Finite Group Databasebeta

← #10692721

Subgroups — #10692721

Order 768.

534 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 89 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#83336C211yesmaximal, characteristic
2#74298C211yesmaximal, characteristic
3C3×#1515C211yesmaximal, characteristic, complemented
4#71837C211yesmaximal, characteristic, complemented
5#70818C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9425—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#874C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5088C412yescomplemented
14#5088C412yescomplemented
15#5067—22no—
16#5001—22no—
17#4937C2211yescharacteristic
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4665—22no—
21#4665—22no—
22#4665—22no—
23C3×#328—22no—
24#4538—22no—
25#4485—22no—
26#3138—31no—
27#2380—31no—
28#1555—31no—
29#1515—31no—
30#1515D311yescharacteristic, complemented
31#1288—31no—
32#1288—31no—
33C22×D12—21no—
34C22×D12—21no—
35C22×D12D412yes—
36C22×C4×D3—22no—
37C3×C4⋊Q8C2×C411yescharacteristic, complemented
38C3×C4⋊Q8C2×C411yescharacteristic, complemented
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×#208—21no—
50#874—21no—
51#874—21no—
52#874C2×C412yes—
53#874D412yes—
54C3×C4⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×#97—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95D412yes—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95D412yes—
63#830—21no—
64#809—24no—
65#809—24no—
66C4⋊(C3⋊C8)—21no—
67#803—21no—
68#803—21no—
69C4×C3⋊C8C2×C412yescomplemented
70#533C2×D311yescharacteristic
71C2×#127—31no—
72#490—62no—
73#486—32no—
74#486—32no—
75#432—31no—
76#432C2×D311yescharacteristic
77#418—62no—
78C4×M16—31no—
79#386—62no—
80#386—62no—
81#386—62no—
82#383—31no—
83#383C2×D311yescharacteristic
84#331—62no—
85#328—22no—
86#328—62no—
87C23×D3—22no—
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C22×C3×C4—22no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—22no—
95C2×D12—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—42no—
101C2×D12—42no—
102C2×D12—81no—
103C2×D12—81no—
104C2×D12—81no—
105C2×D12—81no—
106C2×D12C22⋊C414yes—
107C2×C4×D3—82no—
108C2×C4×D3—82no—
109C2×C4×D3—82no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C42—41no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123(C2×D3)⋊C4—82no—
124(C2×D3)⋊C4—82no—
125(C2×D3)⋊C4—82no—
126#208—42no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C22×D4—32no—
130C22×D4—61no—
131C22×D4—61no—
132C23×C4—62no—
133C2×M16—61no—
134C4⋊Q8C4×D311yescharacteristic, complemented
135C4⋊Q8C4×D311yescharacteristic, complemented
136#127—32no—
137#127—32no—
138#127—61no—
139#127—61no—
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145#117—21no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149C2×C42C22×D311yescharacteristic
150C4⋊C8—21no—
151C4⋊C8—61no—
152#102—64no—
153#102—64no—
154C22⋊C8—21no—
155C22⋊C8—61no—
156#97—21no—
157#97—61no—
158#97—61no—
159C4×C8—32no—
160#95—21no—
161#95—21no—
162#95—32no—
163#95—61no—
164#95—61no—
165#95#6712yes—
166C23×C3#12011yescharacteristic
167C22×D3—24no—
168C22×D3—42no—
169C22×D3—42no—
170C22×D3—42no—
171C22×D3—42no—
172C22×D3—42no—
173C22×D3—42no—
174C3×Q8—81no—
175C3×Q8—81no—
176C2×C3×C4—22no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193D12—44no—
194D12—82no—
195D12—82no—
196D12—82no—
197D12—82no—
198D12—82no—
199C4×D3—88no—
200C3×C8—81no—
201C3⋊C8—42no—
202C3⋊C8—81no—
203C24—62no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—34no—
207C2×D4—62no—
208C2×D4—62no—
209C2×D4—62no—
210C2×D4—62no—
211C2×D4—62no—
212C2×D4—122no—
213C2×D4—122no—
214C2×D4—241no—
215C2×D4—241no—
216C2×D4—241no—
217C2×D4—241no—
218C22×C4—22no—
219C22×C4—62no—
220C22×C4—242no—
221C22×C4—242no—
222C22×C4—242no—
223C22×C4D3×D411yescharacteristic
224C22×C4D3×D411yescharacteristic
225C22×C4C2×#6711yescharacteristic
226M16—122no—
227C2×C8—41no—
228C2×C8—62no—
229C2×C8—121no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—242no—
238C22⋊C4—242no—
239C22⋊C4—242no—
240C42—41no—
241C42C2×C4×D311yescharacteristic
242C42#23511yescharacteristic
243C42D3×D411yescharacteristic
244C42#23711yescharacteristic
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#38611yescharacteristic
248C22×C3#45512yes—
249C2×D3—44no—
250C2×D3—44no—
251C2×D3—44no—
252C2×D3—82no—
253C2×D3—82no—
254C2×D3—82no—
255C2×D3—82no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—82no—
263C3×C4—81no—
264C3×C4—81no—
265Dic3—82no—
266C23—64no—
267C23—122no—
268C23—122no—
269C23—122no—
270C23—122no—
271C23—122no—
272C23—122no—
273C23#93711yescharacteristic
274Q8—81no—
275Q8—81no—
276D4—124no—
277D4—242no—
278D4—242no—
279D4—242no—
280D4—242no—
281D4—242no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—248no—
294C2×C4D3×C22⋊C411yescharacteristic
295C2×C4#67⋊C411yescharacteristic
296C2×C4#88311yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4C4×#6711yescharacteristic
299C2×C4#93811yescharacteristic
300C8—81no—
301C8—122no—
302C8—241no—
303C2×C3—22no—
304C2×C3#188511yescharacteristic
305C2×C3#190411yescharacteristic
306C2×C3#190411yescharacteristic
307D3—84no—
308C22—21no—
309C22—21no—
310C22—124no—
311C22—124no—
312C22—124no—
313C22—242no—
314C22—242no—
315C22—242no—
316C22—242no—
317C22#495611yescharacteristic
318C22#520212yes—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—81no—
327C4—81no—
328C4—242no—
329C3#942511yescharacteristic, complemented, Sylow
330C2—22no—
331C2—244no—
332C2#7004911yescharacteristic
333C2#7034311yescharacteristic
334C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9425
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
3. C3×#1515C2semidirect
4. #71837C2semidirect
6. #67752C2semidirect
7. #67633C2semidirect
13. #5088C4semidirect
14. #5088C4semidirect
30. #1515D3semidirect
37. C3×C4⋊Q8C2×C4semidirect
38. C3×C4⋊Q8C2×C4semidirect
69. C4×C3⋊C8C2×C4semidirect
134. C4⋊Q8C4×D3semidirect
135. C4⋊Q8C4×D3semidirect
329. C3#9425semidirect

Maximal subgroups

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

subgroupindex
#833362
#742982
C3×#15152
#718372
#708182
#677522
#676332
#94253