Finite Group Databasebeta

← #10699207

Subgroups — #10699207

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#83370C211yesmaximal, characteristic
2C3×#1986C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#69879C211yesmaximal, characteristic
5#69850C211yesmaximal, characteristic
6#69486C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10505—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×C4×M16C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5093—21no—
14#5092—21no—
15#5050—22no—
16#5031—21no—
17#5031—21no—
18C2×#810—22no—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#339—21no—
23C3×#339—21no—
24#4554—21no—
25#4554—21no—
26#4547—21no—
27#4546—21no—
28#4458—21no—
29#4457—21no—
30#3209—31no—
31#1997—31no—
32#1986D311yescharacteristic
33#1981—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×Dic6D412yes—
38C2×C3×M16D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52C2×#216—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×#208—21no—
56C22×C3⋊C8—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61C3×#97—21no—
62C3×C4×C8—21no—
63C3×#95—21no—
64#830—21no—
65#810—22no—
66#810—22no—
67#810—44no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74C4×C3⋊C8—21no—
75#533C2×D311yescharacteristic
76C2×C4⋊Q8—31no—
77#494—61no—
78#487—61no—
79#462—61no—
80#462—61no—
81#451—62no—
82#431—31no—
83#423—31no—
84C2×C4⋊C8—31no—
85C2×#103—62no—
86C4×M16C2×D311yescharacteristic
87C4×C4⋊C4C2×D311yescharacteristic
88#343—61no—
89#343—61no—
90#339—21no—
91#339—21no—
92#335—61no—
93#334—61no—
94#333—61no—
95#332—61no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—22no—
103C2×Dic6—22no—
104C2×Dic6—22no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C3×M16—21no—
108C3×M16—21no—
109C3×M16—21no—
110C3×M16Q1612yescomplemented
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129#216—82no—
130#216—82no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C22×Q8—32no—
143C2×Q16—62no—
144C2×Q16—62no—
145C2×M16—61no—
146C2×M16#6711yescharacteristic
147C2×M16#6711yescharacteristic
148C22×C8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8D1211yescharacteristic
154C4⋊Q8D1211yescharacteristic
155#125—21no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C4×Q8—21no—
159C4×Q8—21no—
160#117—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C42—21no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—31no—
167C4⋊C8—31no—
168C4⋊C8—61no—
169C4⋊C8—61no—
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—21no—
180C23×C3#12111yescharacteristic
181C3×Q8—41no—
182C3×Q8—41no—
183C3×Q8—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12311yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4#12411yescharacteristic
198C2×C3×C4C2×Q1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×C3×C4#13711yescharacteristic
202C2×Dic3—42no—
203C2×Dic3—82no—
204Dic6—44no—
205Dic6—42no—
206Dic6—42no—
207C3×C8—22no—
208C3×C8—41no—
209C3⋊C8—41no—
210C3⋊C8—41no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—122no—
218C2×Q8—122no—
219C22×C4—41no—
220C22×C4—62no—
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224Q16—242no—
225Q16—242no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—121no—
230M16—121no—
231M16#21612yescomplemented
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—122no—
237C2×C8—121no—
238C2×C8—121no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42C2×D1211yescharacteristic
256C42#23511yescharacteristic
257C42D3×D411yescharacteristic
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—81no—
269C3×C4—81no—
270C3×C4#45311yescharacteristic
271C3×C4#45511yescharacteristic
272C3×C4#46211yescharacteristic
273C3×C4#46411yescharacteristic
274C3×C4#46711yescharacteristic
275C3×C4#47011yescharacteristic
276Dic3—82no—
277C23#93811yescharacteristic
278Q8—41no—
279Q8—41no—
280Q8—81no—
281Q8—124no—
282Q8—122no—
283Q8—122no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—122no—
293C2×C4—242no—
294C2×C4#88211yescharacteristic
295C2×C4#88311yescharacteristic
296C2×C4#88511yescharacteristic
297C2×C4#88611yescharacteristic
298C2×C4#93011yescharacteristic
299C2×C4#93911yescharacteristic
300C2×C4#94211yescharacteristic
301C2×C4C2×#21611yescharacteristic
302C2×C4#95011yescharacteristic
303C2×C4#95111yescharacteristic
304C8—22no—
305C8—41no—
306C8—121no—
307C8—121no—
308C8—241no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203711yescharacteristic
313C2×C3#204911yescharacteristic
314C2×C3#205611yescharacteristic
315C22—21no—
316C22—21no—
317C22#496111yescharacteristic
318C22#505411yescharacteristic
319C22#509011yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—81no—
325C4—81no—
326C4—242no—
327C4#482211yescharacteristic
328C4#482511yescharacteristic
329C4#502811yescharacteristic
330C4#503111yescharacteristic
331C4#505111yescharacteristic
332C4#505211yescharacteristic
333C3#1050511yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6968111yescharacteristic
337C2#7035911yescharacteristic
338C2#7040311yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1986D3
hypocenterC3#10505
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
110. 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
#833702
C3×#19862
#718302
#698792
#698502
#694862
#676202
#105053