Finite Group Databasebeta

← #10691798

Subgroups — #10691798

Order 768.

416 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 89 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#83318C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic, complemented
3C3×#1496C211yesmaximal, characteristic
4#71812C211yesmaximal, characteristic
5#70836C211yesmaximal, characteristic, complemented
6#67756C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9282—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5239—22no—
14#5092C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5076C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5054—22no—
19#4932C2211yescharacteristic
20#4926C2211yescharacteristic
21#4925C2211yescharacteristic
22#4828—22no—
23#4720—22no—
24#4679—21no—
25C8×Dic6—21no—
26C3×D4⋊C8—21no—
27C3×D4⋊C8—21no—
28#4537—22no—
29#4467—22no—
30#3210—31no—
31#2985—31no—
32#1506—31no—
33#1505—31no—
34#1496D311yescharacteristic
35#1287—31no—
36#1274—31no—
37C2×C3×#41—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#127C2×C411yescharacteristic
40C3×#124—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#216—22no—
47C2×#214—22no—
48#926—21no—
49#924—21no—
50#924D412yes—
51C2×C4×Dic3—21no—
52#870—21no—
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6D412yes—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—21no—
62C3×C4×C8D411yescharacteristic
63C3×C4×C8D411yescharacteristic
64#835—22no—
65#832—22no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#830—21no—
69#830D412yes—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8C2×C411yescharacteristic
79#533—31no—
80#532C2×D311yescharacteristic
81#494—31no—
82#492—31no—
83#487—31no—
84#486—31no—
85#464—62no—
86#452—62no—
87#446—62no—
88#446—61no—
89C8×Q8—61no—
90#433—62no—
91#432—31no—
92#432C2×D311yescharacteristic
93#431—31no—
94#431C2×D311yescharacteristic
95#376—31no—
96#328—62no—
97#327—62no—
98D4⋊C8—21no—
99D4⋊C8—21no—
100C3×#41—82no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C22×C3×C4—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—41no—
111C2×Dic6—21no—
112C2×Dic6C22⋊C412yes—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#217—22no—
125#216—82no—
126#214—82no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133C4×Dic3—22no—
134C4×Dic3—22no—
135C4×Dic3—41no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—22no—
139C2×#41—21no—
140C2×Q16—62no—
141C2×SD16—62no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8C4×D311yescharacteristic
145#127C4×D311yescharacteristic
146#125—61no—
147#124—21no—
148C22⋊Q8—32no—
149C22⋊Q8—61no—
150#121—21no—
151#121—21no—
152C4×Q8—32no—
153C4×Q8—61no—
154C4×D4—21no—
155C4×D4—21no—
156#117—61no—
157C2×C42—61no—
158C2×C42C22×D311yescharacteristic
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—62no—
162C4⋊C8—62no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#103—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—32no—
170C22⋊C8—61no—
171#97—21no—
172#97—62no—
173#97—61no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8—62no—
177C4×C8#6711yescharacteristic
178C4×C8#6711yescharacteristic
179#95—61no—
180#95—61no—
181C23×C3—41no—
182C23×C3#12111yescharacteristic
183C3×Q8—42no—
184C3×D4—42no—
185C3×D4—41no—
186C3×D4—41no—
187C3×D4—81no—
188C3×D4—82no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—82no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—42no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206Dic6—41no—
207Dic6—41no—
208Dic6—42no—
209C3×C8—41no—
210C3×C8—41no—
211C3×C8—81no—
212C3⋊C8—42no—
213C3⋊C8—41no—
214C3⋊C8—41no—
215#41—82no—
216C2×Q8—21no—
217C2×Q8—32no—
218C2×Q8—61no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—41no—
223C22×C4—41no—
224C22×C4—121no—
225C22×C4#23511yescharacteristic
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228Q16—242no—
229SD16—242no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—62no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—32no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—62no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—62no—
247C42—62no—
248C42—62no—
249C42—121no—
250C42C2×C4×D311yescharacteristic
251C42D3×D411yescharacteristic
252C42#23711yescharacteristic
253C42C2×#6711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3#38611yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—81no—
267C3×C4#44311yescharacteristic
268C3×C4#44411yescharacteristic
269Dic3—42no—
270Dic3—42no—
271Dic3—82no—
272C23—41no—
273C23#88411yescharacteristic
274Q8—42no—
275Q8—122no—
276Q8—121no—
277Q8—121no—
278D4—42no—
279D4—41no—
280D4—41no—
281D4—81no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—82no—
291C2×C4—122no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4D3×C22⋊C411yescharacteristic
295C2×C4#67⋊C411yescharacteristic
296C2×C4#88311yescharacteristic
297C2×C4C4×#6711yescharacteristic
298C2×C4#93711yescharacteristic
299C2×C4#93811yescharacteristic
300C8—41no—
301C8—41no—
302C8—81no—
303C8—121no—
304C8—121no—
305C8—122no—
306C2×C3—41no—
307C2×C3—161no—
308C2×C3#187611yescharacteristic
309C2×C3#190511yescharacteristic
310C2×C3#191011yescharacteristic
311C22—21no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#495611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—122no—
325C4—122no—
326C4—242no—
327C4#514411yescharacteristic
328C4#514511yescharacteristic
329C3#928211yescharacteristic, complemented, Sylow
330C2—41no—
331C2—161no—
332C2#7001111yescharacteristic
333C2#7033011yescharacteristic
334C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1496C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1496D3
hypocenterC3#9282
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #79266C2semidirect
5. #70836C2semidirect
6. #67756C2semidirect
7. #67737C2semidirect
14. #5092C4semidirect
15. #5079C4semidirect
16. #5076C4semidirect
17. #5070C4semidirect
329. C3#9282semidirect

Maximal subgroups

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

subgroupindex
#833182
#792662
C3×#14962
#718122
#708362
#677562
#677372
#92823