Finite Group Databasebeta

← #10698270

Subgroups — #10698270

Order 768.

392 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 91 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#74219C211yesmaximal, characteristic
2C3×#1993C211yesmaximal, characteristic
3#71788C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic
5#67659C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic
8#10361—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12C2×#835—21no—
13C2×#835—21no—
14#4982—21no—
15#4982—21no—
16#4936C2211yescharacteristic
17#4936C2211yescharacteristic
18C2×#803C2211yescharacteristic
19C2×#803C2211yescharacteristic
20#4661—21no—
21#4661—21no—
22#4657—21no—
23#4657—21no—
24C3×#340—21no—
25C3×#340—21no—
26#4529—21no—
27#4529—21no—
28#2845—31no—
29#1993—31no—
30#1993D311yescharacteristic
31#1960—31no—
32#1295—31no—
33#1295—31no—
34#1295—31no—
35C22×C3×Q8—21no—
36C22×C3×C8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8Q812yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58C3×#97—21no—
59C3×#97—21no—
60C3×#95—21no—
61C3×#95—21no—
62#835—42no—
63#835—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70#803—21no—
71#803—21no—
72#803—21no—
73#803—21no—
74C2×C4⋊Q8C2×D311yescharacteristic
75C2×#103—61no—
76C2×#103—61no—
77C2×#97—31no—
78C2×#97—31no—
79C2×#97C2×D311yescharacteristic
80#398—61no—
81#398—61no—
82#398—61no—
83#398—61no—
84#398—61no—
85#398—61no—
86#382—31no—
87#382—31no—
88#382C2×D311yescharacteristic
89#340—21no—
90#340—21no—
91#340—61no—
92#340—61no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42C2×Q811yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C3×C42#4111yescharacteristic
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—81no—
127C4⋊Dic3—81no—
128Dic3⋊C4—81no—
129Dic3⋊C4—81no—
130Dic3⋊C4—81no—
131Dic3⋊C4—81no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C22×Q8—21no—
139C22×C8—21no—
140C22×C8—61no—
141C22×C8—61no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8Dic612yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4#6711yescharacteristic
159C2×C4⋊C4#6711yescharacteristic
160C2×C42C22×D311yescharacteristic
161#103—122no—
162#103—122no—
163#97—21no—
164#97—21no—
165#97—61no—
166#97—61no—
167#97—61no—
168#97—61no—
169#95—21no—
170#95—21no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177C23×C3#12011yescharacteristic
178C3×Q8—42no—
179C3×Q8—42no—
180C3×Q8—81no—
181C3×Q8—81no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4C22⋊Q811yescharacteristic
198C2×C3×C4C22⋊Q811yescharacteristic
199C2×C3×C4C22⋊Q811yescharacteristic
200C2×C3×C4#12411yescharacteristic
201C2×C3×C4#12411yescharacteristic
202C2×C3×C4#12411yescharacteristic
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C3×C8—42no—
216C3⋊C8—42no—
217C3⋊C8—42no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4C2×#6711yescharacteristic
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—121no—
240C2×C8—121no—
241C2×C8—121no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C4⋊C4—241no—
256C4⋊C4—241no—
257C4⋊C4—241no—
258C4⋊C4—241no—
259C42C2×Dic611yescharacteristic
260C42#23511yescharacteristic
261C42#23711yescharacteristic
262C42#23711yescharacteristic
263C22×C3#39311yescharacteristic
264C22×C3#45311yescharacteristic
265C22×C3#45411yescharacteristic
266C22×C3#45511yescharacteristic
267C22×C3#45511yescharacteristic
268C22×C3#45611yescharacteristic
269C22×C3#45611yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4—82no—
279Dic3—81no—
280Dic3—81no—
281Dic3—81no—
282Dic3—81no—
283C23#93711yescharacteristic
284Q8—42no—
285Q8—42no—
286Q8—81no—
287Q8—81no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4#87811yescharacteristic
316C2×C4#87811yescharacteristic
317C2×C4#88511yescharacteristic
318C2×C4#88511yescharacteristic
319C2×C4#92411yescharacteristic
320C2×C4#93611yescharacteristic
321C8—42no—
322C8—122no—
323C8—122no—
324C2×C3#166111yescharacteristic
325C2×C3#166111yescharacteristic
326C2×C3#166211yescharacteristic
327C2×C3#166311yescharacteristic
328C2×C3#202711yescharacteristic
329C2×C3#202811yescharacteristic
330C2×C3#202811yescharacteristic
331C22#494711yescharacteristic
332C22#506111yescharacteristic
333C22#509511yescharacteristic
334C22#519811yescharacteristic
335C22#520111yescharacteristic
336C22#520211yescharacteristic
337C22#520511yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—81no—
345C4—81no—
346C4—82no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4—241no—
351C3#1036111yescharacteristic, complemented, Sylow
352C2#6967311yescharacteristic
353C2#7031311yescharacteristic
354C2#7033311yescharacteristic
355C2#7124511yescharacteristic
356C2#7124611yescharacteristic
357C2#7139211yescharacteristic
358C2#7139311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socle, prime radicalC23×C3#120
hypercenter#1993D3
hypocenterC3#10361
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
351. C3#10361semidirect

Maximal subgroups

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

subgroupindex
#742192
C3×#19932
#717882
#717862
#676592
#676302
#103613