic
19#5470C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#431C2211yescharacteristic
24#4850—22no—
25#4849—22no—
26#4847—22no—
27#4846—22no—
28#4845—22no—
29#4829—22no—
30#4739—22no—
31#4737—22no—
32#4736—22no—
33#4733—22no—
34#4723—22no—
35#4722—22no—
36#4699C2211yescharacteristic
37#4699C2211yescharacteristic
38#4699C2211yescharacteristic
39#4694C2211yescharacteristic
40#4694C2211yescharacteristic
41#4694C2211yescharacteristic
42#4694C2211yescharacteristic
43#4694C2211yescharacteristic
44#4693C2211yescharacteristic
45#4682C2211yescharacteristic
46#4682C2211yescharacteristic
47#4682C2211yescharacteristic
48#4682C2211yescharacteristic
49#4682C2211yescharacteristic
50#4682C2211yescharacteristic
51#4681C2211yescharacteristic
52#4681C2211yescharacteristic
53#4681C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4681C2211yescharacteristic
57#4681C2211yescharacteristic
58#4680C2211yescharacteristic
59#4680C2211yescharacteristic
60C3×#322C2211yescharacteristic
61C3×#322C2211yescharacteristic
62C3×#322C2211yescharacteristic
63C3×C82C2211yescharacteristic
64#3231—31no—
65#3230—31no—
66#3228—31no—
67#3226—31no—
68#3210—31no—
69#3209—31no—
70#1723—31no—
71#1723—31no—
72#1722—31no—
73#1720—31no—
74#1719—31no—
75#1718—31no—
76#1713—31no—
77#1711—31no—
78#1454D311yescharacteristic, complemented
79C2×#235—21no—
80C22×Dic6—21no—
81C2×C3×C42C2311yescharacteristic
82#930—21no—
83#930—21no—
84#926—21no—
85#926—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88#924—21no—
89#924—21no—
90#924—21no—
91#924—21no—
92C2×Dic12—22no—
93C2×Dic12—22no—
94C2×Dic12—22no—
95C2×#204—22no—
96#875—21no—
97#874C2311yescharacteristic
98C4×D12—21no—
99C4×D12—21no—
100#870—21no—
101#870—21no—
102#869—21no—
103#869—21no—
104#869—21no—
105#869C2311yescharacteristic
106#869C2311yescharacteristic
107#869C2311yescharacteristic
108#869C2311yescharacteristic
109#869C2311yescharacteristic
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112C4×Dic6—21no—
113C4×Dic6—21no—
114C4×Dic6—21no—
115C4×Dic6—21no—
116C3×C4⋊C8D412yes—
117C3×C4⋊C8D412yes—
118C3×C4⋊C8D412yes—
119C3×C22⋊C8D412yes—
120C3×C22⋊C8D412yes—
121C3×C22⋊C8D412yes—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128#821—22no—
129#821—22no—
130#821—22no—
131#821—22no—
132#821—22no—
133#818—22no—
134#818—22no—
135#818—22no—
136#818—22no—
137#818—22no—
138#818—22no—
139#817—22no—
140#817—22no—
141#816—22no—
142#816—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148#533—31no—
149#533—31no—
150#532—31no—
151C2×C4⋊Q8—31no—
152#500—31no—
153#500—31no—
154#500—31no—
155#500—31no—
156#500—31no—
157#500—31no—
158#498—31no—
159#498—31no—
160#494—31no—
161#494—31no—
162#494—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#485—62no—
177#484—62no—
178#482—62no—
179#480—62no—
180#479—62no—
181#478—62no—
182#477—62no—
183#474—62no—
184#464—62no—
185#462—62no—
186#452—62no—
187#451—62no—
188#431C2×D311yescharacteristic
189#431C2×D311yescharacteristic
190#431C2×D311yescharacteristic
191#322C2×D311yescharacteristic
192#322C2×D311yescharacteristic
193#322C2×D311yescharacteristic
194C82C2×D311yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C2×#67—41no—
199C22×Dic3—41no—
200#235—82no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×C4×D3—41no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—41no—
213C2×Dic6—82no—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D412yes—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D412yes—
219C2×C3×C8C2×D412yes—
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C3×C42C2411yescharacteristic
224#217—41no—
225#217—41no—
226(C2×D3)⋊C4—41no—
227(C2×D3)⋊C4—41no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—41no—
243C4⋊Dic3—41no—
244Dic3⋊C4—41no—
245Dic3⋊C4—41no—
246Dic3⋊C4—41no—
247Dic3⋊C4—41no—
248C4×Dic3—41no—
249C4×Dic3—41no—
250Dic12—82no—
251Dic12—82no—
252Dic12—82no—
253#204—82no—
254C2×#41—61no—
255C22×Q8—61no—
256C2×Q16—62no—
257C2×Q16—62no—
258C2×Q16—62no—
259C2×SD16—62no—
260C4⋊Q8—31no—
261C4⋊Q8—31no—
262C4⋊Q8—31no—
263C4⋊Q8—31no—
264C4⋊Q8—31no—
265C4⋊Q8—31no—
266C4⋊Q8—31no—
267C4⋊Q8—61no—
268C4⋊Q8—61no—
269C4⋊Q8—61no—
270#127—31no—
271#125—61no—
272#125—61no—
273#124—61no—
274C22⋊Q8—61no—
275C22⋊Q8—61no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278#121—61no—
279#121—61no—
280C4×Q8—61no—
281C4×Q8—61no—
282C4×Q8—61no—
283C4×Q8—61no—
284C4×D4—61no—
285C4×D4—61no—
286#117—61no—
287#117—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C42C22×D311yescharacteristic
291#107—62no—
292#107—62no—
293#107—62no—
294#107—62no—
295#107—62no—
296#107—62no—
297#106—62no—
298#106—62no—
299C4⋊C8D1212yes—
300C4⋊C8D1212yes—
301C4⋊C8D1212yes—
302#103—62no—
303#103—62no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314C22⋊C8D1212yes—
315C22⋊C8D1212yes—
316C22⋊C8D1212yes—
317C4×C8C22×D311yescharacteristic
318C4×C8C22×D311yescharacteristic
319C4×C8C22×D311yescharacteristic
320C4×C8C22×D311yescharacteristic
321C4×C8C22×D311yescharacteristic
322C4×C8C22×D311yescharacteristic
323C23×C3#12711yescharacteristic
324C22×D3—41no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4#12711yescharacteristic
329C2×C3×C4#12711yescharacteristic
330C2×C3×C4#12711yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334#67—82no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—82no—
343D12—42no—
344D12—42no—
345D12—42no—
346C4×D3—82no—
347Dic6—42no—
348Dic6—42no—
349Dic6—42no—
350Dic6—42no—
351Dic6—42no—
352Dic6—42no—
353Dic6—42no—
354Dic6—82no—
355C3×C8—22no—
356C3×C8—22no—
357C3×C8—22no—
358C3×C8—22no—
359C3×C8—22no—
360C3×C8—22no—
361#41—242no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—121no—
370C2×Q8—242no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—121no—
375C22×C4—121no—
376C22×C4—121no—
377C22×C4C2×D1211yescharacteristic
378C22×C4C2×D1211yescharacteristic
379C22×C4C2×D1211yescharacteristic
380Q16—242no—
381Q16—242no—
382Q16—242no—
383SD16—242no—
384C2×C8C2×D1212yes—
385C2×C8C2×D1212yes—
386C2×C8C2×D1212yes—
387C2×C8C2×D1212yes—
388C2×C8C2×D1212yes—
389C2×C8C2×D1212yes—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—121no—
405C4⋊C4—121no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C4⋊C4—121no—
409C4⋊C4—121no—
410C22⋊C4—121no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C42—121no—
415C42—121no—
416C42C2×D1211yescharacteristic
417C42C2×D1211yescharacteristic
418C42C2×D1211yescharacteristic
419C42C23×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3C2×#12711yescharacteristic
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4#57612yes—
431C3×C4#57812yes—
432C3×C4#57812yes—
433Dic3—81no—
434Dic3—81no—
435Dic3—81no—
436Dic3—81no—
437Dic3—81no—
438Dic3—81no—
439Dic3—81no—
440C23—121no—
441C23#87411yescharacteristic
442Q8—122no—
443Q8—122no—
444Q8—122no—
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449Q8—242no—
450D4—122no—
451D4—122no—
452D4—122no—
453D4—242no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—242no—
465C2×C4—242no—
466C2×C4#87411yescharacteristic
467C2×C4#87411yescharacteristic
468C2×C4#87411yescharacteristic
469C2×C4C22×D1211yescharacteristic
470C2×C4C22×D1211yescharacteristic
471C2×C4C22×D1211yescharacteristic
472C8—22no—
473C8—22no—
474C8—22no—
475C8—22no—
476C8—22no—
477C8—22no—
478C2×C3—41no—
479C2×C3#315511yescharacteristic
480C2×C3#315611yescharacteristic
481C2×C3#315611yescharacteristic
482D3—161no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22C2×#87411yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—241no—
494C4—241no—
495C4—241no—
496C4—241no—
497C4—241no—
498C4—241no—
499C4—241no—
500C4#575212yes—
501C4#575312yes—
502C4#575312yes—
503C3#1974411yescharacteristic, complemented, Sylow
504C2—41no—
505C2—481no—
506C2#7856511yescharacteristic
507C2#7857611yescharacteristic
508C2#7857611yescharacteristic
subgroupG/H
centerC22C2×#874
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1454C2
socleC22×C3C2×#127
prime radical#5477C22
hypercenter#1454D3
hypocenterC3#19744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1454

Complements

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

NH ≅ G/Ntype
1. #79313C2semidirect
3. #79273C2semidirect
4. #79272C2semidirect
7. C3×#1454C2semidirect
8. #68099C2semidirect
9. #68099C2semidirect
10. #68098C2semidirect
11. #68097C2semidirect
12. #68096C2semidirect
13. #68093C2semidirect
14. #68065C2semidirect
15. #68061C2semidirect
78. #1454D3semidirect
503. C3#19744semidirect

Maximal subgroups

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

subgroupindex
#793132
#793062
#792732
#792722
#792702
#792682
C3×#14542
#680992
#680982
#680972
#680962
#680932
#680652
#680612
#197443