Finite Group Databasebeta

← #10696086

Subgroups — #10696086

Order 768.

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

normal subgroups only

This group has 492 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#83042C211yesmaximal, characteristic, complemented
2#83012C211yesmaximal, characteristic
3#79355C211yesmaximal, characteristic
4C3×#1534C211yesmaximal, characteristic, complemented
5#70835C211yesmaximal, characteristic
6#70802C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10021—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5069—21no—
15#5068—22no—
16#5067—21no—
17#4986—21no—
18#4985—21no—
19#4983—21no—
20#4979—21no—
21#4978—22no—
22#4966—21no—
23#4966—21no—
24#4937C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4850—21no—
27#4848—21no—
28#4738—21no—
29#4736—21no—
30#4710—22no—
31C3×#330—22no—
32#4484—22no—
33#4470—22no—
34#3142—31no—
35#3112—31no—
36#2634—31no—
37#1534D311yescharacteristic, complemented
38#1514—31no—
39#1491—31no—
40#1288—31no—
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52#930—21no—
53#930—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×#208—21no—
59C2×#204—22no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#821—22no—
82#818—22no—
83#816—22no—
84#810—22no—
85#810—22no—
86#809—22no—
87#809—22no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#532—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×Q8—31no—
93#491—61no—
94#490—61no—
95#488—62no—
96#480—61no—
97#477—61no—
98#460—61no—
99#448—61no—
100#440—62no—
101#432C2×D311yescharacteristic
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16—31no—
106#400—61no—
107#400—61no—
108#398—61no—
109#395—61no—
110#387—62no—
111#383—31no—
112#330—22no—
113#327—62no—
114Q8⋊C8—62no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124#235—44no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—42no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
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×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—44no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158#208—42no—
159C2×C3⋊C8—22no—
160#204—44no—
161C2×#41—61no—
162C22×Q8—31no—
163C2×SD16—62no—
164C2×M16—61no—
165C4⋊Q8—31no—
166#127—31no—
167#125—21no—
168#125—21no—
169#125—21no—
170#125—21no—
171#124—61no—
172#121—61no—
173#121—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
186C2×C4⋊C4D1211yescharacteristic
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200#97—21no—
201#97—61no—
202#97#6712yes—
203C4×C8—32no—
204#95—32no—
205#95—62no—
206#95—62no—
207#95—62no—
208#95—61no—
209#95—61no—
210C23×C3#12011yescharacteristic
211C22×D3—41no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225#67—82no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235D12—42no—
236D12—42no—
237D12—81no—
238C4×D3—82no—
239Dic6—22no—
240Dic6—22no—
241Dic6—41no—
242Dic6—41no—
243Dic6—42no—
244C3×C8—42no—
245C3×C8—81no—
246C3⋊C8—42no—
247#41—124no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—61no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265SD16—124no—
266M16—122no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—124no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42—121no—
288C42—121no—
289C42#23511yescharacteristic
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42C2×#6711yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#39211yescharacteristic
296C22×C3#45412yes—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—81no—
307C3×C4—81no—
308Dic3—42no—
309Dic3—82no—
310Dic3—81no—
311Dic3—81no—
312C23—121no—
313C23#88211yescharacteristic
314Q8—62no—
315Q8—62no—
316Q8—121no—
317Q8—121no—
318Q8—122no—
319D4—122no—
320D4—122no—
321D4—241no—
322D4—242no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—62no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—242no—
340C2×C4#88211yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#93011yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#94011yescharacteristic
346C8—42no—
347C8—81no—
348C8—122no—
349C2×C3—22no—
350C2×C3#200311yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#201011yescharacteristic
353D3—161no—
354C22—21no—
355C22—21no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#475312yes—
360C22#496011yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—122no—
370C4—242no—
371C4—241no—
372C4—241no—
373C3#1002111yescharacteristic, complemented, Sylow
374C2—22no—
375C2—481no—
376C2#7011611yescharacteristic
377C2#7040611yescharacteristic
378C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1534C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1534D3
hypocenterC3#10021
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
1. #83042C2semidirect
4. C3×#1534C2semidirect
6. #70802C2semidirect
7. #67752C2semidirect
37. #1534D3semidirect
373. C3#10021semidirect

Maximal subgroups

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

subgroupindex
#830422
#830122
#793552
C3×#15342
#708352
#708022
#677522
#100213