Finite Group Databasebeta

← #10692407

Subgroups — #10692407

Order 768.

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

normal subgroups only

This group has 390 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#83362C211yesmaximal, characteristic
2#78510C211yesmaximal, characteristic
3C3×#1485C211yesmaximal, characteristic, complemented
4#71837C211yesmaximal, characteristic, complemented
5#70791C211yesmaximal, characteristic
6#67747C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9369—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5479C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5128—21no—
14#5127—21no—
15#5089C411yescharacteristic, complemented
16#5089C411yescharacteristic, complemented
17#5081C411yescharacteristic, complemented
18#5081C411yescharacteristic, complemented
19#5066—21no—
20#5063—21no—
21#5003—21no—
22#4996—21no—
23#4937C2211yescharacteristic
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26#4706—21no—
27#4701—21no—
28#4690—21no—
29#4689—21no—
30C3×Q8⋊C8—21no—
31C3×Q8⋊C8—21no—
32#4538—21no—
33#4538—21no—
34#4470—21no—
35#4467—21no—
36#3240—31no—
37#2967—31no—
38#1521—31no—
39#1515—31no—
40#1485D311yescharacteristic, complemented
41#1288—31no—
42#1287—31no—
43C2×C3×M16—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C42C2311yescharacteristic
53#930D411yescharacteristic
54#929—21no—
55C4×#67—21no—
56#924D411yescharacteristic
57C2×ΓL(1,𝔽25)—21no—
58C2×D3×C8—21no—
59#876—21no—
60#875C2×C411yescharacteristic
61C4×D12D411yescharacteristic
62#872—21no—
63#870C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×C4×C8—21no—
69C3×C4×C8D411yescharacteristic
70C3×C4×C8D411yescharacteristic
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74#830—21no—
75#830D411yescharacteristic
76#830D411yescharacteristic
77(C2×D3)⋊C8—21no—
78(C2×D3)⋊C8—21no—
79#815—21no—
80Dic3⋊C8—21no—
81Dic3⋊C8—21no—
82C8×Dic3—21no—
83#810—21no—
84#810—21no—
85#809—21no—
86#809—21no—
87#808—21no—
88#808—21no—
89#807—21no—
90#807—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)—21no—
93#803—21no—
94#803C2×C411yescharacteristic
95#803C2×C411yescharacteristic
96#533C2×D311yescharacteristic
97#529—31no—
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×M16C2×D311yescharacteristic
116#383—31no—
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121Q8⋊C8—21no—
122Q8⋊C8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—21no—
129C22×Dic3—41no—
130C2×D12C22⋊C411yescharacteristic
131C2×C4×D3—21no—
132C2×Dic6C22⋊C411yescharacteristic
133C3×M16—42no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154C4⋊Dic3C22⋊C411yescharacteristic
155C4⋊Dic3C22⋊C411yescharacteristic
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160C4×Dic3—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165ΓL(1,𝔽25)—42no—
166D3×C8—42no—
167C2×M16—21no—
168C2×M16—61no—
169C22×C8—61no—
170C4⋊Q8C4×D311yescharacteristic
171C4⋊Q8C4×D311yescharacteristic
172#126—61no—
173#125—21no—
174#125—31no—
175#124—31no—
176#123—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—31no—
180#121—31no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—31no—
184C4×D4—31no—
185C4×D4—61no—
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—21no—
209#97—31no—
210#97—31no—
211#97—61no—
212#97—61no—
213C4×C8—21no—
214C4×C8—61no—
215C4×C8#6711yescharacteristic
216C4×C8#6711yescharacteristic
217#95—61no—
218#95—61no—
219#95—61no—
220C23×C3#12111yescharacteristic
221C22×D3—21no—
222C3×Q8—42no—
223C3×Q8—81no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235#67—81no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242D12—41no—
243C4×D3—41no—
244C4×D3—41no—
245Dic6—41no—
246C3×C8—22no—
247C3×C8—41no—
248C3×C8—81no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8—41no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—31no—
256C2×D4—31no—
257C2×D4—61no—
258C22×C4—61no—
259C22×C4—121no—
260C22×C4#23511yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263M16—42no—
264M16—122no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—41no—
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—31no—
278C4⋊C4—31no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C22⋊C4—41no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—41no—
292C42—61no—
293C42C2×C4×D311yescharacteristic
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C2×#6711yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#38611yescharacteristic
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—81no—
311C3×C4—81no—
312Dic3—41no—
313Dic3—41no—
314Dic3—41no—
315Dic3—161no—
316C23—61no—
317C23#88411yescharacteristic
318Q8—42no—
319Q8—81no—
320Q8—121no—
321D4—121no—
322D4—241no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—241no—
334C2×C4—241no—
335C2×C4—241no—
336C2×C4D3×C22⋊C411yescharacteristic
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#93811yescharacteristic
342C8—22no—
343C8—41no—
344C8—81no—
345C8—121no—
346C8—121no—
347C8—121no—
348C8—121no—
349C2×C3—41no—
350C2×C3#188111yescharacteristic
351C2×C3#190611yescharacteristic
352C2×C3#190911yescharacteristic
353D3—81no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22#495611yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4—481no—
373C3#936911yescharacteristic, complemented, Sylow
374C2—41no—
375C2—241no—
376C2#7001211yescharacteristic
377C2#7035011yescharacteristic
378C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1485C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1485D3
hypocenterC3#9369
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
3. C3×#1485C2semidirect
4. #71837C2semidirect
6. #67747C2semidirect
7. #67666C2semidirect
15. #5089C4semidirect
16. #5089C4semidirect
17. #5081C4semidirect
18. #5081C4semidirect
40. #1485D3semidirect
373. C3#9369semidirect

Maximal subgroups

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

subgroupindex
#833622
#785102
C3×#14852
#718372
#707912
#677472
#676662
#93693