Finite Group Databasebeta

← #10708337

Subgroups — #10708337

Order 768.

396 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 55 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#76956C211yesmaximal, characteristic
2C3×#1599C211yesmaximal, characteristic
3#71905C211yesmaximal, characteristic, complemented
4#71899C211yesmaximal, characteristic, complemented
5#67647C211yesmaximal, characteristic
6#67459C211yesmaximal, characteristic, complemented
7#67455C211yesmaximal, characteristic, complemented
8#12271—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#450—21no—
11C3×#394C2211yescharacteristic
12#5220—21no—
13#5044—21no—
14#5004—21no—
15#4986C2211yescharacteristic
16#4973C2211yescharacteristic
17#4951—21no—
18#4947—21no—
19#4943—21no—
20Dic3×C22⋊C4—21no—
21#4651—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2572—31no—
29#2028—31no—
30#1617—31no—
31#1599D311yescharacteristic
32#1353—31no—
33#1345—31no—
34#1319—31no—
35C22×C3×D4D411yescharacteristic
36C23×Dic3—21no—
37C22×Dic6D411yescharacteristic
38C2×C3×SD16—41no—
39C3×C4⋊Q8—41no—
40C3×#125—41no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4C2311yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54C3×#102—41no—
55C3×C22⋊C8—21no—
56C3×#95D411yescharacteristic
57#835—41no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#810—41no—
68#810—41no—
69#807—41no—
70#544C2×D311yescharacteristic
71#484—61no—
72#450—21no—
73#419—61no—
74#400—31no—
75#394C2×D311yescharacteristic
76#393—61no—
77#391—31no—
78#388—61no—
79#386—61no—
80#383—61no—
81#381—61no—
82C4×C22⋊C4—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C24×C3—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×D4—41no—
93C2×C3×D4—42no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—42no—
103C22×Dic3—42no—
104C22×Dic3—81no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C3×SD16—82no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123#217—42no—
124#217—81no—
125#217—81no—
126C4⋊Dic3—41no—
127C4×Dic3—42no—
128C4×Dic3—41no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8—31no—
134C22×D4#6711yescharacteristic
135C23×C4—61no—
136C2×SD16—41no—
137C2×M16—61no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142#117—21no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4C22×D311yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C42—31no—
151C2×C42—61no—
152#107—121no—
153#103—121no—
154#103—121no—
155#103—121no—
156#102—41no—
157C22⋊C8—21no—
158C22⋊C8—61no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95#6711yescharacteristic
168C23×C3—22no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3#12011yescharacteristic
176C3×Q8—81no—
177C3×Q8—81no—
178C3×D4—42no—
179C3×D4—81no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—84no—
206Dic6—42no—
207Dic6—81no—
208C3×C8—81no—
209C3⋊C8—81no—
210C3⋊C8—81no—
211C24—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×D4—41no—
219C2×D4—42no—
220C22×C4—21no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—122no—
226C22×C4—122no—
227C22×C4—241no—
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4C2×#6711yescharacteristic
231SD16—82no—
232M16—121no—
233M16—121no—
234C2×C8—41no—
235C2×C8—121no—
236C2×C8—121no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—121no—
245C22⋊C4—42no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—122no—
250C22⋊C4—241no—
251C22⋊C4—241no—
252C42—41no—
253C42—121no—
254C42—122no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#45211yescharacteristic
265C22×C3#45511yescharacteristic
266C22×C3B(4,𝔽2)11yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—81no—
274Dic3—42no—
275Dic3—81no—
276Dic3—81no—
277Dic3—81no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#93711yescharacteristic
286Q8—81no—
287Q8—81no—
288Q8—122no—
289Q8—241no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—62no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—244no—
316C2×C4#91511yescharacteristic
317C2×C4#91611yescharacteristic
318C8—81no—
319C8—241no—
320C8—241no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—82no—
324C2×C3#160611yescharacteristic
325C2×C3#219711yescharacteristic
326C2×C3#220411yescharacteristic
327C22—21no—
328C22—21no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#503211yescharacteristic
337C22#517311yescharacteristic
338C22#520211yescharacteristic
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—122no—
347C4—241no—
348C4—241no—
349C4—241no—
350C3#1227111yescharacteristic, complemented, Sylow
351C2—21no—
352C2—21no—
353C2—82no—
354C2#7097311yescharacteristic
355C2#7098811yescharacteristic
356C2#7099811yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1599C2
socleC22×C3B(4,𝔽2)
prime radicalC22×C3×D4D4
hypercenter#1599D3
hypocenterC3#12271
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1599

Complements

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

NH ≅ G/Ntype
3. #71905C2semidirect
4. #71899C2semidirect
6. #67459C2semidirect
7. #67455C2semidirect
350. C3#12271semidirect

Maximal subgroups

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

subgroupindex
#769562
C3×#15992
#719052
#718992
#676472
#674592
#674552
#122713