Finite Group Databasebeta

← #10699164

Subgroups — #10699164

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#83326C211yesmaximal, characteristic
2C3×#1559C211yesmaximal, characteristic
3#71821C211yesmaximal, characteristic
4#70812C211yesmaximal, characteristic
5#70525C211yesmaximal, characteristic
6#70492C211yesmaximal, characteristic
7#67754C211yesmaximal, characteristic
8#10499—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5082—21no—
14#5081—21no—
15#5044—21no—
16#5043—21no—
17#5022—21no—
18#5022—21no—
19#4926C2211yescharacteristic
20C4×#208C2211yescharacteristic
21C4×#208C2211yescharacteristic
22C3×#333—21no—
23C3×#332—21no—
24#4554—21no—
25#4554—21no—
26#4543—21no—
27#4543—21no—
28#4480—21no—
29#4479—21no—
30#4465—21no—
31#4465—21no—
32#3236—31no—
33#1998—31no—
34#1569—31no—
35#1559D311yescharacteristic
36#1528—31no—
37#1526—31no—
38#1291—31no—
39C2×C3×M16—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×#208—21no—
54C2×#208—21no—
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#97—21no—
64#835—21no—
65#835—21no—
66#835—21no—
67#835—21no—
68#830—21no—
69#808—21no—
70#808—21no—
71#807—21no—
72#807—21no—
73C4⋊(C3⋊C8)D411yescharacteristic
74C4⋊(C3⋊C8)D411yescharacteristic
75#803—21no—
76#803—21no—
77#803D411yescharacteristic
78#803D411yescharacteristic
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81#533C2×D311yescharacteristic
82C2×#125—31no—
83#501—61no—
84#491—61no—
85#484—61no—
86#483—61no—
87#479—61no—
88#479—61no—
89#432—31no—
90#432C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92C4×M16—31no—
93C4×M16—31no—
94#343—61no—
95#343—61no—
96#335—61no—
97#334—61no—
98#333—21no—
99#332—21no—
100#330—61no—
101#330—61no—
102#330—61no—
103#330—61no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C3×M16—42no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134#208—22no—
135#208—42no—
136#208—41no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—41no—
142C2×M16—21no—
143C2×M16—61no—
144C2×M16—61no—
145C4⋊Q8#6711yescharacteristic
146C4⋊Q8#6711yescharacteristic
147#125—21no—
148#125—61no—
149#125—61no—
150#125—61no—
151#125—61no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156#117—21no—
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—21no—
168C4⋊C8—21no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171#103—61no—
172#103—61no—
173#103—61no—
174#103—61no—
175C22⋊C8—21no—
176C22⋊C8—61no—
177#97—21no—
178#97—31no—
179#97—31no—
180#97—61no—
181#97—61no—
182C4×C8—61no—
183C4×C8—61no—
184C23×C3#12311yescharacteristic
185C3×Q8—41no—
186C3×Q8—41no—
187C3×Q8—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×C3×C4#12411yescharacteristic
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C3×C8—81no—
205C3×C8—81no—
206C3⋊C8—22no—
207C3⋊C8—41no—
208C3⋊C8—42no—
209C3⋊C8—41no—
210C3⋊C8—41no—
211C2×Q8—21no—
212C2×Q8—21no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4#23511yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218M16—42no—
219M16—62no—
220M16—122no—
221M16—121no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—121no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C42—41no—
245C42#23511yescharacteristic
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42C2×#6711yescharacteristic
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—81no—
259C3×C4—81no—
260C3×C4#45411yescharacteristic
261C3×C4#45611yescharacteristic
262Dic3—81no—
263Dic3—81no—
264C23#88311yescharacteristic
265Q8—41no—
266Q8—41no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4—241no—
278C2×C4#88411yescharacteristic
279C2×C4#88411yescharacteristic
280C2×C4#88511yescharacteristic
281C2×C4#92911yescharacteristic
282C2×C4#93711yescharacteristic
283C2×C4#93911yescharacteristic
284C8—62no—
285C8—81no—
286C8—81no—
287C8—121no—
288C8—122no—
289C8—121no—
290C8—121no—
291C2×C3—41no—
292C2×C3#203811yescharacteristic
293C2×C3#204911yescharacteristic
294C2×C3#205011yescharacteristic
295C22—21no—
296C22—21no—
297C22—21no—
298C22#495911yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—81no—
305C4—81no—
306C4—241no—
307C4—241no—
308C4#520111yescharacteristic
309C4#520511yescharacteristic
310C3#1049911yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7014411yescharacteristic
313C2#7030911yescharacteristic
314C2#7035211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1559C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1559D3
hypocenterC3#10499
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1559

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
#833262
C3×#15592
#718212
#708122
#705252
#704922
#677542
#104993