Finite Group Databasebeta

← #10699163

Subgroups — #10699163

Order 768.

326 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#3236C211yesmaximal, characteristic
2#71833C211yesmaximal, characteristic
3#71821C211yesmaximal, characteristic
4#71742C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic
6#70795C211yesmaximal, characteristic
7#67753C211yesmaximal, characteristic
8#10499—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C3×#501—21no—
12C3×#491—21no—
13C3×#484—21no—
14C3×#483—21no—
15C3×#479—21no—
16C3×#479—21no—
17C3×#432C2211yescharacteristic
18#4926C2211yescharacteristic
19#4924C2211yescharacteristic
20C4×#208C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4547—21no—
23#4546—21no—
24#4543—21no—
25#4543—21no—
26#4543—21no—
27#4543—21no—
28#4530—21no—
29#4530—21no—
30#4480—21no—
31#4479—21no—
32#3236D311yescharacteristic
33#1998—31no—
34#1569—31no—
35#1559—31no—
36#1528—31no—
37#1526—31no—
38#1291—31no—
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×#208—21no—
57C2×#208—21no—
58C2×#208—21no—
59C3×#107—21no—
60C3×#107—21no—
61C3×#106—21no—
62C3×#106—21no—
63C3×C4⋊C8D411yescharacteristic
64C3×C4⋊C8D411yescharacteristic
65C3×#103—21no—
66C3×#103—21no—
67C3×#103—21no—
68C3×#103—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71#830—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77#803D411yescharacteristic
78#803D411yescharacteristic
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81#533C2×D311yescharacteristic
82C2×#125C2×D311yescharacteristic
83#501—21no—
84#491—21no—
85#484—21no—
86#483—21no—
87#479—21no—
88#479—21no—
89#432—31no—
90#432C2×D311yescharacteristic
91#423—31no—
92C4×M16—31no—
93C4×M16—31no—
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—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#208—22no—
134#208—42no—
135#208—41no—
136#208—42no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×M16—61no—
143C2×M16—61no—
144C2×M16—61no—
145C4⋊Q8#6711yescharacteristic
146C4⋊Q8#6711yescharacteristic
147#125—21no—
148#125—21no—
149#125—21no—
150#125—21no—
151#125—21no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156#117—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4#6711yescharacteristic
160C2×C4⋊C4#6711yescharacteristic
161C2×C42C22×D311yescharacteristic
162#107—21no—
163#107—21no—
164#106—21no—
165#106—21no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8#6711yescharacteristic
170C4⋊C8#6711yescharacteristic
171#103—21no—
172#103—21no—
173#103—21no—
174#103—21no—
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—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C3×C8—41no—
205C3×C8—41no—
206C3⋊C8—22no—
207C3⋊C8—41no—
208C3⋊C8—42no—
209C3⋊C8—81no—
210C3⋊C8—81no—
211C2×Q8—21no—
212C2×Q8—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4#23711yescharacteristic
216C22×C4#23711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218M16—62no—
219M16—122no—
220M16—121no—
221M16—122no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—121no—
227C2×C8—121no—
228C2×C8—121no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C42—41no—
245C42D3×D411yescharacteristic
246C42#23711yescharacteristic
247C42C2×#6711yescharacteristic
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—81no—
261C3×C4—81no—
262C3×C4#45411yescharacteristic
263C3×C4#45611yescharacteristic
264C23#93511yescharacteristic
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—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#93511yescharacteristic
279C2×C4#93611yescharacteristic
280C2×C4#93811yescharacteristic
281C2×C4#93911yescharacteristic
282C2×C4#93911yescharacteristic
283C2×C4#95311yescharacteristic
284C8—41no—
285C8—41no—
286C8—62no—
287C8—121no—
288C8—122no—
289C8—241no—
290C8—241no—
291C2×C3—41no—
292C2×C3#203811yescharacteristic
293C2×C3#204911yescharacteristic
294C2×C3#205011yescharacteristic
295C22—21no—
296C22—21no—
297C22—21no—
298C22#522211yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4#524511yescharacteristic
309C4#524611yescharacteristic
310C3#1049911yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7186311yescharacteristic
313C2#7210411yescharacteristic
314C2#7211711yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3236C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#3236D3
hypocenterC3#10499
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

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
C3×#32362
#718332
#718212
#717422
#717352
#707952
#677532
#104993