Finite Group Databasebeta

← #10696210

Subgroups — #10696210

Order 768.

478 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 79 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#83368C211yesmaximal, characteristic, complemented
2#83358C211yesmaximal, characteristic
3#79355C211yesmaximal, characteristic
4C3×#1537C211yesmaximal, characteristic, complemented
5#70830C211yesmaximal, characteristic, complemented
6#70801C211yesmaximal, characteristic
7#67756C211yesmaximal, characteristic, complemented
8#10041—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5091—21no—
15#5090—21no—
16#5089—21no—
17#5086—21no—
18#5050—21no—
19#5047—22no—
20#5044—21no—
21#5031—21no—
22C4×#216—22no—
23#5027—21no—
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4850—21no—
27#4848—21no—
28#4738—21no—
29#4736—21no—
30#4710—22no—
31C3×#330—22no—
32#4483—22no—
33#4469—22no—
34#3216—31no—
35#3112—31no—
36#3104—31no—
37#1537D311yescharacteristic, complemented
38#1496—31no—
39#1467—31no—
40#1287—31no—
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
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
52C2×#216—22no—
53C2×#216—22no—
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#204—22no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75C3×#97D412yes—
76#835—22no—
77#830—21no—
78#830D412yes—
79#821—22no—
80#818—22no—
81#816—22no—
82#810—22no—
83#810—22no—
84#809—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91#532—31no—
92C2×C4×Q8—31no—
93#497—61no—
94#489—61no—
95#484—61no—
96#480—61no—
97#477—61no—
98#477—61no—
99#462—61no—
100#462—61no—
101#460—61no—
102#460—61no—
103#452—62no—
104#451—61no—
105#448—61no—
106#440—62no—
107C4×Q16—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110#432C2×D311yescharacteristic
111#431—31no—
112#330—22no—
113D4⋊C8—62no—
114#324—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—41no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123#235—82no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—42no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146#216—44no—
147#216—82no—
148#215—82no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161#204—44no—
162C2×#41—61no—
163C22×Q8—31no—
164C2×Q16—62no—
165C2×Q16—62no—
166C2×SD16—62no—
167C2×SD16—62no—
168C4⋊Q8—31no—
169C4⋊Q8D1211yescharacteristic
170C4⋊Q8D1211yescharacteristic
171#127—31no—
172#125—21no—
173#124—61no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176#121—61no—
177#121—61no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186#117—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#107—62no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—62no—
198#103—62no—
199#103—62no—
200#103—62no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207#97—21no—
208#97—61no—
209#97#6712yes—
210C4×C8—32no—
211C23×C3#12111yescharacteristic
212C22×D3—41no—
213C3×Q8—41no—
214C3×Q8—42no—
215C3×Q8—41no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227#67—82no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234D12—42no—
235D12—42no—
236D12—42no—
237C4×D3—82no—
238Dic6—22no—
239Dic6—22no—
240Dic6—41no—
241Dic6—41no—
242Dic6—42no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—42no—
247#41—242no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—122no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—121no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4#23511yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264Q16—124no—
265Q16—242no—
266SD16—124no—
267SD16—242no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—62no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42—41no—
288C42—121no—
289C42—121no—
290C42C2×D1211yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3#39211yescharacteristic
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4#47312yes—
309Dic3—42no—
310Dic3—81no—
311Dic3—81no—
312C23—121no—
313C23#88411yescharacteristic
314Q8—41no—
315Q8—42no—
316Q8—41no—
317Q8—62no—
318Q8—62no—
319Q8—121no—
320Q8—121no—
321Q8—122no—
322D4—122no—
323D4—122no—
324D4—122no—
325D4—242no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—62no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—242no—
337C2×C4—241no—
338C2×C4#88211yescharacteristic
339C2×C4#88211yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4#93011yescharacteristic
342C2×C4#93711yescharacteristic
343C2×C4#94011yescharacteristic
344C8—42no—
345C8—81no—
346C8—122no—
347C8—122no—
348C2×C3—41no—
349C2×C3#200811yescharacteristic
350C2×C3#201011yescharacteristic
351C2×C3#202011yescharacteristic
352D3—161no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#496011yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—122no—
368C4—241no—
369C4—241no—
370C4#475512yes—
371C3#1004111yescharacteristic, complemented, Sylow
372C2—41no—
373C2—481no—
374C2#7012511yescharacteristic
375C2#7040611yescharacteristic
376C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1537C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1537D3
hypocenterC3#10041
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
1. #83368C2semidirect
4. C3×#1537C2semidirect
5. #70830C2semidirect
7. #67756C2semidirect
37. #1537D3semidirect
371. C3#10041semidirect

Maximal subgroups

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

subgroupindex
#833682
#833582
#793552
C3×#15372
#708302
#708012
#677562
#100413