Finite Group Databasebeta

← #10698367

Subgroups — #10698367

Order 768.

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

normal subgroups only

This group has 360 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×#2605C211yesmaximal, characteristic
2#69566C211yesmaximal, characteristic
3#69493C211yesmaximal, characteristic
4#69433C211yesmaximal, characteristic
5#67633C211yesmaximal, characteristic
6#67621C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic
8#10379—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C3×#400—21no—
11C3×#395—21no—
12C3×#391—21no—
13C3×#391—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#383C2211yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×#376C2211yescharacteristic
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4523—21no—
23#4523—21no—
24#4452—21no—
25#4452—21no—
26#4452—21no—
27#4452—21no—
28#2605D311yescharacteristic
29#1986—31no—
30#1515—31no—
31#1506—31no—
32#1300—31no—
33#1288—31no—
34#1285—31no—
35C22×C3×Q8—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42—21no—
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45#924—21no—
46#924—21no—
47C2×#208—21no—
48#870—21no—
49#869Q811yescharacteristic, complemented
50#869Q811yescharacteristic, complemented
51C4×Dic6—21no—
52C4×Dic6—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62C3×#95D411yescharacteristic
63C3×#95D411yescharacteristic
64C3×#95D411yescharacteristic
65#830—21no—
66#830—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69#803—21no—
70#803—21no—
71#803—21no—
72C4×C3⋊C8—21no—
73#533—31no—
74#432—31no—
75#432—31no—
76C4×M16—31no—
77#400—21no—
78#395—21no—
79#391—21no—
80#391—21no—
81#387—21no—
82#387—21no—
83#383C2×D311yescharacteristic
84C4×C4⋊C4C2×D311yescharacteristic
85#376C2×D311yescharacteristic
86#339—61no—
87#339—61no—
88#328—61no—
89#328—61no—
90#328—61no—
91#328—61no—
92C2×C3×Q8—42no—
93C2×C3×Q8—42no—
94C2×C3×Q8—81no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C2×Dic6—21no—
105C2×Dic6—21no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—81no—
110C3×C42—41no—
111C3×C42—41no—
112C3×C42—42no—
113C3×C42C2×Q811yescharacteristic
114C3×C42#4111yescharacteristic
115C3×C42#4111yescharacteristic
116C3×C42#4111yescharacteristic
117#217—41no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124C4×Dic3—41no—
125#208—22no—
126#208—41no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—21no—
131C2×M16—61no—
132C4⋊Q8—31no—
133C4⋊Q8—31no—
134#125—61no—
135C22⋊Q8—61no—
136C22⋊Q8—61no—
137C4×Q8—61no—
138C4×Q8—61no—
139#117—61no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C42—21no—
145C2×C42#6711yescharacteristic
146C2×C42#6711yescharacteristic
147C2×C42C22×D311yescharacteristic
148C4⋊C8—61no—
149C4⋊C8—61no—
150C22⋊C8—61no—
151C22⋊C8—61no—
152#97—61no—
153#97—61no—
154#97—61no—
155C4×C8—61no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95#6711yescharacteristic
165#95#6711yescharacteristic
166#95#6711yescharacteristic
167#95#6711yescharacteristic
168C23×C3#12011yescharacteristic
169C3×Q8—84no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
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—41no—
192C2×C3×C4C22⋊Q811yescharacteristic
193C2×C3×C4C22⋊Q811yescharacteristic
194C2×C3×C4C22⋊Q811yescharacteristic
195C2×C3×C4#12411yescharacteristic
196C2×C3×C4#12411yescharacteristic
197C2×C3×C4#12411yescharacteristic
198C2×Dic3—41no—
199C2×Dic3—41no—
200Dic6—81no—
201Dic6—81no—
202C3⋊C8—42no—
203C3⋊C8—81no—
204C3⋊C8—81no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—81no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4C2×#6711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—62no—
220M16—121no—
221C2×C8—121no—
222C2×C8—121no—
223C2×C8—121no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—81no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C22⋊C4—121no—
235C42—41no—
236C42—41no—
237C42—42no—
238C42—121no—
239C42D3×Q811yescharacteristic
240C42#23911yescharacteristic
241C42#23911yescharacteristic
242C42#23911yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3#39311yescharacteristic
246C22×C3#45511yescharacteristic
247C22×C3#45611yescharacteristic
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—42no—
255C3×C4—42no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—81no—
259C3×C4—81no—
260Dic3—81no—
261Dic3—81no—
262C23#95311yescharacteristic
263Q8—84no—
264Q8—241no—
265Q8—241no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4#94611yescharacteristic
291C2×C4#94611yescharacteristic
292C2×C4#94611yescharacteristic
293C2×C4#94711yescharacteristic
294C2×C4#94711yescharacteristic
295C2×C4#94711yescharacteristic
296C8—122no—
297C8—241no—
298C8—241no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3#202811yescharacteristic
302C2×C3#203111yescharacteristic
303C2×C3#203111yescharacteristic
304C22—21no—
305C22—21no—
306C22#523111yescharacteristic
307C22#524211yescharacteristic
308C22#524611yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—241no—
322C4—241no—
323C3#1037911yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#7184211yescharacteristic
327C2#7184211yescharacteristic
328C2#7190511yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2605C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#2605D3
hypocenterC3#10379
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2605

Complements

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

NH ≅ G/Ntype
49. #869Q8semidirect
50. #869Q8semidirect
323. C3#10379semidirect

Maximal subgroups

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

subgroupindex
C3×#26052
#695662
#694932
#694332
#676332
#676212
#676152
#103793