Finite Group Databasebeta

← #10696502

Subgroups — #10696502

Order 768.

484 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C3×#491C211yesmaximal, characteristic
2C2×#5081C211yesmaximal, characteristic
3#83055C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic
5C2×#4543C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10084—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#491—22no—
13C3×#491—22no—
14C2×C3×#103—21no—
15C2×C3×#103—21no—
16C2×C3×#97C2211yescharacteristic
17#5081—22no—
18#5081—22no—
19#4986—21no—
20#4985—21no—
21#4983—21no—
22#4982—22no—
23#4981—21no—
24C2×#808—21no—
25C2×#807—21no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4529—22no—
31#4465—22no—
32C2×#501—31no—
33C2×#491D311yescharacteristic
34#2822—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#330—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D412yes—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#103—24no—
69C3×#103—24no—
70C3×#97—21no—
71C3×#97D412yes—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#808—24no—
78#807—24no—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#125—31no—
85C2×#125C2×D311yescharacteristic
86#501—62no—
87#501—62no—
88#491—22no—
89#491—22no—
90C2×#107—61no—
91C2×#106—61no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#97—31no—
95C2×#97—31no—
96C2×#97C2×D311yescharacteristic
97#400—61no—
98#400—61no—
99#398—61no—
100#398—62no—
101#398—61no—
102#382—31no—
103#340—62no—
104#330—62no—
105#330—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—44no—
108C2×C3×Q8—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—42no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—82no—
146C2×C3⋊C8—24no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C22×Q8—21no—
152C22×C8—21no—
153C22×C8—61no—
154C22×C8—61no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8#6712yes—
159#125—21no—
160#125—21no—
161#125—21no—
162#125—21no—
163#125—61no—
164#125—61no—
165#125—61no—
166#125—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4#6711yescharacteristic
178C2×C4⋊C4#6711yescharacteristic
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#106—64no—
182#103—24no—
183#103—24no—
184#97—21no—
185#97—32no—
186#97—61no—
187#97—61no—
188#97—61no—
189#97#6712yes—
190#95—62no—
191#95—62no—
192#95—62no—
193#95—61no—
194#95—61no—
195C23×C3#12011yescharacteristic
196C3×Q8—44no—
197C3×Q8—82no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C3×C8—42no—
224C3⋊C8—24no—
225C3⋊C8—42no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—62no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—64no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—121no—
244C2×C8—121no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—122no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—242no—
267C42D3×D411yescharacteristic
268C42#23711yescharacteristic
269C42C2×#6711yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3#39211yescharacteristic
272C22×C3#45412yes—
273C22×C3#45511yescharacteristic
274C22×C3#45611yescharacteristic
275C22×C3#45612yes—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—82no—
283C3×C4—81no—
284C3×C4—81no—
285Dic3—82no—
286Dic3—81no—
287Dic3—81no—
288C23#93711yescharacteristic
289Q8—44no—
290Q8—82no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—122no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4#93711yescharacteristic
311C2×C4#93811yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#94011yescharacteristic
315C2×C4#95311yescharacteristic
316C8—42no—
317C8—64no—
318C8—122no—
319C2×C3#166312yes—
320C2×C3#166412yes—
321C2×C3#200311yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#200711yescharacteristic
324C22#506212yes—
325C22#520211yescharacteristic
326C22#520511yescharacteristic
327C22#520512yes—
328C22#522311yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—241no—
339C4—242no—
340C4—241no—
341C3#1008411yescharacteristic, complemented, Sylow
342C2#7125212yes—
343C2#7125312yes—
344C2#7173011yescharacteristic
345C2#7210811yescharacteristic
346C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#491C2
socle, prime radicalC23×C3#120
hypercenterC2×#491D3
hypocenterC3#10084
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
341. C3#10084semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4912
C2×#50812
#830552
#717882
C2×#45432
C2×#44652
#676592
#100843