Finite Group Databasebeta

← #10696507

Subgroups — #10696507

Order 768.

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 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#83055C211yesmaximal, characteristic
2C2×#5064C211yesmaximal, characteristic
3C2×#4703C211yesmaximal, characteristic
4C2×C3×#330C211yesmaximal, characteristic
5#69828C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10084—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×#816—21no—
15C2×#816—21no—
16#5064—22no—
17#5064—22no—
18#4986—21no—
19#4985—21no—
20#4983—21no—
21#4982—22no—
22#4981—21no—
23C2×#808—21no—
24C2×#807—21no—
25#4936C2211yescharacteristic
26C2×#803C2211yescharacteristic
27#4703—22no—
28#4703—22no—
29C3×#330—22no—
30#4551—22no—
31#4465—22no—
32C2×#501—31no—
33C2×#491—31no—
34#2822—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#330D311yescharacteristic
38#1295—31no—
39C22×Dic6—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D412yes—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—22no—
59C22×C3⋊C8—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
68C3×#97—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#816—24no—
78#816—24no—
79#808—24no—
80#807—24no—
81#803—21no—
82#803D412yes—
83C2×C4⋊Q8—31no—
84C2×#125—31no—
85C2×#125C2×D311yescharacteristic
86#501—62no—
87#501—62no—
88#491—62no—
89#491—62no—
90C2×#107—61no—
91C2×#106—61no—
92C2×#103—61no—
93C2×#103—61no—
94C2×#97—31no—
95C2×#97C2×D311yescharacteristic
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—22no—
105#330—62no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—41no—
115C2×Dic6—22no—
116C2×Dic6—44no—
117C2×Dic6—41no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8—42no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—42no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—82no—
149C2×C3⋊C8—24no—
150C2×C3⋊C8—22no—
151C22×Q8—61no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—61no—
155C4⋊Q8—32no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
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—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4D1211yescharacteristic
178C2×C4⋊C4D1211yescharacteristic
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#106—64no—
182#103—64no—
183#103—64no—
184#97—21no—
185#97—21no—
186#97—21no—
187#97—32no—
188#97—61no—
189#97#6712yes—
190#95—62no—
191#95—62no—
192#95—62no—
193#95—61no—
194#95—61no—
195C23×C3#12011yescharacteristic
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12711yescharacteristic
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—82no—
221Dic6—44no—
222Dic6—82no—
223C3×C8—42no—
224C3×C8—42no—
225C3⋊C8—24no—
226C2×Q8—62no—
227C2×Q8—124no—
228C2×Q8—121no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—121no—
235C22×C4C2×D1211yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—64no—
244C2×C8—62no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—122no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—242no—
267C42#23511yescharacteristic
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3#39211yescharacteristic
272C22×C3#45412yes—
273C22×C3#45511yescharacteristic
274C22×C3#45612yes—
275C22×C3#45611yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—81no—
283C3×C4—81no—
284Dic3—82no—
285Dic3—82no—
286Dic3—81no—
287Dic3—81no—
288C23#88211yescharacteristic
289Q8—124no—
290Q8—242no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—242no—
310C2×C4#88211yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#88411yescharacteristic
313C2×C4#93011yescharacteristic
314C2×C4#93711yescharacteristic
315C2×C4#94011yescharacteristic
316C8—42no—
317C8—42no—
318C8—64no—
319C2×C3#166312yes—
320C2×C3#166412yes—
321C2×C3#200311yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#200711yescharacteristic
324C22#475212yes—
325C22#482511yescharacteristic
326C22#482611yescharacteristic
327C22#482712yes—
328C22#496011yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—81no—
336C4—81no—
337C4—242no—
338C4—241no—
339C4—242no—
340C4—241no—
341C3#1008411yescharacteristic, complemented, Sylow
342C2#6837612yes—
343C2#6837812yes—
344C2#7011911yescharacteristic
345C2#7038811yescharacteristic
346C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socle, prime radicalC23×C3#120
hypercenterC2×#330D3
hypocenterC3#10084
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

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
#830552
C2×#50642
C2×#47032
C2×C3×#3302
#698282
C2×#44652
#676592
#100843