Finite Group Databasebeta

← #10692402

Subgroups — #10692402

Order 768.

390 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 91 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#83127C211yesmaximal, characteristic
2#83125C211yesmaximal, characteristic
3C3×#1521C211yesmaximal, characteristic
4#70769C211yesmaximal, characteristic, complemented
5#69566C211yesmaximal, characteristic, complemented
6#67747C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9369—31nomaximal, Sylow
9C3×#529C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5064C411yescharacteristic, complemented
14#5064C411yescharacteristic, complemented
15#5055C411yescharacteristic, complemented
16#5055C411yescharacteristic, complemented
17#5045—21no—
18#5035—21no—
19#5024—21no—
20#5022—21no—
21#5012—21no—
22#5011—21no—
23#5011—21no—
24#5009—21no—
25#5000—21no—
26#4997—21no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29C4×#208C2211yescharacteristic
30C3×#328—21no—
31C3×#327—21no—
32#4484—21no—
33#4484—21no—
34#4452—21no—
35#4452—21no—
36#3240—31no—
37#2967—31no—
38#1521D311yescharacteristic
39#1515—31no—
40#1485—31no—
41#1288—31no—
42#1287—31no—
43C3×#126—21no—
44C3×#125C2×C411yescharacteristic
45C3×#124C2×C411yescharacteristic
46C3×#123—21no—
47C3×C22⋊Q8D411yescharacteristic
48C3×#121D411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C3×C4×D4—21no—
51C3×C4×D4D411yescharacteristic
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58C2×#208—21no—
59C22×C3⋊C8—21no—
60#870—21no—
61#869C2×C411yescharacteristic
62#869C2×C411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71#835—21no—
72#835—21no—
73#832—21no—
74#832—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830D411yescharacteristic
79#830D411yescharacteristic
80#808—21no—
81#808—21no—
82#807—21no—
83#807—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#803—21no—
91#803C2×C411yescharacteristic
92#803C2×C411yescharacteristic
93C4×C3⋊C8—21no—
94C4×C3⋊C8D411yescharacteristic
95C4×C3⋊C8D411yescharacteristic
96#533—31no—
97#529C2×D311yescharacteristic
98#501—31no—
99#501—31no—
100#489—31no—
101#489—31no—
102#485—61no—
103#482—61no—
104#479—61no—
105#478—61no—
106#446—61no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#432—31no—
112#432—31no—
113#431C2×D311yescharacteristic
114#405—61no—
115C4×M16—31no—
116#383C2×D311yescharacteristic
117#328—21no—
118#328—61no—
119#328—61no—
120#327—21no—
121Q8⋊C8—61no—
122Q8⋊C8—61no—
123C2×C3×Q8C22⋊C411yescharacteristic
124C2×C3×D4—21no—
125C2×C3×D4C22⋊C411yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4C22⋊C411yescharacteristic
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C42—21no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158#208—42no—
159#208—42no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—42no—
167C2×M16—61no—
168C2×M16—61no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#126—21no—
173#125—61no—
174#125C4×D311yescharacteristic
175#124C4×D311yescharacteristic
176#123—21no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C22⋊Q8#6711yescharacteristic
180#121#6711yescharacteristic
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8D1211yescharacteristic
184C4×D4—21no—
185C4×D4D1211yescharacteristic
186#117—21no—
187#117—61no—
188C2×C42C22×D311yescharacteristic
189#107—61no—
190#107—61no—
191#106—61no—
192#106—61no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198#103—61no—
199#103—61no—
200#102—61no—
201#102—61no—
202C22⋊C8—21no—
203C22⋊C8—31no—
204C22⋊C8—31no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#97—31no—
209#97—31no—
210#97—61no—
211#97—61no—
212#97—61no—
213C4×C8—21no—
214C4×C8—31no—
215C4×C8—31no—
216C4×C8—61no—
217#95—21no—
218#95—21no—
219#95—21no—
220C23×C3—21no—
221C23×C3#12111yescharacteristic
222C3×Q8—41no—
223C3×D4—41no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×Dic3—41no—
243C2×Dic3—41no—
244Dic6—42no—
245Dic6—81no—
246C3×C8—81no—
247C3⋊C8—22no—
248C3⋊C8—41no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8—41no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8(C2×D3)⋊C411yescharacteristic
256C2×D4—21no—
257C2×D4(C2×D3)⋊C411yescharacteristic
258C22×C4—21no—
259C22×C4—41no—
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263M16—122no—
264M16—122no—
265C2×C8—41no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—122no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4(C2×D3)⋊C411yescharacteristic
284C4⋊C4(C2×D3)⋊C411yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—121no—
291C42—21no—
292C42—121no—
293C42C2×C4×D311yescharacteristic
294C42C2×D1211yescharacteristic
295C42D3×D411yescharacteristic
296C42#23711yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3#38611yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—161no—
314Dic3—81no—
315Dic3—81no—
316C23—21no—
317C23#93911yescharacteristic
318Q8—41no—
319Q8—122no—
320Q8—241no—
321D4—41no—
322D4—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4#67⋊C411yescharacteristic
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88611yescharacteristic
340C2×C4C2×(C2×D3)⋊C411yescharacteristic
341C2×C4#93911yescharacteristic
342C8—62no—
343C8—81no—
344C8—121no—
345C8—121no—
346C8—121no—
347C8—121no—
348C8—121no—
349C2×C3—41no—
350C2×C3—81no—
351C2×C3#188111yescharacteristic
352C2×C3#190611yescharacteristic
353C2×C3#190911yescharacteristic
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22#495811yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—161no—
371C4—241no—
372C4—241no—
373C3#936911yescharacteristic, complemented, Sylow
374C2—41no—
375C2—81no—
376C2#7004611yescharacteristic
377C2#7029311yescharacteristic
378C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1521C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#1521D3
hypocenterC3#9369
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1521

Complements

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

NH ≅ G/Ntype
4. #70769C2semidirect
5. #69566C2semidirect
6. #67747C2semidirect
7. #67633C2semidirect
13. #5064C4semidirect
14. #5064C4semidirect
15. #5055C4semidirect
16. #5055C4semidirect
373. C3#9369semidirect

Maximal subgroups

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

subgroupindex
#831272
#831252
C3×#15212
#707692
#695662
#677472
#676332
#93693