Finite Group Databasebeta

← #10712002

Subgroups — #10712002

Order 768.

424 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78667C211yesmaximal, characteristic
2#70060C211yesmaximal, characteristic
3#70045C211yesmaximal, characteristic
4#70044C211yesmaximal, characteristic, complemented
5#70002C211yesmaximal, characteristic, complemented
6C3×#1391C211yesmaximal, characteristic, complemented
7#67696C211yesmaximal, characteristic, complemented
8#12968—31nomaximal, Sylow
9#5832C2211yescharacteristic
10#5819C2211yescharacteristic
11C3×#410C2211yescharacteristic
12C3×#410C2211yescharacteristic
13C3×#408C2211yescharacteristic
14#5240C2211yescharacteristic
15#5234C2211yescharacteristic
16#5128—21no—
17C8×#67—21no—
18#5102—21no—
19#5005—21no—
20#4999—21no—
21#4729—21no—
22#4727—21no—
23#67⋊C8—21no—
24#4718—21no—
25#4559—21no—
26#4533—21no—
27#4532—21no—
28#4474C412yescomplemented
29#4473C412yescomplemented
30#2979—31no—
31#2126—31no—
32#1884—31no—
33#1868—31no—
34#1868—31no—
35#1391—31no—
36#1391D311yescharacteristic, complemented
37C2×C3×#41C2311yescharacteristic
38C2×#235D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C22×C3×C8D411yescharacteristic
41C3×#117—21no—
42C3×#117—21no—
43#948—42no—
44#947Q811yescharacteristic
45#944Q811yescharacteristic
46#940C2×C411yescharacteristic
47#939—21no—
48#939—21no—
49#936C2×C411yescharacteristic
50C4×#67—21no—
51C4×#67—21no—
52#926—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×#208—21no—
55C2×#208—21no—
56C2×#208—21no—
57C2×#208D411yescharacteristic
58C22×C3⋊C8D411yescharacteristic
59C2×ΓL(1,𝔽25)—21no—
60C2×D3×C8—21no—
61C3×#99—21no—
62C3×#99—21no—
63C3×#99—21no—
64C3×#99C2×C412yes—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67#834—21no—
68#834C2×C412yes—
69#830—21no—
70#830—21no—
71(C2×D3)⋊C8—21no—
72(C2×D3)⋊C8—21no—
73Dic3⋊C8—21no—
74Dic3⋊C8—21no—
75Dic3⋊C8—42no—
76C8×Dic3—21no—
77C8×Dic3—21no—
78#805—42no—
79C4⋊(C3⋊C8)—42no—
80#568—31no—
81#532—31no—
82#436—61no—
83#436—61no—
84C8×D4—61no—
85C8×D4—61no—
86#431—61no—
87#424—61no—
88#424—61no—
89#421—61no—
90#410—31no—
91#410C2×D311yescharacteristic
92#410C2×D311yescharacteristic
93#408—31no—
94#408—61no—
95#408C2×D311yescharacteristic
96#354—32no—
97#353—32no—
98#344—61no—
99#344—61no—
100#339—61no—
101C3×#41—41no—
102C3×#41—41no—
103C2×C3×Q8C2×Q811yescharacteristic
104C2×C3×D4C22×C411yescharacteristic
105C2×C3×D4#4111yescharacteristic
106C2×C3×D4#4111yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—21no—
111C2×#67—21no—
112C2×#67—21no—
113C22×Dic3—21no—
114#235—41no—
115#235—41no—
116C2×D12C4⋊C411yescharacteristic
117C2×C4×D3—21no—
118C2×Dic6C4⋊C411yescharacteristic
119C3×M16—41no—
120C2×C3×C8—21no—
121C2×C3×C8—21no—
122C2×C3×C8—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42—41no—
129#217—21no—
130#217—21no—
131#217—22no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134C4⋊Dic3—41no—
135Dic3⋊C4—21no—
136Dic3⋊C4—21no—
137C4×Dic3—21no—
138C4×Dic3—41no—
139C4×Dic3C4⋊C411yescharacteristic
140C4×Dic3C4⋊C411yescharacteristic
141#208—22no—
142#208—41no—
143#208—41no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152ΓL(1,𝔽25)—81no—
153D3×C8—81no—
154C2×#41—31no—
155C2×#41C22×D311yescharacteristic
156#131—122no—
157C2×M16—31no—
158C2×M16—61no—
159C2×M16—61no—
160C2×M16—61no—
161C2×M16—61no—
162C2×M16D1211yescharacteristic
163C22×C8—31no—
164C22×C8—61no—
165C22×C8D1211yescharacteristic
166C4⋊Q8—31no—
167#127—31no—
168#124—31no—
169#124—31no—
170#121—61no—
171#121—61no—
172C4×D4—61no—
173C4×D4—61no—
174#117—21no—
175#117—21no—
176#117—61no—
177C2×C42—31no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—122no—
181C4⋊C8—122no—
182#104—122no—
183#99—21no—
184#99—21no—
185#99—21no—
186#99—32no—
187#99—61no—
188#99C4×D312yes—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3C4×D411yescharacteristic
198C23×C3C4×D411yescharacteristic
199C23×C3#12311yescharacteristic
200C22×D3—21no—
201C3×Q8—41no—
202C3×D4—41no—
203C3×D4—41no—
204C3×D4—41no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—42no—
212C2×C3×C4—81no—
213C2×C3×C4C2×C4⋊C411yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4C22⋊Q811yescharacteristic
216C2×C3×C4C22⋊Q811yescharacteristic
217#67—42no—
218#67—42no—
219#67—81no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225D12—41no—
226C4×D3—41no—
227C4×D3—41no—
228Dic6—41no—
229C3×C8—41no—
230C3×C8—41no—
231C3⋊C8—41no—
232C3⋊C8—42no—
233C3⋊C8—41no—
234#41—41no—
235#41—41no—
236#41—121no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8D3×Q811yescharacteristic
240C2×D4—31no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4C2×C4×D311yescharacteristic
245C2×D4#23511yescharacteristic
246C2×D4#23711yescharacteristic
247C22×C4—61no—
248C22×C4—61no—
249C22×C4C2×D1211yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252M16—41no—
253M16—62no—
254M16—121no—
255M16—121no—
256M16—121no—
257M16—121no—
258M16—241no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—121no—
267C2×C8—121no—
268C2×C8—241no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—62no—
281C42—31no—
282C42—31no—
283C42—41no—
284C42—41no—
285C42—61no—
286C42—121no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3#38511yescharacteristic
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—82no—
301C3×C4—161no—
302C3×C4B(4,𝔽2)11yescharacteristic
303C3×C4#45811yescharacteristic
304Dic3—22no—
305Dic3—41no—
306Dic3—41no—
307Dic3—82no—
308C23—61no—
309C23C4×D1211yescharacteristic
310C23#67⋊C411yescharacteristic
311C23#88311yescharacteristic
312Q8—41no—
313Q8—121no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—121no—
318D4—122no—
319D4—122no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—81no—
332C2×C4—122no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4#88211yescharacteristic
337C2×C4D3×C4⋊C411yescharacteristic
338C2×C4#89611yescharacteristic
339C2×C4#89711yescharacteristic
340C8—41no—
341C8—41no—
342C8—122no—
343C8—121no—
344C8—121no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3#186711yescharacteristic
350D3—81no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22#466811yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—62no—
365C4—82no—
366C4—121no—
367C4—121no—
368C4—161no—
369C4—242no—
370C4#474111yescharacteristic
371C4#474211yescharacteristic
372C3#1296811yescharacteristic, complemented, Sylow
373C2—21no—
374C2—41no—
375C2—41no—
376C2—41no—
377C2—241no—
378C2#6896111yescharacteristic
subgroupG/H
centerC4#4741
commutatorC2×C3×D4C22×C4
FrattiniC2×#41C22×D3
FittingC3×#1391C2
socleC2×C3#1867
prime radical#5832C22
cosocleC2×C3×#41C23
hypercenter#1391D3
hypocenterC3#12968
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
4. #70044C2semidirect
5. #70002C2semidirect
6. C3×#1391C2semidirect
7. #67696C2semidirect
28. #4474C4semidirect
29. #4473C4semidirect
36. #1391D3semidirect
372. C3#12968semidirect

Maximal subgroups

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

subgroupindex
#786672
#700602
#700452
#700442
#700022
C3×#13912
#676962
#129683