Finite Group Databasebeta

← #10696876

Subgroups — #10696876

Order 768.

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

normal subgroups only

This group has 468 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×#501C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#83019C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71737C211yesmaximal, characteristic, complemented
6#69798C211yesmaximal, characteristic
7#67662C211yesmaximal, characteristic, complemented
8#10138—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#501—22no—
13C3×#501—22no—
14C2×C3×#107—21no—
15C2×C3×#106—21no—
16C2×C3×#97C2211yescharacteristic
17#5227—22no—
18#5085—21no—
19#5083—22no—
20#5079—21no—
21#5067—21no—
22#5066—22no—
23#5064—21no—
24#4968—21no—
25#4967—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4547—22no—
30#4528—22no—
31#4464—22no—
32#3234—31no—
33#3206—31no—
34C2×#501D311yescharacteristic, complemented
35#1556—31no—
36#1530—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#215—24no—
58#930—21no—
59#930—21no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C3×#107—24no—
66C3×#106—24no—
67C3×#97—21no—
68C3×#97D412yes—
69#835—24no—
70#830—21no—
71#830—21no—
72#830D412yes—
73#809—22no—
74#809—22no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—22no—
87#501—22no—
88#501—61no—
89#492—61no—
90#490—61no—
91#487—61no—
92#481—61no—
93#480—61no—
94#478—62no—
95#461—62no—
96#448—62no—
97#432—31no—
98#432—31no—
99#431—31no—
100C2×#107—21no—
101C2×#106—21no—
102C2×#97C2×D311yescharacteristic
103#332—62no—
104#329—62no—
105#324—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—41no—
116#235—82no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×C3×C8—24no—
123C2×C3×C8—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#215—84no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×#41—61no—
149C22×Q8—21no—
150C2×SD16—64no—
151C22×C8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8#6712yes—
157#127—31no—
158#125—21no—
159#125—21no—
160#125—21no—
161#125—21no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4#6711yescharacteristic
172C2×C4⋊C4#6711yescharacteristic
173C2×C42C22×D311yescharacteristic
174#107—24no—
175#107—62no—
176#106—24no—
177#106—62no—
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#103—64no—
182#102—62no—
183#102—62no—
184C22⋊C8—32no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187#97—21no—
188#97—61no—
189#97—61no—
190#97#6712yes—
191C4×C8—61no—
192C23×C3#12111yescharacteristic
193C22×D3—41no—
194C3×Q8—44no—
195C3×Q8—82no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4—81no—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12711yescharacteristic
211#67—82no—
212C2×Dic3—41no—
213D12—42no—
214D12—42no—
215D12—81no—
216C4×D3—82no—
217Dic6—81no—
218C3×C8—24no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—242no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—61no—
226C2×Q8—82no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—121no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—121no—
235C22×C4D3×D411yescharacteristic
236C22×C4#23711yescharacteristic
237C22×C4C2×#6711yescharacteristic
238SD16—244no—
239C2×C8—24no—
240C2×C8—22no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—121no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42D3×D411yescharacteristic
261C42D3×D411yescharacteristic
262C42C2×#6711yescharacteristic
263C42C2×#6711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3#39211yescharacteristic
267C22×C3#47112yes—
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4—82no—
278C3×C4—81no—
279C3×C4#45612yes—
280Dic3—81no—
281C23—121no—
282C23#93911yescharacteristic
283Q8—44no—
284Q8—82no—
285Q8—241no—
286D4—122no—
287D4—122no—
288D4—241no—
289D4—242no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—121no—
300C2×C4—242no—
301C2×C4#93711yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#93811yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#94011yescharacteristic
306C2×C4#95311yescharacteristic
307C8—24no—
308C8—122no—
309C8—122no—
310C8—241no—
311C2×C3—22no—
312C2×C3#200511yescharacteristic
313C2×C3#200811yescharacteristic
314C2×C3#201911yescharacteristic
315D3—161no—
316C22—21no—
317C22—21no—
318C22—241no—
319C22—241no—
320C22—241no—
321C22#520312yes—
322C22#522311yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—81no—
330C4—81no—
331C4—241no—
332C4#509512yes—
333C3#1013811yescharacteristic, complemented, Sylow
334C2—22no—
335C2—481no—
336C2#7179511yescharacteristic
337C2#7209911yescharacteristic
338C2#7210711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#501C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#501D3
hypocenterC3#10138
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C3×#501C2semidirect
4. #71776C2semidirect
5. #71737C2semidirect
7. #67662C2semidirect
34. C2×#501D3semidirect
333. C3#10138semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5012
#832132
#830192
#717762
#717372
#697982
#676622
#101383