Finite Group Databasebeta

← #10697457

Subgroups — #10697457

Order 768.

484 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 97 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#79313C211yesmaximal, characteristic, complemented
2#79269C211yesmaximal, characteristic
3#78497C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic
6#69828C211yesmaximal, characteristic, complemented
7C3×#1281C211yesmaximal, characteristic, complemented
8#10223—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15#5113—22no—
16#4936C2211yescharacteristic
17#4849—22no—
18#4842—22no—
19#4739—22no—
20#4735—22no—
21#4714—21no—
22#4714—21no—
23#4713—21no—
24#4703—21no—
25#4699—21no—
26#4694—21no—
27#4681—21no—
28#4680—21no—
29#4551—22no—
30#4470—22no—
31#4467—22no—
32#3227—31no—
33#3209—31no—
34#3135—31no—
35#1993—31no—
36#1518—31no—
37#1517—31no—
38#1281D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C22×C3×C8D412yes—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×Dic12—24no—
52#875—21no—
53#874D411yescharacteristic
54C4×D12—21no—
55C4×D12—21no—
56#870—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×C22⋊C8D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#821—24no—
77#821—22no—
78#821—22no—
79#817—24no—
80#816—24no—
81#816—22no—
82#816—22no—
83#533—31no—
84#532—31no—
85C2×C4⋊Q8—31no—
86#498—61no—
87#494—61no—
88#491—61no—
89#490—61no—
90#489—61no—
91#489—61no—
92#487—61no—
93#486—61no—
94#481—62no—
95#476—62no—
96#462—62no—
97#451—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100#416—62no—
101C2×#97C2×D311yescharacteristic
102#382—31no—
103#340—62no—
104#328—62no—
105#327—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C2×#67—41no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112#235—42no—
113#235—42no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×Dic6—22no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#217—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—42no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150Dic12—84no—
151C2×#41—61no—
152C22×Q8—61no—
153C2×Q16—64no—
154C22×C8D1212yes—
155C4⋊Q8—32no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#127—31no—
163#125—61no—
164#124—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167#121—61no—
168#121—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—61no—
172C4×D4—61no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#106—64no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181#103—62no—
182#103—62no—
183#103—64no—
184#102—62no—
185#102—62no—
186#102—64no—
187C22⋊C8D1212yes—
188C22⋊C8D1212yes—
189#97—21no—
190#97—21no—
191C4×C8—21no—
192C4×C8—21no—
193#95—61no—
194#95—61no—
195C23×C3#12711yescharacteristic
196C22×D3—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#13512yes—
206C2×C3×C4#13712yes—
207#67—81no—
208#67—81no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—82no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217D12—42no—
218D12—42no—
219D12—81no—
220C4×D3—82no—
221Dic6—42no—
222Dic6—44no—
223Dic6—42no—
224Dic6—82no—
225Dic6—81no—
226C3×C8—42no—
227C3×C8—42no—
228C3×C8—42no—
229#41—122no—
230#41—122no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—121no—
236C2×Q8—242no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—121no—
241C22×C4—121no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4C2×D1211yescharacteristic
245C22×C4C2×D1211yescharacteristic
246C22×C4C2×D1211yescharacteristic
247Q16—244no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—121no—
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42#23911yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#49511yescharacteristic
278C22×C3#49711yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4#45212yes—
287C3×C4#46412yes—
288Dic3—82no—
289Dic3—81no—
290Dic3—81no—
291Dic3—81no—
292Dic3—161no—
293C23—121no—
294C23#87411yescharacteristic
295Q8—122no—
296Q8—124no—
297Q8—122no—
298Q8—241no—
299Q8—242no—
300D4—122no—
301D4—122no—
302D4—241no—
303D4—241no—
304D4—241no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—122no—
311C2×C4—242no—
312C2×C4—242no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#89511yescharacteristic
320C2×C4#89511yescharacteristic
321C2×C4#89511yescharacteristic
322C2×C4#90412yes—
323C2×C4#90912yes—
324C8—42no—
325C8—42no—
326C8—42no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#200811yescharacteristic
330C2×C3#200911yescharacteristic
331C2×C3#201511yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#467111yescharacteristic
339C22#469611yescharacteristic
340C22#471511yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—241no—
346C4—242no—
347C4—241no—
348C4—241no—
349C4—481no—
350C4#473312yes—
351C4#484512yes—
352C3#1022311yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—481no—
356C2#6867711yescharacteristic
357C2#6902811yescharacteristic
358C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1281D3
hypocenterC3#10223
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #79313C2semidirect
4. #70793C2semidirect
6. #69828C2semidirect
7. C3×#1281C2semidirect
38. #1281D3semidirect
352. C3#10223semidirect

Maximal subgroups

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

subgroupindex
#793132
#792692
#784972
#707932
#707902
#698282
C3×#12812
#102233