Finite Group Databasebeta

← #10696869

Subgroups — #10696869

Order 768.

484 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 85 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×#489C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic, complemented
3#82949C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic
6#69801C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#10137—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#489—22no—
13C3×#489—22no—
14C2×C3×#103—21no—
15C2×C3×#102—21no—
16C2×C3×#97C2211yescharacteristic
17#5228—21no—
18#5228—22no—
19#5218—21no—
20#5092—21no—
21#5090—22no—
22#5079—21no—
23#5060—21no—
24#5055—21no—
25#5054—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4547—22no—
30#4525—22no—
31#4465—22no—
32#3215—31no—
33#3209—31no—
34C2×#489D311yescharacteristic
35#1556—31no—
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#216—22no—
57C2×#216—24no—
58C2×#214—22no—
59#926—21no—
60#924—21no—
61#924—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#103—24no—
68C3×#102—24no—
69C3×#97—21no—
70C3×#97D412yes—
71#835—22no—
72#835—24no—
73#832—22no—
74#830—21no—
75#830—21no—
76#830D412yes—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124C2×D311yescharacteristic
86#497—61no—
87#494—61no—
88#489—22no—
89#489—22no—
90#489—61no—
91#487—61no—
92#464—62no—
93#462—62no—
94#451—62no—
95#451—61no—
96#450—61no—
97#432—31no—
98#432—31no—
99#431—31no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#97C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#323—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×Q8—82no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—42no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×C3×C8—22no—
124C2×C3×C8—42no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138#216—84no—
139#216—82no—
140#214—82no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×Q8—21no—
152C22×Q8#6711yescharacteristic
153C22×D4#6711yescharacteristic
154C2×Q16—64no—
155C2×Q16—62no—
156C2×SD16—62no—
157C22×C8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8#6712yes—
164#125—61no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C22×D311yescharacteristic
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#103—24no—
183#103—64no—
184#103—62no—
185#102—24no—
186#102—62no—
187C22⋊C8—32no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—21no—
191#97—61no—
192#97—61no—
193#97#6712yes—
194C4×C8—61no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3#12111yescharacteristic
200C3×Q8—44no—
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—82no—
204C3×D4—42no—
205C3×D4—42no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—82no—
212C2×C3×C4—81no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—41no—
220C2×Dic3—41no—
221Dic6—42no—
222Dic6—42no—
223C3×C8—42no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—21no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—82no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—42no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244Q16—244no—
245Q16—242no—
246SD16—242no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—121no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—121no—
267C42—121no—
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42C2×#6711yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—162no—
278C22×C3#39211yescharacteristic
279C22×C3#47312yes—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—81no—
286C3×C4—82no—
287C3×C4#45512yes—
288Dic3—81no—
289Dic3—81no—
290C23—41no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23#93911yescharacteristic
295Q8—44no—
296Q8—42no—
297Q8—42no—
298Q8—82no—
299Q8—122no—
300Q8—122no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4#93711yescharacteristic
313C2×C4#93711yescharacteristic
314C2×C4#93811yescharacteristic
315C2×C4#93911yescharacteristic
316C2×C4#94011yescharacteristic
317C2×C4#95311yescharacteristic
318C8—42no—
319C8—122no—
320C8—122no—
321C8—241no—
322C2×C3—22no—
323C2×C3—161no—
324C2×C3#200411yescharacteristic
325C2×C3#200811yescharacteristic
326C2×C3#202011yescharacteristic
327C22—21no—
328C22—21no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—162no—
333C22#520412yes—
334C22#522311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—82no—
342C4—241no—
343C4—241no—
344C4#506112yes—
345C3#1013711yescharacteristic, complemented, Sylow
346C2—22no—
347C2—161no—
348C2#7173111yescharacteristic
349C2#7209911yescharacteristic
350C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#489C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenterC2×#489D3
hypocenterC3#10137
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. #83222C2semidirect
4. #71776C2semidirect
6. #69801C2semidirect
7. #67662C2semidirect
345. C3#10137semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832222
#829492
#717762
#717202
#698012
#676622
#101373