Finite Group Databasebeta

← #10699209

Subgroups — #10699209

Order 768.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#83222C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71778C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic
5#69868C211yesmaximal, characteristic
6#69850C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10505—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5228—22no—
14C2×#835—22no—
15#5092—21no—
16#5090—21no—
17#5090—21no—
18#5079—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#343—21no—
23C3×#343—21no—
24#4555—21no—
25#4555—21no—
26#4486—21no—
27#4481—21no—
28#4458—21no—
29#4457—21no—
30#3209—31no—
31#1997D311yescharacteristic
32#1986—31no—
33#1981—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8D412yes—
38C2×C3×M16—21no—
39C2×C3×M16D411yescharacteristic
40C2×C3×M16D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#216—22no—
49C2×#216—22no—
50#926—21no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56#870—21no—
57#869D411yescharacteristic
58#869D411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×#97—21no—
64C3×C4×C8—21no—
65#835—22no—
66#835—22no—
67#835—44no—
68C3⋊#95—21no—
69#830—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74C4×C3⋊C8—21no—
75#533—31no—
76C2×C4⋊Q8C2×D311yescharacteristic
77#494—61no—
78#487—61no—
79#462—61no—
80#462—61no—
81#451—62no—
82#431—31no—
83#423C2×D311yescharacteristic
84C2×C4⋊C8—31no—
85C2×#103—62no—
86C4×M16C2×D311yescharacteristic
87C4×C4⋊C4—31no—
88#343—21no—
89#343—21no—
90#339—61no—
91#339—61no—
92#335—61no—
93#334—61no—
94#333—61no—
95#332—61no—
96C2×C3×Q8—22no—
97C2×C3×Q8—22no—
98C2×C3×Q8—42no—
99C2×C3×Q8—42no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—41no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C3×M16—21no—
108C3×M16—21no—
109C3×M16—21no—
110C3×M16—41no—
111C3×M16—41no—
112C3×M16Q1612yescomplemented
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125#216—82no—
126#216—82no—
127C4⋊Dic3—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137C4×Dic3—41no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—41no—
142C22×Q8D1212yes—
143C2×Q16—62no—
144C2×Q16—62no—
145C2×M16—21no—
146C2×M16#6711yescharacteristic
147C2×M16#6711yescharacteristic
148C22×C8—61no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155#125—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160#117—61no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—61no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—31no—
169C4⋊C8—31no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—62no—
173#103—62no—
174#103—124no—
175C22⋊C8—61no—
176#97—21no—
177C4×C8—21no—
178C4×C8—61no—
179#95—61no—
180C23×C3#12111yescharacteristic
181C3×Q8—44no—
182C3×Q8—42no—
183C3×Q8—42no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—82no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4C2×Q1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207C3×C8—22no—
208C3×C8—41no—
209C3×C8—81no—
210C3⋊C8—41no—
211C3⋊C8—41no—
212C3⋊C8—81no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—61no—
218C2×Q8—61no—
219C22×C4—22no—
220C22×C4—121no—
221C22×C4C2×D1211yescharacteristic
222C22×C4#23511yescharacteristic
223C22×C4C2×#6711yescharacteristic
224Q16—242no—
225Q16—242no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—41no—
230M16—41no—
231M16#21612yescomplemented
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—122no—
238C2×C8—121no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—62no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C42—121no—
253C42—121no—
254C42—121no—
255C42D3×D411yescharacteristic
256C42D3×D411yescharacteristic
257C42#23711yescharacteristic
258C42#23711yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3#39411yescharacteristic
262C22×C3#46211yescharacteristic
263C22×C3#46411yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—82no—
268C3×C4#45311yescharacteristic
269C3×C4#45511yescharacteristic
270C3×C4#46211yescharacteristic
271C3×C4#46411yescharacteristic
272C3×C4#46711yescharacteristic
273C3×C4#47011yescharacteristic
274Dic3—82no—
275Dic3—81no—
276Dic3—81no—
277C23#93011yescharacteristic
278Q8—44no—
279Q8—42no—
280Q8—42no—
281Q8—121no—
282Q8—121no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—122no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4#88211yescharacteristic
295C2×C4#88311yescharacteristic
296C2×C4#88511yescharacteristic
297C2×C4#88611yescharacteristic
298C2×C4#93811yescharacteristic
299C2×C4#93911yescharacteristic
300C2×C4#94211yescharacteristic
301C2×C4C2×#21611yescharacteristic
302C2×C4#95011yescharacteristic
303C2×C4#95111yescharacteristic
304C8—22no—
305C8—41no—
306C8—81no—
307C8—121no—
308C8—121no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203711yescharacteristic
313C2×C3#204911yescharacteristic
314C2×C3#205611yescharacteristic
315C22—21no—
316C22—21no—
317C22#496111yescharacteristic
318C22#502811yescharacteristic
319C22#503111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—242no—
325C4—241no—
326C4—241no—
327C4#474611yescharacteristic
328C4#474711yescharacteristic
329C4#505111yescharacteristic
330C4#505211yescharacteristic
331C4#505411yescharacteristic
332C4#509011yescharacteristic
333C3#1050511yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6968111yescharacteristic
337C2#7006611yescharacteristic
338C2#7011011yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10505
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
112. C3×M16Q16semidirect
231. M16#216semidirect
333. C3#10505semidirect

Maximal subgroups

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

subgroupindex
#832222
C3×#19972
#717782
#705092
#698682
#698502
#677152
#105053