Finite Group Databasebeta

← #10699169

Subgroups — #10699169

Order 768.

326 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 73 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#79320C211yesmaximal, characteristic
2#70825C211yesmaximal, characteristic
3#70812C211yesmaximal, characteristic
4#70795C211yesmaximal, characteristic
5#69880C211yesmaximal, characteristic
6#69862C211yesmaximal, characteristic
7C3×#1291C211yesmaximal, characteristic
8#10499—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#4924C2211yescharacteristic
15C4×#208C2211yescharacteristic
16#4829—21no—
17#4829—21no—
18#4722—21no—
19#4721—21no—
20#4703—21no—
21#4702—21no—
22#4554—21no—
23#4554—21no—
24#4486—21no—
25#4481—21no—
26#4480—21no—
27#4479—21no—
28#4465—21no—
29#4465—21no—
30#4465—21no—
31#4465—21no—
32#3236—31no—
33#1998—31no—
34#1569—31no—
35#1559—31no—
36#1528—31no—
37#1526—31no—
38#1291D311yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C42C2311yescharacteristic
41#926—21no—
42C2×C4⋊Dic3D411yescharacteristic
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×#208—21no—
49C2×#208—21no—
50#870—21no—
51#870—21no—
52#870—21no—
53#870—21no—
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D411yescharacteristic
61C3×C4⋊C8D411yescharacteristic
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#818—21no—
69#818—21no—
70#817—21no—
71#817—21no—
72#816—21no—
73#816—21no—
74#816—21no—
75#816—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#803D411yescharacteristic
79#803D411yescharacteristic
80C4×C3⋊C8—21no—
81#533—31no—
82C2×#125—31no—
83#501—61no—
84#491—61no—
85#484—61no—
86#483—61no—
87#479—61no—
88#479—61no—
89#432C2×D311yescharacteristic
90#432C2×D311yescharacteristic
91#423—31no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#343—61no—
95#343—61no—
96#335—61no—
97#334—61no—
98#333—61no—
99#332—61no—
100#330—61no—
101#330—61no—
102#330—61no—
103#330—61no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C3×M16—22no—
112C3×M16—41no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—41no—
137#208—42no—
138#208—42no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—41no—
142C2×M16—21no—
143C2×M16—61no—
144C2×M16—61no—
145C4⋊Q8—31no—
146C4⋊Q8—31no—
147#125—61no—
148#125—61no—
149#125—61no—
150#125—61no—
151#125—61no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156#117—61no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C42C22×D311yescharacteristic
162#107—61no—
163#107—61no—
164#106—61no—
165#106—61no—
166C4⋊C8—21no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8D1211yescharacteristic
170C4⋊C8D1211yescharacteristic
171#103—61no—
172#103—61no—
173#103—61no—
174#103—61no—
175C22⋊C8—21no—
176C22⋊C8—21no—
177#97—21no—
178#97—21no—
179#97—21no—
180#97—31no—
181#97—31no—
182C4×C8—21no—
183C4×C8—61no—
184C23×C3#12311yescharacteristic
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201Dic6—41no—
202Dic6—41no—
203Dic6—81no—
204C3×C8—42no—
205C3×C8—41no—
206C3×C8—41no—
207C3×C8—81no—
208C3⋊C8—22no—
209C3⋊C8—41no—
210C3⋊C8—81no—
211C2×Q8—61no—
212C2×Q8—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4#23511yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4#23911yescharacteristic
218M16—22no—
219M16—41no—
220M16—122no—
221M16—122no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—121no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—121no—
244C42—121no—
245C42C2×D1211yescharacteristic
246C42#23511yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39411yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4#45411yescharacteristic
259C3×C4#45611yescharacteristic
260Dic3—81no—
261Dic3—81no—
262Dic3—81no—
263Dic3—81no—
264C23#89411yescharacteristic
265Q8—121no—
266Q8—121no—
267Q8—241no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4—241no—
278C2×C4#87511yescharacteristic
279C2×C4#88211yescharacteristic
280C2×C4#88411yescharacteristic
281C2×C4#88411yescharacteristic
282C2×C4#89411yescharacteristic
283C2×C4#89511yescharacteristic
284C8—42no—
285C8—41no—
286C8—41no—
287C8—62no—
288C8—81no—
289C8—121no—
290C8—241no—
291C2×C3—41no—
292C2×C3#203811yescharacteristic
293C2×C3#204911yescharacteristic
294C2×C3#205011yescharacteristic
295C22—21no—
296C22—21no—
297C22—21no—
298C22#467211yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—241no—
305C4—241no—
306C4—241no—
307C4—241no—
308C4#482611yescharacteristic
309C4#482711yescharacteristic
310C3#1049911yescharacteristic, complemented, Sylow
311C2—41no—
312C2#6871911yescharacteristic
313C2#6873411yescharacteristic
314C2#6901911yescharacteristic
subgroupG/H
centerC22#4672
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1291D3
hypocenterC3#10499
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
310. C3#10499semidirect

Maximal subgroups

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

subgroupindex
#793202
#708252
#708122
#707952
#698802
#698622
C3×#12912
#104993