Finite Group Databasebeta

← #10699211

Subgroups — #10699211

Order 768.

402 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 95 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
2#70746C211yesmaximal, characteristic
3#70524C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6#69486C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic
8#10505—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#810—22no—
15C2×C4⋊(C3⋊C8)C2211yescharacteristic
16C4×#208C2211yescharacteristic
17#4849—21no—
18#4849—21no—
19#4739—22no—
20#4699—21no—
21#4681—21no—
22#4554—21no—
23#4554—21no—
24#4523—21no—
25#4523—21no—
26#4486—21no—
27#4481—21no—
28#4480—21no—
29#4479—21no—
30#3209—31no—
31#1997—31no—
32#1986—31no—
33#1981—31no—
34#1565—31no—
35#1550—31no—
36#1303D311yescharacteristic
37C22×Dic6D412yes—
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45#924—21no—
46#924—21no—
47C2×#208D411yescharacteristic
48C2×#208D411yescharacteristic
49C22×C3⋊C8—21no—
50C2×Dic12—22no—
51C2×Dic12—22no—
52#870—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869D411yescharacteristic
58#869D411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D411yescharacteristic
64C3×C4⋊C8D411yescharacteristic
65C3×C22⋊C8—21no—
66C3×C4×C8—21no—
67C3×#95—21no—
68#816—22no—
69#816—22no—
70#810—44no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75#533—31no—
76C2×C4⋊Q8—31no—
77#494—61no—
78#487—61no—
79#462—61no—
80#462—61no—
81#451—62no—
82#431C2×D311yescharacteristic
83#423C2×D311yescharacteristic
84C2×C4⋊C8—31no—
85C2×#103—62no—
86C4×M16—31no—
87C4×C4⋊C4C2×D311yescharacteristic
88#343—61no—
89#343—61no—
90#339—61no—
91#339—61no—
92#335—61no—
93#334—61no—
94#333—61no—
95#332—61no—
96C22×C3×C4—41no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×Dic3—22no—
101C2×Dic6—22no—
102C2×Dic6—21no—
103C2×Dic6—22no—
104C2×Dic6—21no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C3×M16—41no—
108C3×M16—41no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8—41no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—41no—
114C3×C42—41no—
115C3×C42—41no—
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120#217—41no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—21no—
133#208—21no—
134#208—21no—
135#208Q1612yescomplemented
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—41no—
140Dic12—82no—
141Dic12—82no—
142C22×Q8—32no—
143C2×Q16—62no—
144C2×Q16—62no—
145C2×M16—21no—
146C2×M16—31no—
147C2×M16—31no—
148C22×C8—61no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
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—61no—
163C2×C4⋊C4—61no—
164C2×C42—21no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8D1211yescharacteristic
171C4⋊C8D1211yescharacteristic
172#103—62no—
173#103—62no—
174#103—124no—
175C22⋊C8—21no—
176#97—61no—
177C4×C8—21no—
178C4×C8—61no—
179#95—21no—
180C23×C3#12111yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4C2×Q1611yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—82no—
201Dic6—44no—
202Dic6—41no—
203Dic6—42no—
204Dic6—41no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—41no—
208C3×C8—41no—
209C3×C8—81no—
210C3⋊C8—22no—
211C3⋊C8—41no—
212C3⋊C8—81no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—122no—
218C2×Q8—122no—
219C22×C4—41no—
220C22×C4—62no—
221C22×C4#23511yescharacteristic
222C22×C4D3×D411yescharacteristic
223C22×C4D3×D411yescharacteristic
224Q16—242no—
225Q16—242no—
226M16—32no—
227M16—41no—
228M16—41no—
229M16—61no—
230M16—61no—
231M16—61no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—122no—
238C2×C8—121no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C42—41no—
253C42—41no—
254C42—121no—
255C42C2×D1211yescharacteristic
256C42#23511yescharacteristic
257C42D3×D411yescharacteristic
258C42#23911yescharacteristic
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—81no—
275Dic3—82no—
276Dic3—81no—
277C23#88411yescharacteristic
278Q8—124no—
279Q8—121no—
280Q8—122no—
281Q8—121no—
282Q8—122no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—122no—
293C2×C4—242no—
294C2×C4#87511yescharacteristic
295C2×C4#88211yescharacteristic
296C2×C4#88411yescharacteristic
297C2×C4#89411yescharacteristic
298C2×C4#89411yescharacteristic
299C2×C4#89511yescharacteristic
300C2×C4#91211yescharacteristic
301C2×C4#91511yescharacteristic
302C2×C4#91511yescharacteristic
303C2×C4D3×Q1611yescharacteristic
304C8—41no—
305C8—41no—
306C8—62no—
307C8—81no—
308C8—121no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203711yescharacteristic
313C2×C3#204911yescharacteristic
314C2×C3#205611yescharacteristic
315C22—21no—
316C22—21no—
317C22#467211yescharacteristic
318C22#476711yescharacteristic
319C22#479511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—242no—
325C4—241no—
326C4—241no—
327C4#476711yescharacteristic
328C4#479511yescharacteristic
329C4#482211yescharacteristic
330C4#482511yescharacteristic
331C4#486711yescharacteristic
332C4#488711yescharacteristic
333C3#1050511yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6873311yescharacteristic
337C2#6886411yescharacteristic
338C2#6901911yescharacteristic
subgroupG/H
centerC22#4672
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1303D3
hypocenterC3#10505
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
135. #208Q16semidirect
333. C3#10505semidirect

Maximal subgroups

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

subgroupindex
#793132
#707462
#705242
#705092
#694932
#694862
C3×#13032
#105053