Finite Group Databasebeta

← #10691834

Subgroups — #10691834

Order 768.

410 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 107 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#83224C211yesmaximal, characteristic
2#79045C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic
6#67740C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#9288—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5122—21no—
14#5119—21no—
15#5093C412yescomplemented
16#5091—22no—
17#5078C412yescomplemented
18#4970—22no—
19#4936C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×#803C2211yescharacteristic
22#4828—22no—
23#4828—22no—
24C3×#333—22no—
25#4529—21no—
26#4529—21no—
27#4467—22no—
28#3230—31no—
29#2961—31no—
30#1993—31no—
31#1548D311yescharacteristic
32#1518—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46#924—21no—
47#924D412yes—
48C22×C3⋊C8D411yescharacteristic
49C22×C3⋊C8D411yescharacteristic
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8D412yes—
58C3×C22⋊C8—21no—
59C3×C4×C8—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#830—21no—
63#815—22no—
64Dic3⋊C8—24no—
65C8×Dic3—22no—
66#810—24no—
67#807—24no—
68C4⋊(C3⋊C8)—21no—
69#803—21no—
70#803—21no—
71C4×C3⋊C8C2×C412yes—
72#533—31no—
73C2×C4⋊Q8C2×D311yescharacteristic
74#500—32no—
75#487—32no—
76#484—62no—
77#477—62no—
78#446—62no—
79#446—62no—
80#431—31no—
81#431C2×D311yescharacteristic
82#424—61no—
83#423C2×D311yescharacteristic
84#405—61no—
85C2×#97—31no—
86#382—31no—
87#340—61no—
88#340—61no—
89#333—22no—
90#328—62no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—82no—
94C22×C3×C4—41no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C22×Dic3—41no—
99C2×Dic6—21no—
100C2×Dic6C22⋊C412yes—
101C3×M16—41no—
102C3×M16—41no—
103C2×C3×C8—22no—
104C2×C3×C8—41no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42C22×C411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4111yescharacteristic
113#217—22no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119C4⋊Dic3C22⋊C412yes—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122C4×Dic3—22no—
123C2×C3⋊C8—21no—
124C2×C3⋊C8—21no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—21no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8Q1612yes—
130C22×Q8—21no—
131C2×M16—21no—
132C22×C8—31no—
133C22×C8—31no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D312yes—
140#125—61no—
141C22⋊Q8—32no—
142C22⋊Q8—61no—
143C4×Q8—32no—
144C4×Q8—61no—
145#117—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—61no—
149C2×C42C22×D311yescharacteristic
150#107—64no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—64no—
154C4⋊C8—61no—
155C4⋊C8#6712yes—
156#103—64no—
157C22⋊C8—21no—
158C22⋊C8—61no—
159#97—62no—
160#97—61no—
161#97—61no—
162C4×C8—21no—
163C4×C8—32no—
164C4×C8—62no—
165#95—61no—
166#95—61no—
167C23×C3#12111yescharacteristic
168C3×Q8—82no—
169C3×Q8—82no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—82no—
174C2×C3×C4C2×C22⋊C411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4C4×D411yescharacteristic
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12111yescharacteristic
179C2×C3×C4#12311yescharacteristic
180C2×C3×C4C2×Q1611yescharacteristic
181C2×C3×C4#13511yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×C3×C4#13711yescharacteristic
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189Dic6—42no—
190Dic6—81no—
191C3×C8—42no—
192C3×C8—81no—
193C3⋊C8—22no—
194C3⋊C8—42no—
195C3⋊C8—41no—
196C2×Q8—22no—
197C2×Q8—32no—
198C2×Q8—41no—
199C2×Q8—61no—
200C2×Q8—82no—
201C22×C4—41no—
202C22×C4—121no—
203C22×C4#23511yescharacteristic
204C22×C4D3×D411yescharacteristic
205C22×C4D3×D411yescharacteristic
206M16—41no—
207M16—41no—
208C2×C8—22no—
209C2×C8—32no—
210C2×C8—41no—
211C2×C8—61no—
212C2×C8—61no—
213C2×C8—62no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—61no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—32no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—62no—
225C4⋊C4—61no—
226C4⋊C4—62no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—62no—
230C42—62no—
231C42C2×C4×D311yescharacteristic
232C42D3×D411yescharacteristic
233C42#23711yescharacteristic
234C42C2×#6711yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#38611yescharacteristic
238C22×C3#46211yescharacteristic
239C22×C3#46411yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—82no—
245C3×C4C4×Q1611yescharacteristic
246C3×C4#44111yescharacteristic
247C3×C4#45111yescharacteristic
248C3×C4#45211yescharacteristic
249Dic3—42no—
250Dic3—42no—
251Dic3—161no—
252C23#88411yescharacteristic
253Q8—82no—
254Q8—82no—
255Q8—122no—
256Q8—241no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—82no—
263C2×C4—241no—
264C2×C4—241no—
265C2×C4—241no—
266C2×C4D3×C22⋊C411yescharacteristic
267C2×C4#67⋊C411yescharacteristic
268C2×C4#88311yescharacteristic
269C2×C4#91511yescharacteristic
270C2×C4#91611yescharacteristic
271C2×C4D3×Q1611yescharacteristic
272C2×C4#91811yescharacteristic
273C2×C4C4×#6711yescharacteristic
274C2×C4#93711yescharacteristic
275C2×C4#93811yescharacteristic
276C8—42no—
277C8—62no—
278C8—81no—
279C8—122no—
280C8—121no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3#187811yescharacteristic
284C2×C3#190311yescharacteristic
285C2×C3#190911yescharacteristic
286C22—21no—
287C22—21no—
288C22#479511yescharacteristic
289C22#479811yescharacteristic
290C22#495611yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—82no—
296C4—122no—
297C4—122no—
298C4—481no—
299C4#478911yescharacteristic
300C4#479011yescharacteristic
301C4#517311yescharacteristic
302C4#518611yescharacteristic
303C3#928811yescharacteristic, complemented, Sylow
304C2—21no—
305C2—21no—
306C2#7008711yescharacteristic
307C2#7028111yescharacteristic
308C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9288
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
15. #5093C4semidirect
17. #5078C4semidirect
303. C3#9288semidirect

Maximal subgroups

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

subgroupindex
#832242
#790452
C3×#15482
#717882
#707932
#677402
#677132
#92883