Finite Group Databasebeta

← #10696798

Subgroups — #10696798

Order 768.

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

normal subgroups only

This group has 454 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#79352C211yesmaximal, characteristic
2#79318C211yesmaximal, characteristic
3#78507C211yesmaximal, characteristic, complemented
4#70831C211yesmaximal, characteristic, complemented
5#70803C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic
7C3×#1291C211yesmaximal, characteristic, complemented
8#10126—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5121—21no—
16#5121—21no—
17#4925C2211yescharacteristic
18#4849—21no—
19#4848—21no—
20#4847—22no—
21#4736—21no—
22#4735—21no—
23#4733—22no—
24#4715—21no—
25#4714—21no—
26#4713—21no—
27#4702—21no—
28#4702—21no—
29#4702—21no—
30#4683—22no—
31#4484—22no—
32#4469—22no—
33#4459—22no—
34#3234—31no—
35#3216—31no—
36#3166—31no—
37#1497—31no—
38#1484—31no—
39#1470—31no—
40#1291D311yescharacteristic, complemented
41C2×#235—21no—
42C2×C3×M16—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46#926—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic12—22no—
54C2×#204—22no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#870—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×#97—21no—
75C3×#97—21no—
76C3×C4×C8D412yes—
77#830—21no—
78#821—22no—
79#821—22no—
80#821—22no—
81#818—22no—
82#818—22no—
83#818—22no—
84#817—22no—
85#817—22no—
86#817—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8—21no—
90#533—31no—
91#532—31no—
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#501—61no—
96#499—62no—
97#497—61no—
98#490—61no—
99#489—61no—
100#481—61no—
101#480—61no—
102#478—62no—
103#462—61no—
104#460—61no—
105#452—62no—
106#432C2×D311yescharacteristic
107#432C2×D311yescharacteristic
108#431—31no—
109#428—61no—
110#428—61no—
111C4×M16C2×D311yescharacteristic
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121#235—82no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×M16—24no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158Dic12—82no—
159#204—82no—
160C2×#41—61no—
161C2×Q16—62no—
162C2×SD16—62no—
163C2×M16—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167#127—31no—
168#125—61no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#124—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176#121—61no—
177#121—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#107—62no—
190#107—62no—
191#106—62no—
192#106—62no—
193#106—62no—
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8D1212yes—
197#103—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8D1212yes—
204#97—21no—
205#97—21no—
206#97—21no—
207C4×C8—61no—
208C4×C8D1212yes—
209C23×C3#12111yescharacteristic
210C22×D3—41no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220#67—82no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228D12—42no—
229D12—42no—
230D12—42no—
231C4×D3—82no—
232Dic6—41no—
233Dic6—41no—
234Dic6—42no—
235Dic6—42no—
236C3×C8—24no—
237C3×C8—42no—
238C3×C8—42no—
239C3⋊C8—81no—
240#41—242no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4C2×D1211yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4#23911yescharacteristic
254Q16—242no—
255SD16—242no—
256M16—24no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—121no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—121no—
281C42C2×D1211yescharacteristic
282C42C2×D1211yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#39211yescharacteristic
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4#45612yes—
298Dic3—81no—
299Dic3—81no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23—121no—
304C23#89511yescharacteristic
305Q8—121no—
306Q8—121no—
307Q8—122no—
308Q8—122no—
309D4—122no—
310D4—122no—
311D4—122no—
312D4—242no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—122no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—242no—
323C2×C4—241no—
324C2×C4#87411yescharacteristic
325C2×C4#88211yescharacteristic
326C2×C4#88211yescharacteristic
327C2×C4#88211yescharacteristic
328C2×C4#89511yescharacteristic
329C2×C4#89511yescharacteristic
330C8—24no—
331C8—42no—
332C8—42no—
333C8—241no—
334C2×C3—41no—
335C2×C3#200511yescharacteristic
336C2×C3#200811yescharacteristic
337C2×C3#201711yescharacteristic
338D3—161no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22#467111yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4—241no—
355C4—241no—
356C4#482612yes—
357C3#1012611yescharacteristic, complemented, Sylow
358C2—41no—
359C2—481no—
360C2#6872711yescharacteristic
361C2#6876811yescharacteristic
362C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1291D3
hypocenterC3#10126
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
3. #78507C2semidirect
4. #70831C2semidirect
5. #70803C2semidirect
7. C3×#1291C2semidirect
40. #1291D3semidirect
357. C3#10126semidirect

Maximal subgroups

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

subgroupindex
#793522
#793182
#785072
#708312
#708032
#698532
C3×#12912
#101263