Finite Group Databasebeta

← #10699208

Subgroups — #10699208

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#83222C211yesmaximal, characteristic
2C3×#1981C211yesmaximal, characteristic
3#71790C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#70420C211yesmaximal, characteristic
6#69493C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10505—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#103—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14#5228—22no—
15#5092—21no—
16#5090—21no—
17#5090—21no—
18#5079—21no—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4530—21no—
23#4530—21no—
24#4523—21no—
25#4523—21no—
26#4482—21no—
27#4481—21no—
28#4480—21no—
29#4479—21no—
30#3209—31no—
31#1997—31no—
32#1986—31no—
33#1981D311yescharacteristic
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8D412yes—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#216—22no—
49C2×#216—22no—
50#926—21no—
51#924—21no—
52#924—21no—
53C2×#208—21no—
54C2×#208D411yescharacteristic
55C2×#208D411yescharacteristic
56#870—21no—
57#869D411yescharacteristic
58#869D411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×#103—44no—
64C3×#95—21no—
65#835—22no—
66#835—22no—
67#830—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)D411yescharacteristic
71C4⋊(C3⋊C8)D411yescharacteristic
72#803—21no—
73C4×C3⋊C8—21no—
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#423—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×#103—22no—
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—
96C2×C3×Q8—22no—
97C2×C3×Q8—22no—
98C2×C3×Q8—42no—
99C2×C3×Q8—42no—
100C22×C3×C4—22no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×C3×C8—42no—
108C2×C3×C8—41no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C42—41no—
116C3×C42—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122#216—82no—
123#216—82no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—41no—
131#208—21no—
132#208—21no—
133#208—21no—
134#208—41no—
135#208—41no—
136#208Q1612yescomplemented
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—41no—
142C22×Q8#6712yes—
143C2×Q16—62no—
144C2×Q16—62no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16—61no—
148C22×C8—21no—
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—21no—
164C2×C42—21no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—31no—
169C4⋊C8—31no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—44no—
173#103—62no—
174#103—62no—
175C22⋊C8—61no—
176#97—61no—
177C4×C8—61no—
178C4×C8—61no—
179#95—21no—
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—42no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—82no—
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—41no—
203C2×Dic3—41no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207C3×C8—81no—
208C3⋊C8—22no—
209C3⋊C8—41no—
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—41no—
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224Q16—242no—
225Q16—242no—
226M16—32no—
227M16—61no—
228M16—61no—
229M16—61no—
230M16—121no—
231M16—121no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—121no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C42—41no—
253C42—41no—
254C42—121no—
255C42#23511yescharacteristic
256C42#23511yescharacteristic
257C42D3×D411yescharacteristic
258C42D3×D411yescharacteristic
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—82no—
269C3×C4#45311yescharacteristic
270C3×C4#45511yescharacteristic
271C3×C4#46211yescharacteristic
272C3×C4#46411yescharacteristic
273C3×C4#46711yescharacteristic
274C3×C4#47011yescharacteristic
275Dic3—81no—
276Dic3—81no—
277C23#93911yescharacteristic
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—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4#88311yescharacteristic
295C2×C4#88411yescharacteristic
296C2×C4#88411yescharacteristic
297C2×C4#88511yescharacteristic
298C2×C4#91511yescharacteristic
299C2×C4#91611yescharacteristic
300C2×C4D3×Q1611yescharacteristic
301C2×C4#91811yescharacteristic
302C2×C4#92911yescharacteristic
303C2×C4#93711yescharacteristic
304C8—62no—
305C8—81no—
306C8—121no—
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#495911yescharacteristic
318C22#516611yescharacteristic
319C22#518211yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—82no—
325C4—241no—
326C4—241no—
327C4#479511yescharacteristic
328C4#479811yescharacteristic
329C4#481311yescharacteristic
330C4#481611yescharacteristic
331C4#506111yescharacteristic
332C4#507711yescharacteristic
333C3#1050511yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6966511yescharacteristic
337C2#7028511yescharacteristic
338C2#7035211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1981D3
hypocenterC3#10505
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
136. #208Q16semidirect
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×#19812
#717902
#704382
#704202
#694932
#676202
#105053