Finite Group Databasebeta

← #10691807

Subgroups — #10691807

Order 768.

396 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 95 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#83359C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3C3×#1531C211yesmaximal, characteristic, complemented
4#70801C211yesmaximal, characteristic
5#69798C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9284—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13#5088C411yescharacteristic
14#5088C411yescharacteristic
15#5080C411yescharacteristic
16#5078C411yescharacteristic
17#5066—22no—
18#5024—22no—
19#5024—21no—
20Q8×C3⋊C8—21no—
21#5003—22no—
22#5000—22no—
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×#330—22no—
27#4483—21no—
28#4483—21no—
29#4464—22no—
30#3228—31no—
31#2985—31no—
32#1531D311yescharacteristic, complemented
33#1530—31no—
34#1496—31no—
35#1287—31no—
36#1275—31no—
37C2×#235—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125D412yes—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117D412yes—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50#875—21no—
51#874C2×C411yescharacteristic
52C4×D12—21no—
53C4×D12—21no—
54#869C2×C411yescharacteristic
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×#97—21no—
58C3×#97—21no—
59C3×#97—21no—
60#830—21no—
61#830—21no—
62#830—21no—
63#809—22no—
64#809—22no—
65#808—22no—
66#807—22no—
67C4⋊(C3⋊C8)—22no—
68C4⋊(C3⋊C8)—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—22no—
73#803D412yes—
74C4×C3⋊C8—22no—
75C4×C3⋊C8C2×C411yescharacteristic
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533C2×D311yescharacteristic
80#532—31no—
81#500—31no—
82#498—31no—
83#486—31no—
84#486—31no—
85#482—62no—
86#478—62no—
87#446—61no—
88#446—62no—
89C8×Q8—61no—
90#433—62no—
91#432—31no—
92#432C2×D311yescharacteristic
93#431—31no—
94#431—31no—
95C2×#97C2×D311yescharacteristic
96#330—22no—
97#329—62no—
98D4⋊C8—61no—
99D4⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×#67—41no—
106#235—82no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×C4×D3—41no—
111C2×Dic6—21no—
112C2×C3×C8—22no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C42—22no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—21no—
137C2×#41—61no—
138C22×C8—21no—
139C4⋊Q8—31no—
140C4⋊Q8C4×D311yescharacteristic
141C4⋊Q8C4×D311yescharacteristic
142#127—31no—
143#125#6712yes—
144#124—61no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147#121—61no—
148#121—61no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×D4—61no—
152C4×D4—61no—
153#117D1212yes—
154C2×C42C22×D311yescharacteristic
155#107—62no—
156#106—62no—
157C4⋊C8—21no—
158C4⋊C8—62no—
159C4⋊C8—62no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#102—62no—
164#102—62no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—21no—
170#97—21no—
171#97—21no—
172#97—32no—
173#97—62no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8—31no—
178C4×C8—62no—
179C23×C3#12311yescharacteristic
180C22×D3—41no—
181C3×Q8—41no—
182C3×Q8—41no—
183C3×Q8—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196#67—82no—
197C2×Dic3—41no—
198D12—42no—
199D12—41no—
200D12—41no—
201D12—81no—
202C4×D3—82no—
203Dic6—81no—
204C3×C8—42no—
205C3×C8—81no—
206C3⋊C8—41no—
207C3⋊C8—42no—
208C3⋊C8—41no—
209C3⋊C8—41no—
210C3⋊C8—41no—
211#41—242no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—61no—
218C2×D4—121no—
219C22×C4—121no—
220C22×C4C2×D1211yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4#23711yescharacteristic
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—62no—
230C2×C8—61no—
231C2×C8—61no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4(C2×D3)⋊C412yes—
239C4⋊C4(C2×D3)⋊C412yes—
240C22⋊C4—22no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C42—22no—
244C42C2×C4×D311yescharacteristic
245C42D3×D411yescharacteristic
246C42D3×D411yescharacteristic
247C42C2×#6711yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#38611yescharacteristic
252C2×D3—81no—
253C2×D3—81no—
254C2×D3—81no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—42no—
261C3×C4—42no—
262C3×C4#44311yescharacteristic
263C3×C4#44411yescharacteristic
264Dic3—81no—
265C23—121no—
266C23#88611yescharacteristic
267Q8—41no—
268Q8—41no—
269Q8—81no—
270Q8—241no—
271D4—122no—
272D4—121no—
273D4—121no—
274D4—241no—
275D4—242no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—121no—
283C2×C4—242no—
284C2×C4#67⋊C411yescharacteristic
285C2×C4#67⋊C411yescharacteristic
286C2×C4#88211yescharacteristic
287C2×C4C2×(C2×D3)⋊C411yescharacteristic
288C2×C4#93911yescharacteristic
289C2×C4#93911yescharacteristic
290C8—42no—
291C8—81no—
292C8—121no—
293C8—121no—
294C8—122no—
295C8—121no—
296C8—121no—
297C2×C3—41no—
298C2×C3#187611yescharacteristic
299C2×C3#190311yescharacteristic
300C2×C3#191211yescharacteristic
301D3—161no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—241no—
306C22—241no—
307C22—241no—
308C22#495811yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—241no—
317C4#482311yescharacteristic
318C4#482411yescharacteristic
319C3#928411yescharacteristic, complemented, Sylow
320C2—41no—
321C2—481no—
322C2#7010511yescharacteristic
323C2#7037711yescharacteristic
324C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9284
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
3. C3×#1531C2semidirect
6. #67757C2semidirect
7. #67660C2semidirect
32. #1531D3semidirect
319. C3#9284semidirect

Maximal subgroups

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

subgroupindex
#833592
#833312
C3×#15312
#708012
#697982
#677572
#676602
#92843