Finite Group Databasebeta

← #10696563

Subgroups — #10696563

Order 768.

496 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 83 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#83283C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3#73410C211yesmaximal, characteristic, complemented
4C3×#1505C211yesmaximal, characteristic
5#70823C211yesmaximal, characteristic, complemented
6#70812C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10093—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5232—22no—
15#5081—21no—
16#5079—21no—
17#5071—21no—
18#5070—21no—
19#5055—22no—
20#5034—21no—
21#5033—21no—
22#5009—22no—
23#4936—21no—
24#4935—21no—
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4661—22no—
28#4656—22no—
29C3×#327—22no—
30#4482—22no—
31#4465—22no—
32#3234—31no—
33#3141—31no—
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1505D311yescharacteristic
38#1300—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×#127D411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—24no—
52C2×C4⋊Dic3D412yes—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×#208—21no—
58#870—21no—
59#870—21no—
60#870—21no—
61#870—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#95—21no—
70C3×#95D412yes—
71#835—24no—
72#832—24no—
73#832—22no—
74#832—22no—
75C3⋊#95—24no—
76#830—21no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#125—31no—
86#501—61no—
87#490—61no—
88#489—62no—
89#487—61no—
90#486—61no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432C2×D311yescharacteristic
96#417—62no—
97C4×M16—31no—
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#327—22no—
106C3×#41—44no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—21no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—24no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—84no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146#208—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×#41—21no—
150C22×Q8—61no—
151C2×M16—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8D1211yescharacteristic
157#127D1211yescharacteristic
158#125—61no—
159#125—61no—
160#125—61no—
161#125—61no—
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—64no—
174C2×C42—21no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180#103—64no—
181#102—64no—
182#102—62no—
183#102—62no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186#97—21no—
187#97—32no—
188#97—61no—
189C4×C8—61no—
190#95—21no—
191#95—64no—
192#95#6712yes—
193C23×C3—41no—
194C23×C3#12011yescharacteristic
195C3×Q8—42no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—81no—
199C3×D4—82no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×Dic3—42no—
215C2×Dic3—44no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220Dic6—82no—
221Dic6—82no—
222C3×C8—81no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225#41—44no—
226C2×Q8—21no—
227C2×Q8—121no—
228C2×Q8—121no—
229C2×Q8—242no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—61no—
237C22×C4—62no—
238C22×C4—61no—
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4C2×#6711yescharacteristic
242M16—122no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—62no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—62no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—62no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—244no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—24no—
264C42C2×D1211yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3#39211yescharacteristic
274C22×C3#45412yes—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4—81no—
282C3×C4#45612yes—
283Dic3—81no—
284Dic3—82no—
285Dic3—81no—
286C23—41no—
287C23#93711yescharacteristic
288Q8—42no—
289Q8—242no—
290Q8—242no—
291D4—42no—
292D4—42no—
293D4—81no—
294D4—82no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—82no—
303C2×C4—122no—
304C2×C4—124no—
305C2×C4—122no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#88411yescharacteristic
313C2×C4#93011yescharacteristic
314C2×C4#94011yescharacteristic
315C8—81no—
316C8—122no—
317C8—122no—
318C2×C3—22no—
319C2×C3—161no—
320C2×C3#200311yescharacteristic
321C2×C3#200511yescharacteristic
322C2×C3#200811yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#496011yescharacteristic
329C22#520112yes—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—44no—
336C4—81no—
337C4—242no—
338C4—241no—
339C4—241no—
340C4#482612yes—
341C3#1009311yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7005011yescharacteristic
345C2#7032611yescharacteristic
346C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1505D3
hypocenterC3#10093
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
3. #73410C2semidirect
5. #70823C2semidirect
6. #70812C2semidirect
7. #67615C2semidirect
341. C3#10093semidirect

Maximal subgroups

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

subgroupindex
#832832
#832732
#734102
C3×#15052
#708232
#708122
#676152
#100933