Finite Group Databasebeta

← #10696087

Subgroups — #10696087

Order 768.

492 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 81 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#79355C211yesmaximal, characteristic
2#78529C211yesmaximal, characteristic
3#74278C211yesmaximal, characteristic, complemented
4#70814C211yesmaximal, characteristic, complemented
5#69857C211yesmaximal, characteristic, complemented
6#69564C211yesmaximal, characteristic
7C3×#1288C211yesmaximal, characteristic, complemented
8#10021—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#383C2211yescharacteristic
15#5133—21no—
16#5133—21no—
17#4926C2211yescharacteristic
18#4850—21no—
19#4848—21no—
20#4738—21no—
21#4736—21no—
22#4713—21no—
23#4710—22no—
24#4703—21no—
25#4697—22no—
26#4661—21no—
27#4659—21no—
28#4659—21no—
29#4656—21no—
30#4653—22no—
31#4465—22no—
32#4456—22no—
33#4451—22no—
34#3142—31no—
35#3112—31no—
36#2634—31no—
37#1534—31no—
38#1514—31no—
39#1491—31no—
40#1288D311yescharacteristic, complemented
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×#204—22no—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×C4×C8D412yes—
74C3×#95—21no—
75C3×#95—21no—
76C3×#95D412yes—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—21no—
81#821—22no—
82#821—22no—
83#821—22no—
84#818—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#532—31no—
91C2×#125—31no—
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#432—31no—
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16C2×D311yescharacteristic
106#400—61no—
107#400—61no—
108#398—61no—
109#395—61no—
110#387—62no—
111#383C2×D311yescharacteristic
112#330—62no—
113#327—62no—
114Q8⋊C8—62no—
115C22×C3×C4—22no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—41no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
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—
133C3×M16—42no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—44no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—41no—
160#204—44no—
161C2×#41—61no—
162C22×Q8—31no—
163C2×SD16—62no—
164C2×M16—21no—
165C4⋊Q8—31no—
166#127—31no—
167#125—61no—
168#125—61no—
169#125—61no—
170#125—61no—
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—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200#97—21no—
201#97—61no—
202#97D1212yes—
203C4×C8D1212yes—
204#95—21no—
205#95—21no—
206#95—62no—
207#95—62no—
208#95—62no—
209#95D1212yes—
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—42no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224#67—82no—
225C2×Dic3—22no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
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—42no—
246C3⋊C8—81no—
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—22no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265SD16—124no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—121no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—124no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42—121no—
288C42—121no—
289C42C2×D1211yescharacteristic
290C42C2×D1211yescharacteristic
291C42D3×D411yescharacteristic
292C42#23911yescharacteristic
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—82no—
307Dic3—42no—
308Dic3—81no—
309Dic3—81no—
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—42no—
329C2×C4—62no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—122no—
339C2×C4—242no—
340C2×C4#87411yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#89511yescharacteristic
344C2×C4#89511yescharacteristic
345C2×C4#89511yescharacteristic
346C8—42no—
347C8—42no—
348C8—241no—
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#467111yescharacteristic
360C22#482712yes—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—122no—
369C4—241no—
370C4—241no—
371C4—241no—
372C4—241no—
373C3#1002111yescharacteristic, complemented, Sylow
374C2—22no—
375C2—481no—
376C2#6870311yescharacteristic
377C2#6876711yescharacteristic
378C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1288D3
hypocenterC3#10021
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
3. #74278C2semidirect
4. #70814C2semidirect
5. #69857C2semidirect
7. C3×#1288C2semidirect
40. #1288D3semidirect
373. C3#10021semidirect

Maximal subgroups

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

subgroupindex
#793552
#785292
#742782
#708142
#698572
#695642
C3×#12882
#100213