Finite Group Databasebeta

← #10692401

Subgroups — #10692401

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#79326C211yesmaximal, characteristic
3C3×#1515C211yesmaximal, characteristic, complemented
4#71825C211yesmaximal, characteristic, complemented
5#69565C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9369—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5479C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5089C411yescharacteristic, complemented
14#5089C411yescharacteristic, complemented
15#5081C411yescharacteristic, complemented
16#5081C411yescharacteristic, complemented
17#5066—21no—
18#5063—21no—
19#5003—21no—
20#4996—21no—
21#4926C2211yescharacteristic
22#4925C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4836—21no—
25#4828—21no—
26#4729—21no—
27#4728—21no—
28#4709—21no—
29#4708—21no—
30C3×#328—21no—
31C3×#328—21no—
32#4535—21no—
33#4535—21no—
34#4452—21no—
35#4451—21no—
36#3240—31no—
37#2967—31no—
38#1521—31no—
39#1515D311yescharacteristic, complemented
40#1485—31no—
41#1288—31no—
42#1287—31no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52#930D411yescharacteristic
53#929—21no—
54C4×#67—21no—
55#924D411yescharacteristic
56C2×#208—21no—
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⋊C8D411yescharacteristic
67C3×C22⋊C8D411yescharacteristic
68C3×#97—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72#830—21no—
73#830—21no—
74(C2×D3)⋊C8—21no—
75(C2×D3)⋊C8—21no—
76#815—21no—
77Dic3⋊C8—21no—
78Dic3⋊C8—21no—
79C8×Dic3—21no—
80#810—21no—
81#810—21no—
82#809—21no—
83#809—21no—
84#808—21no—
85#808—21no—
86#807—21no—
87#807—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91#803C2×C411yescharacteristic
92#803C2×C411yescharacteristic
93C4×C3⋊C8—21no—
94C4×C3⋊C8D411yescharacteristic
95C4×C3⋊C8D411yescharacteristic
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#432C2×D311yescharacteristic
113#431—31no—
114#405—61no—
115C4×M16—31no—
116#383C2×D311yescharacteristic
117#328—21no—
118#328—21no—
119#328—61no—
120#327—61no—
121Q8⋊C8—61no—
122Q8⋊C8—61no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—21no—
130C2×D12C22⋊C411yescharacteristic
131C2×C4×D3—21no—
132C2×Dic6C22⋊C411yescharacteristic
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152C4⋊Dic3C22⋊C411yescharacteristic
153C4⋊Dic3C22⋊C411yescharacteristic
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158C4×Dic3—21no—
159#208—42no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—41no—
165ΓL(1,𝔽25)—42no—
166D3×C8—42no—
167C2×M16—61no—
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—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8#6711yescharacteristic
207C22⋊C8#6711yescharacteristic
208#97—21no—
209#97—31no—
210#97—31no—
211#97—61no—
212#97—61no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—61no—
216C4×C8—61no—
217#95—21no—
218#95—21no—
219#95—21no—
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×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238#67—81no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242D12—41no—
243C4×D3—41no—
244C4×D3—41no—
245Dic6—41no—
246C3×C8—41no—
247C3×C8—41no—
248C3⋊C8—22no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8—81no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—31no—
256C2×D4—31no—
257C2×D4—61no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263M16—122no—
264M16—122no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—122no—
272C2×C8—121no—
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
294C42#23511yescharacteristic
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
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—
312C3×C4—161no—
313Dic3—41no—
314Dic3—41no—
315Dic3—41no—
316C23—61no—
317C23#93811yescharacteristic
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—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4D3×C22⋊C411yescharacteristic
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93711yescharacteristic
342C8—41no—
343C8—41no—
344C8—62no—
345C8—121no—
346C8—121no—
347C8—121no—
348C8—241no—
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—161no—
373C3#936911yescharacteristic, complemented, Sylow
374C2—41no—
375C2—241no—
376C2#7005511yescharacteristic
377C2#7035011yescharacteristic
378C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9369
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
3. C3×#1515C2semidirect
4. #71825C2semidirect
6. #67757C2semidirect
7. #67633C2semidirect
13. #5089C4semidirect
14. #5089C4semidirect
15. #5081C4semidirect
16. #5081C4semidirect
39. #1515D3semidirect
373. C3#9369semidirect

Maximal subgroups

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

subgroupindex
#833622
#793262
C3×#15152
#718252
#695652
#677572
#676332
#93693