Finite Group Databasebeta

← #10698368

Subgroups — #10698368

Order 768.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#74275C211yesmaximal, characteristic
2C3×#1515C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#67749C211yesmaximal, characteristic
6#67748C211yesmaximal, characteristic
7#67633C211yesmaximal, characteristic
8#10379—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#432C2211yescharacteristic
11C3×#383C2211yescharacteristic
12C4×C4⋊Dic3C2211yescharacteristic
13#4932C2211yescharacteristic
14#4926C2211yescharacteristic
15C4×#208C2211yescharacteristic
16#4660—21no—
17#4656—21no—
18#4655—21no—
19#4654—21no—
20#4646—21no—
21#4646—21no—
22C3×#328—21no—
23C3×#328—21no—
24#4538—21no—
25#4538—21no—
26#4532—21no—
27#4532—21no—
28#2605—31no—
29#1986—31no—
30#1515D311yescharacteristic
31#1506—31no—
32#1300—31no—
33#1288—31no—
34#1285—31no—
35C22×Dic6—21no—
36C3×C4⋊Q8Q811yescharacteristic, complemented
37C3×C4⋊Q8Q811yescharacteristic, complemented
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C42C2311yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3D411yescharacteristic
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53C3×C4⋊C8—21no—
54C3×C22⋊C8—21no—
55C3×#97—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95D411yescharacteristic
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95D411yescharacteristic
67C3⋊#95D411yescharacteristic
68#830—21no—
69C4⋊(C3⋊C8)—21no—
70#803—21no—
71#803—21no—
72C4×C3⋊C8—21no—
73#533C2×D311yescharacteristic
74#432—31no—
75#432C2×D311yescharacteristic
76C4×M16—31no—
77#400—61no—
78#395—61no—
79#391—61no—
80#391—61no—
81#387—61no—
82#387—61no—
83#383C2×D311yescharacteristic
84C4×C4⋊C4—31no—
85#376—31no—
86#339—61no—
87#339—61no—
88#328—21no—
89#328—21no—
90#328—61no—
91#328—61no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×Dic6—42no—
104C2×Dic6—42no—
105C2×Dic6—81no—
106C2×C3×C8—41no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—41no—
115C3×C42C2×Q811yescharacteristic
116C3×C42#4111yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119C4⋊Dic3—42no—
120C4⋊Dic3—42no—
121Dic3⋊C4—42no—
122Dic3⋊C4—81no—
123C4×Dic3—42no—
124C4×Dic3—41no—
125C4×Dic3—41no—
126#208—22no—
127#208—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—61no—
131C2×M16—61no—
132C4⋊Q8Dic611yescharacteristic, complemented
133C4⋊Q8Dic611yescharacteristic, complemented
134#125—21no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139#117—21no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—61no—
147C2×C42C22×D311yescharacteristic
148C4⋊C8—21no—
149C4⋊C8—61no—
150C22⋊C8—21no—
151C22⋊C8—61no—
152#97—21no—
153#97—61no—
154#97—61no—
155C4×C8—61no—
156#95—21no—
157#95—21no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95#6711yescharacteristic
167#95#6711yescharacteristic
168C23×C3#12011yescharacteristic
169C3×Q8—81no—
170C3×Q8—81no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
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×C4C22⋊Q811yescharacteristic
184C2×C3×C4C22⋊Q811yescharacteristic
185C2×C3×C4C22⋊Q811yescharacteristic
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#12411yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201Dic6—84no—
202C3×C8—81no—
203C3⋊C8—42no—
204C3⋊C8—81no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—122no—
208C2×Q8—122no—
209C2×Q8—241no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4D3×D411yescharacteristic
217C22×C4D3×D411yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—62no—
220M16—121no—
221C2×C8—41no—
222C2×C8—121no—
223C2×C8—121no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C4⋊C4—241no—
234C22⋊C4—41no—
235C42—41no—
236C42—121no—
237C42—121no—
238C42—122no—
239C42C2×Dic611yescharacteristic
240C42#23511yescharacteristic
241C42#23711yescharacteristic
242C42#23711yescharacteristic
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—81no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4—81no—
258Dic3—42no—
259Dic3—42no—
260Dic3—81no—
261Dic3—81no—
262C23#93711yescharacteristic
263Q8—81no—
264Q8—81no—
265Q8—244no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
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—62no—
279C2×C4—62no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4#87811yescharacteristic
291C2×C4#87811yescharacteristic
292C2×C4#88511yescharacteristic
293C2×C4#88511yescharacteristic
294C2×C4#92411yescharacteristic
295C2×C4#93611yescharacteristic
296C8—81no—
297C8—122no—
298C8—241no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3#202811yescharacteristic
302C2×C3#203111yescharacteristic
303C2×C3#203111yescharacteristic
304C22—21no—
305C22—21no—
306C22#494711yescharacteristic
307C22#520211yescharacteristic
308C22#520511yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—122no—
320C4—122no—
321C4—241no—
322C4—241no—
323C3#1037911yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#7003811yescharacteristic
327C2#7033311yescharacteristic
328C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1515D3
hypocenterC3#10379
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
36. C3×C4⋊Q8Q8semidirect
37. C3×C4⋊Q8Q8semidirect
132. C4⋊Q8Dic6semidirect
133. C4⋊Q8Dic6semidirect
323. C3#10379semidirect

Maximal subgroups

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

subgroupindex
#742752
C3×#15152
#718412
#718362
#677492
#677482
#676332
#103793