Finite Group Databasebeta

← #10696849

Subgroups — #10696849

Order 768.

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

normal subgroups only

This group has 456 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
1C3×#3227C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3C2×#5082C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic, complemented
6#70787C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#10134—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#498—21no—
13C3×#486—21no—
14C3×#481—22no—
15C3×#476—22no—
16C3×#431C2211yescharacteristic
17C2×#835—21no—
18C2×#835—21no—
19#5082—22no—
20#5082—22no—
21#5081—21no—
22#5071—21no—
23#5034—21no—
24#5033—21no—
25#5009—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4469—22no—
32#3234—31no—
33#3227D311yescharacteristic
34C2×#491—31no—
35#1556—31no—
36#1530—31no—
37#1470—31no—
38#1302—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D412yes—
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C22×C3⋊C8—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64C3×#106—24no—
65C3×C4⋊C8—21no—
66C3×#102—24no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69#835—24no—
70#835—24no—
71#832—22no—
72#832—22no—
73#830—21no—
74#830—21no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—61no—
87#498—21no—
88#491—62no—
89#491—62no—
90#490—61no—
91#486—21no—
92#481—22no—
93#481—61no—
94#480—61no—
95#478—62no—
96#476—22no—
97#432—31no—
98#432—31no—
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#103—61no—
102C2×#97—31no—
103#332—62no—
104#329—62no—
105#324—62no—
106C3×#41—82no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—44no—
110C2×C3×Q8—41no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C2×C3×C8—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147C2×#41—21no—
148C22×Q8—21no—
149C22×C8—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8#6712yes—
154C4⋊Q8#6711yescharacteristic
155#127#6711yescharacteristic
156#125—61no—
157#125—61no—
158#125—61no—
159#125—61no—
160#124—21no—
161#121—21no—
162#121—21no—
163C4×D4—21no—
164C4×D4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173#106—24no—
174#106—62no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8—61no—
178#103—64no—
179#103—64no—
180#102—24no—
181#102—62no—
182#102—62no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185C22⋊C8#6712yes—
186#97—32no—
187#97—61no—
188#97—61no—
189#97—61no—
190C4×C8—21no—
191C23×C3—41no—
192C23×C3#12111yescharacteristic
193C3×Q8—44no—
194C3×Q8—81no—
195C3×Q8—82no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—81no—
199C3×D4—82no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—82no—
206C2×C3×C4—82no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—81no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221#41—82no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—44no—
225C2×Q8—41no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4D3×D411yescharacteristic
235C22×C4#23711yescharacteristic
236C22×C4C2×#6711yescharacteristic
237C2×C8—22no—
238C2×C8—62no—
239C2×C8—62no—
240C2×C8—122no—
241C2×C8—121no—
242C4⋊C4—21no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42D3×D411yescharacteristic
259C42D3×D411yescharacteristic
260C42C2×#6711yescharacteristic
261C42C2×#6711yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3#39211yescharacteristic
268C22×C3#47112yes—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—82no—
275C3×C4—81no—
276C3×C4#45412yes—
277Dic3—81no—
278Dic3—81no—
279C23—41no—
280C23#93811yescharacteristic
281Q8—44no—
282Q8—81no—
283Q8—82no—
284D4—42no—
285D4—42no—
286D4—81no—
287D4—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—122no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—241no—
299C2×C4#93711yescharacteristic
300C2×C4#93711yescharacteristic
301C2×C4#93911yescharacteristic
302C2×C4#93911yescharacteristic
303C2×C4#94011yescharacteristic
304C2×C4#95311yescharacteristic
305C8—42no—
306C8—122no—
307C8—122no—
308C8—241no—
309C2×C3—22no—
310C2×C3—161no—
311C2×C3#200311yescharacteristic
312C2×C3#200911yescharacteristic
313C2×C3#201911yescharacteristic
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22#517712yes—
320C22#522311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—82no—
328C4—241no—
329C4—241no—
330C4#520112yes—
331C3#1013411yescharacteristic, complemented, Sylow
332C2—22no—
333C2—161no—
334C2#7186911yescharacteristic
335C2#7205911yescharacteristic
336C2#7209611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3227C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3227D3
hypocenterC3#10134
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
3. C2×#5082C2semidirect
5. #71776C2semidirect
6. #70787C2semidirect
7. #67742C2semidirect
331. C3#10134semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#832732
C2×#50822
#717982
#717762
#707872
#677422
#101343