Finite Group Databasebeta

← #10694607

Subgroups — #10694607

Order 768.

374 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 93 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#83344C211yesmaximal, characteristic
2C3×#1531C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic
5#70794C211yesmaximal, characteristic
6#69806C211yesmaximal, characteristic
7#69801C211yesmaximal, characteristic
8#9765—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5092C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5051—21no—
18#5051—21no—
19C4×#216—21no—
20C4×#216—21no—
21#4932C2211yescharacteristic
22#4926C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#330—22no—
25#4550—21no—
26#4550—21no—
27#4538—22no—
28#4480—21no—
29#4479—21no—
30#4467—22no—
31#4465—22no—
32#3170—31no—
33#1746—31no—
34#1568—31no—
35#1531—31no—
36#1531D311yescharacteristic
37#1506—31no—
38#1506—31no—
39C22×C3×C8—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—21no—
50C2×#216—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57#870—21no—
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97D412yes—
67#835—21no—
68#835—21no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#810—21no—
73#810—21no—
74#807—21no—
75#807—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#533—31no—
83#533C2×D311yescharacteristic
84#500—31no—
85#494—31no—
86#487—31no—
87#487—31no—
88#467—61no—
89#467—61no—
90C4×Q16—61no—
91C4×Q16—61no—
92#432—31no—
93#432C2×D311yescharacteristic
94C2×#97C2×D311yescharacteristic
95C2×C4×C8—31no—
96#376—31no—
97#336—61no—
98#336—61no—
99#335—61no—
100#334—61no—
101#330—22no—
102#330—62no—
103#328—62no—
104#328—62no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—22no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#217—41no—
130#216—42no—
131#216—42no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139C4×Dic3—41no—
140C4×Dic3—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8C22⋊C411yescharacteristic
145C2×C3⋊C8C22⋊C411yescharacteristic
146C2×C3⋊C8C22⋊C411yescharacteristic
147C2×C3⋊C8C22⋊C411yescharacteristic
148C2×Q16—61no—
149C2×Q16—61no—
150C22×C8—21no—
151C22×C8—31no—
152C22×C8—31no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8C4×D311yescharacteristic
156C4⋊Q8C4×D311yescharacteristic
157#125—21no—
158#125—61no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—61no—
166C4×Q8—61no—
167#117—21no—
168#117—61no—
169C2×C42—61no—
170C2×C42C22×D311yescharacteristic
171#107—61no—
172#107—61no—
173C4⋊C8—21no—
174C4⋊C8—61no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179C22⋊C8—21no—
180C22⋊C8—61no—
181#97—21no—
182#97—21no—
183#97—61no—
184#97#6712yes—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—31no—
189#95—32no—
190#95—61no—
191C23×C3#12111yescharacteristic
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—81no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12411yescharacteristic
206C2×C3×C4#12711yescharacteristic
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212Dic6—41no—
213Dic6—41no—
214Dic6—81no—
215C3×C8—42no—
216C3×C8—81no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C3⋊C8—21no—
221C3⋊C8—81no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—61no—
225C2×Q8—61no—
226C22×C4—62no—
227C22×C4#23511yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230Q16—122no—
231Q16—122no—
232C2×C8—22no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—121no—
241C2×C8—121no—
242C2×C8—121no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—41no—
256C22⋊C4—121no—
257C42—41no—
258C42—121no—
259C42—121no—
260C42—121no—
261C42C2×C4×D311yescharacteristic
262C42D3×D411yescharacteristic
263C42#23711yescharacteristic
264C42C2×#6711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39011yescharacteristic
268C22×C3#47312yes—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4#42111yescharacteristic
277C3×C4#42111yescharacteristic
278Dic3—82no—
279Dic3—81no—
280Dic3—81no—
281C23#88411yescharacteristic
282Q8—41no—
283Q8—41no—
284Q8—81no—
285Q8—121no—
286Q8—121no—
287Q8—241no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4D3×C22⋊C411yescharacteristic
300C2×C4#67⋊C411yescharacteristic
301C2×C4#88511yescharacteristic
302C2×C4C4×#6711yescharacteristic
303C2×C4#93911yescharacteristic
304C2×C4#94011yescharacteristic
305C8—42no—
306C8—61no—
307C8—61no—
308C8—61no—
309C8—61no—
310C8—81no—
311C8—241no—
312C2×C3—22no—
313C2×C3#196311yescharacteristic
314C2×C3#197611yescharacteristic
315C2×C3#197911yescharacteristic
316C22—21no—
317C22—21no—
318C22#475512yes—
319C22#495711yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—81no—
326C4—81no—
327C4—242no—
328C4—241no—
329C4—241no—
330C4#482111yescharacteristic
331C4#482111yescharacteristic
332C3#976511yescharacteristic, complemented, Sylow
333C2—22no—
334C2#7010811yescharacteristic
335C2#7035711yescharacteristic
336C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9765
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
332. C3#9765semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#15312
#718362
#708362
#707942
#698062
#698012
#97653