Finite Group Databasebeta

← #10699217

Subgroups — #10699217

Order 768.

374 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 99 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#79272C211yesmaximal, characteristic
2C3×#1565C211yesmaximal, characteristic
3#70459C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10506—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C4×C4⋊Dic3C2211yescharacteristic
15#4924C2211yescharacteristic
16#4850—21no—
17#4850—21no—
18#4722—22no—
19#4682—21no—
20#4681—21no—
21C3×#335—21no—
22C3×#334—21no—
23#4555—21no—
24#4555—21no—
25#4554—21no—
26#4554—21no—
27#4551—22no—
28#4482—21no—
29#4481—21no—
30#3230—31no—
31#1997—31no—
32#1987—31no—
33#1986—31no—
34#1565D311yescharacteristic
35#1550—31no—
36#1303—31no—
37C22×Dic6—21no—
38C2×C3×M16D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D412yes—
46#924—21no—
47#924—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50#870—21no—
51#869—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×C4×C8—21no—
66C3×C4×C8—21no—
67C3⋊#95—21no—
68#818—22no—
69#818—22no—
70#816—22no—
71#816—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#533—31no—
75C2×C4⋊Q8—31no—
76#500—61no—
77#487—61no—
78#484—62no—
79#477—61no—
80#477—61no—
81#431C2×D311yescharacteristic
82#423—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84C4×M16C2×D311yescharacteristic
85C4×C4⋊C4—31no—
86#343—61no—
87#343—61no—
88#340—62no—
89#339—61no—
90#339—61no—
91#335—21no—
92#334—21no—
93#333—61no—
94#332—61no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C22×Dic3—22no—
99C22×Dic3—41no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6—41no—
103C2×Dic6—41no—
104C2×Dic6—82no—
105C3×M16—21no—
106C3×M16—21no—
107C3×M16—21no—
108C3×M16Q1612yes—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—42no—
114C2×C3×C8—41no—
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119#217—41no—
120C4⋊Dic3—22no—
121C4⋊Dic3—22no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—22no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—41no—
133C4×Dic3—41no—
134C4×Dic3—41no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—41no—
138C22×Q8—61no—
139C2×M16—61no—
140C2×M16D1211yescharacteristic
141C2×M16D1211yescharacteristic
142C22×C8—21no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145C4⋊Q8—61no—
146C4⋊Q8—61no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149#125—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154#117—61no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C42—61no—
159C2×C42C22×D311yescharacteristic
160#107—62no—
161#107—62no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8#6711yescharacteristic
167C4⋊C8#6711yescharacteristic
168#103—62no—
169#103—62no—
170C22⋊C8—21no—
171#97—21no—
172C4×C8—21no—
173C4×C8—21no—
174#95—61no—
175C23×C3#12111yescharacteristic
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12311yescharacteristic
183C2×C3×C4#12311yescharacteristic
184C2×C3×C4#12411yescharacteristic
185C2×C3×C4C2×Q1611yescharacteristic
186C2×C3×C4#13511yescharacteristic
187C2×C3×C4#13711yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C2×Dic3—42no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—42no—
193C2×Dic3—81no—
194C2×Dic3—81no—
195C2×Dic3—82no—
196Dic6—41no—
197Dic6—41no—
198Dic6—82no—
199Dic6—82no—
200Dic6—81no—
201C3×C8—22no—
202C3×C8—41no—
203C3×C8—41no—
204C3×C8—41no—
205C3×C8—81no—
206C3⋊C8—81no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—121no—
210C2×Q8—121no—
211C2×Q8—242no—
212C22×C4—62no—
213C22×C4—121no—
214C22×C4C2×D1211yescharacteristic
215C22×C4D3×D411yescharacteristic
216C22×C4#23911yescharacteristic
217M16—21no—
218M16—21no—
219M16—21no—
220M16—121no—
221M16—121no—
222M16Dic1212yes—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—121no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—121no—
243C42—121no—
244C42—121no—
245C42—121no—
246C42#23511yescharacteristic
247C42D3×D411yescharacteristic
248C42#23911yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39411yescharacteristic
253C22×C3#46211yescharacteristic
254C22×C3#46411yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4#45411yescharacteristic
259C3×C4#45611yescharacteristic
260C3×C4#46211yescharacteristic
261C3×C4#46411yescharacteristic
262C3×C4#46711yescharacteristic
263C3×C4#47011yescharacteristic
264Dic3—82no—
265Dic3—82no—
266Dic3—81no—
267Dic3—81no—
268C23#89511yescharacteristic
269Q8—121no—
270Q8—121no—
271Q8—241no—
272Q8—242no—
273Q8—242no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—122no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—122no—
281C2×C4—241no—
282C2×C4—241no—
283C2×C4—242no—
284C2×C4#89411yescharacteristic
285C2×C4#89411yescharacteristic
286C2×C4#89511yescharacteristic
287C2×C4#90411yescharacteristic
288C2×C4C2×Dic1211yescharacteristic
289C2×C4#90911yescharacteristic
290C2×C4#90911yescharacteristic
291C2×C4#93011yescharacteristic
292C2×C4#93711yescharacteristic
293C2×C4#94711yescharacteristic
294C8—22no—
295C8—41no—
296C8—41no—
297C8—41no—
298C8—81no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#203811yescharacteristic
303C2×C3#204811yescharacteristic
304C2×C3#205611yescharacteristic
305C22—21no—
306C22—21no—
307C22#484511yescharacteristic
308C22#484911yescharacteristic
309C22#499111yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—242no—
314C4—241no—
315C4—242no—
316C4—241no—
317C4#484511yescharacteristic
318C4#484911yescharacteristic
319C4#511611yescharacteristic
320C4#513711yescharacteristic
321C4#520111yescharacteristic
322C4#520511yescharacteristic
323C3#1050611yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#7048711yescharacteristic
327C2#7067811yescharacteristic
328C2#7073611yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1565C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1565D3
hypocenterC3#10506
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
323. C3#10506semidirect

Maximal subgroups

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

subgroupindex
#792722
C3×#15652
#704592
#704382
#698792
#698682
#677462
#105063