Finite Group Databasebeta

← #10697383

Subgroups — #10697383

Order 768.

406 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#3237C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic
3#83327C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#71837C211yesmaximal, characteristic
6#70837C211yesmaximal, characteristic
7C3⋊#1308C211yesmaximal, characteristic
8#10211—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—21no—
13C3×#491—21no—
14C3×#485—22no—
15C3×#477—21no—
16C3×#475—21no—
17C3×#432C2211yescharacteristic
18#5094—21no—
19#5091—21no—
20#5082—21no—
21#5081—21no—
22#5069—21no—
23#5064—21no—
24#5045—22no—
25#5021—21no—
26#5020—21no—
27#4996—22no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4538—22no—
32#4538—22no—
33#4467—22no—
34#3237—31no—
35#3237—31no—
36#3237D311yescharacteristic
37#1515—31no—
38#1515—31no—
39#1515—31no—
40#1308—31no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58#926—21no—
59#924—21no—
60#924—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×#107—22no—
67C3×#106—22no—
68C3×C4⋊C8—21no—
69C3×#103—22no—
70C3×#103—22no—
71C3×C22⋊C8D412yes—
72C3×#97—21no—
73#835—22no—
74#835—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830D412yes—
79#830D412yes—
80#810—22no—
81#810—22no—
82#808—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#533—31no—
91#533C2×D311yescharacteristic
92#533C2×D311yescharacteristic
93#501—21no—
94#501—61no—
95#501—61no—
96#491—21no—
97#491—61no—
98#491—61no—
99#485—22no—
100#485—62no—
101#485—62no—
102#477—21no—
103#477—61no—
104#477—61no—
105#475—21no—
106#475—61no—
107#475—61no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#383—31no—
112#328—62no—
113#328—62no—
114#328—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—41no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#217—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162#125—21no—
163#125—21no—
164#125—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—61no—
176C4×Q8—61no—
177#117—21no—
178#117—21no—
179#117—61no—
180C2×C42C22×D311yescharacteristic
181#107—22no—
182#107—62no—
183#107—62no—
184#106—22no—
185#106—62no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—22no—
191#103—22no—
192#103—62no—
193#103—62no—
194#103—62no—
195#103—62no—
196C22⋊C8—32no—
197C22⋊C8—32no—
198C22⋊C8#6712yes—
199#97—21no—
200#97—61no—
201#97—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3#12711yescharacteristic
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—81no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228Dic6—42no—
229Dic6—81no—
230C3×C8—42no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—61no—
238C2×Q8—61no—
239C22×C4—121no—
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4C2×#6711yescharacteristic
243C2×C8—22no—
244C2×C8—62no—
245C2×C8—62no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—121no—
267C42—41no—
268C42—41no—
269C42—121no—
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39211yescharacteristic
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—81no—
288Dic3—81no—
289Dic3—81no—
290Dic3—161no—
291C23#94011yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—81no—
295Q8—81no—
296Q8—122no—
297Q8—241no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—241no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4#93711yescharacteristic
311C2×C4#93711yescharacteristic
312C2×C4#93811yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#93911yescharacteristic
315C2×C4#95311yescharacteristic
316C8—42no—
317C8—122no—
318C8—122no—
319C2×C3—41no—
320C2×C3#200911yescharacteristic
321C2×C3#200911yescharacteristic
322C2×C3#200911yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22#522311yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—241no—
338C4—241no—
339C4—481no—
340C3#1021111yescharacteristic, complemented, Sylow
341C2—41no—
342C2#7186911yescharacteristic
343C2#7211111yescharacteristic
344C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3237C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3237D3
hypocenterC3#10211
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3237

Complements

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

NH ≅ G/Ntype
340. C3#10211semidirect

Maximal subgroups

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

subgroupindex
C3×#32372
#833722
#833272
#718372
#708372
C3⋊#13082
#102113