Finite Group Databasebeta

← #10696225

Subgroups — #10696225

Order 768.

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

normal subgroups only

This group has 476 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#79344C211yesmaximal, characteristic
2#79321C211yesmaximal, characteristic, complemented
3#79298C211yesmaximal, characteristic
4#70827C211yesmaximal, characteristic
5#70815C211yesmaximal, characteristic, complemented
6#70771C211yesmaximal, characteristic, complemented
7C3×#1287C211yesmaximal, characteristic, complemented
8#10043—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#4926C2211yescharacteristic
16#4851—21no—
17#4850—21no—
18#4846—21no—
19#4842—21no—
20#4841—21no—
21#4841—21no—
22#4731—21no—
23#4731—21no—
24#4723—22no—
25#4722—21no—
26#4721—21no—
27#4710—22no—
28#4703—21no—
29#4702—21no—
30C4×#204—22no—
31#4468—22no—
32#4465—22no—
33#4456—22no—
34#3237—31no—
35#3110—31no—
36#3101—31no—
37#1537—31no—
38#1497—31no—
39#1466—31no—
40#1287D311yescharacteristic, complemented
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—21no—
45#926—21no—
46#926—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50#924—21no—
51#924—21no—
52C2×#204—22no—
53C2×#204—22no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58#870—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×#97—21no—
75C3×#97D412yes—
76C3×C4×C8D412yes—
77#830—21no—
78#821—22no—
79#821—22no—
80#818—22no—
81#818—22no—
82#817—22no—
83#817—22no—
84#816—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#533—31no—
91#533—31no—
92C2×C4×D4—31no—
93#501—61no—
94#491—61no—
95#485—62no—
96#484—61no—
97#483—61no—
98#477—61no—
99#476—61no—
100#475—61no—
101#474—61no—
102#461—61no—
103#461—61no—
104#450—61no—
105#450—61no—
106#440—62no—
107C4×SD16—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110#432C2×D311yescharacteristic
111#431C2×D311yescharacteristic
112#330—62no—
113Q8⋊C8—62no—
114#323—62no—
115C23×D3—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C4×D3—82no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
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#217—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
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—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158#204—44no—
159#204—44no—
160C22×D4—31no—
161C23×C4—61no—
162C2×SD16—62no—
163C2×SD16—62no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168#125—61no—
169#125—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—61no—
183#117—61no—
184C2×C4⋊C4—31no—
185C2×C22⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188#107—62no—
189#106—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195#103—62no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8D1212yes—
205#97—21no—
206#97—61no—
207#97D1212yes—
208C4×C8D1212yes—
209C23×C3#12111yescharacteristic
210C22×D3—22no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—81no—
214C22×D3—81no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—81no—
230D12—22no—
231D12—22no—
232D12—41no—
233D12—41no—
234C4×D3—82no—
235C4×D3—82no—
236Dic6—42no—
237Dic6—42no—
238Dic6—42no—
239Dic6—42no—
240C3×C8—42no—
241C3×C8—42no—
242C3×C8—42no—
243C3⋊C8—81no—
244C24—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—122no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4—242no—
256C22×C4C2×D1211yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4#23911yescharacteristic
259SD16—124no—
260SD16—124no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—121no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42—121no—
284C42—121no—
285C42C2×D1211yescharacteristic
286C42C2×D1211yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#39211yescharacteristic
293C2×D3—42no—
294C2×D3—82no—
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—82no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4#47112yes—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308Dic3—81no—
309C23—62no—
310C23—121no—
311C23—121no—
312C23—241no—
313C23—241no—
314C23#89511yescharacteristic
315Q8—122no—
316Q8—122no—
317Q8—122no—
318Q8—122no—
319D4—62no—
320D4—62no—
321D4—121no—
322D4—121no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4—241no—
334C2×C4#87411yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88211yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#89511yescharacteristic
340C8—42no—
341C8—42no—
342C8—42no—
343C8—241no—
344C2×C3—41no—
345C2×C3#200911yescharacteristic
346C2×C3#201011yescharacteristic
347C2×C3#201911yescharacteristic
348D3—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—122no—
353C22—242no—
354C22—242no—
355C22—241no—
356C22—241no—
357C22#467111yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—241no—
364C4—241no—
365C4—241no—
366C4—241no—
367C4—241no—
368C4#489812yes—
369C3#1004311yescharacteristic, complemented, Sylow
370C2—41no—
371C2—242no—
372C2#6870311yescharacteristic
373C2#6894311yescharacteristic
374C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1287D3
hypocenterC3#10043
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
2. #79321C2semidirect
5. #70815C2semidirect
6. #70771C2semidirect
7. C3×#1287C2semidirect
40. #1287D3semidirect
369. C3#10043semidirect

Maximal subgroups

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

subgroupindex
#793442
#793212
#792982
#708272
#708152
#707712
C3×#12872
#100433