Finite Group Databasebeta

← #10697196

Subgroups — #10697196

Order 768.

476 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 103 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#82431C211yesmaximal, characteristic
2C2×#5021C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic
4C3×#1461C211yesmaximal, characteristic
5#70780C211yesmaximal, characteristic
6#70768C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic
8#10183—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5225—21no—
15C2×#835—21no—
16#5022—22no—
17#5021—22no—
18#5021—21no—
19#5021—22no—
20#5020—21no—
21#4964—21no—
22C2×#807—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4849—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29C3×#324—22no—
30#4482—22no—
31#4456—22no—
32#3209—31no—
33#3078—31no—
34C2×#477—31no—
35#1549—31no—
36#1481—31no—
37#1461D311yescharacteristic
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C2×#208—21no—
55C22×C3⋊C8—21no—
56C2×Dic12—24no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8D412yes—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#835—24no—
73#835—22no—
74#835—22no—
75#816—24no—
76#808—22no—
77#807—24no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#494—61no—
87#487—61no—
88#479—62no—
89#477—62no—
90#477—62no—
91#477—61no—
92#475—61no—
93#462—62no—
94#451—62no—
95#431C2×D311yescharacteristic
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#107—61no—
99#423—31no—
100C2×C4⋊C8—31no—
101#419—61no—
102C2×#103—61no—
103#332—62no—
104Q8⋊C8—62no—
105#324—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8Q1612yescomplemented
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—41no—
118C2×Dic6—22no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42—41no—
130C3×C42—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135#217—41no—
136C4⋊Dic3—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153Dic12—84no—
154C22×Q8—61no—
155C22×Q8D1211yescharacteristic
156C2×Q16—64no—
157C2×M16—61no—
158C22×C8—61no—
159C4⋊Q8—32no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—61no—
173C4×Q8—61no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4D1211yescharacteristic
179C2×C42—21no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—64no—
191#103—64no—
192#103—62no—
193#103—62no—
194C22⋊C8—21no—
195C22⋊C8#6712yes—
196C4×C8—21no—
197C4×C8—21no—
198C23×C3#12111yescharacteristic
199C3×Q8—22no—
200C3×Q8—22no—
201C3×Q8—41no—
202C3×Q8—41no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×C3×C4C2×Q1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—82no—
224Dic6—42no—
225Dic6—44no—
226Dic6—42no—
227Dic6—82no—
228C3×C8—42no—
229C3×C8—81no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—61no—
240C2×Q8—121no—
241C2×Q8—242no—
242C2×Q8Dic1212yescomplemented
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—121no—
246C22×C4C2×D1211yescharacteristic
247C22×C4#23511yescharacteristic
248C22×C4C2×#6711yescharacteristic
249Q16—244no—
250M16—121no—
251M16—121no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—121no—
257C2×C8—121no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—121no—
273C42—41no—
274C42—41no—
275C42—121no—
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3#39211yescharacteristic
283C22×C3#46711yescharacteristic
284C22×C3#47011yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—42no—
289C3×C4—81no—
290C3×C4#45111yescharacteristic
291C3×C4#45111yescharacteristic
292C3×C4#45211yescharacteristic
293C3×C4#45211yescharacteristic
294C3×C4#47312yes—
295Dic3—82no—
296Dic3—81no—
297Dic3—81no—
298C23#93011yescharacteristic
299Q8—22no—
300Q8—22no—
301Q8—41no—
302Q8—41no—
303Q8—122no—
304Q8—124no—
305Q8—122no—
306Q8—242no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—242no—
318C2×C4#88211yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#90411yescharacteristic
323C2×C4C2×Dic1211yescharacteristic
324C2×C4#90911yescharacteristic
325C2×C4#90911yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#94011yescharacteristic
328C8—42no—
329C8—81no—
330C8—122no—
331C8—122no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3#200111yescharacteristic
335C2×C3#200811yescharacteristic
336C2×C3#202011yescharacteristic
337C22—21no—
338C22—21no—
339C22#496011yescharacteristic
340C22#511611yescharacteristic
341C22#513711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—242no—
348C4—241no—
349C4—241no—
350C4#473311yescharacteristic
351C4#473311yescharacteristic
352C4#473911yescharacteristic
353C4#473911yescharacteristic
354C4#475512yes—
355C3#1018311yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#6996411yescharacteristic
359C2#7005011yescharacteristic
360C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1461C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1461D3
hypocenterC3#10183
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
111. C2×C3×Q8Q16semidirect
242. C2×Q8Dic12semidirect
355. C3#10183semidirect

Maximal subgroups

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

subgroupindex
#824312
C2×#50212
#793132
C3×#14612
#707802
#707682
#677452
#101833