Finite Group Databasebeta

← #10691786

Subgroups — #10691786

Order 768.

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

normal subgroups only

This group has 456 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#83304C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic, complemented
3C3×#1505C211yesmaximal, characteristic, complemented
4#70801C211yesmaximal, characteristic, complemented
5#69432C211yesmaximal, characteristic, complemented
6#67756C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9281—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5234—22no—
14#5093C411yescharacteristic, complemented
15#5088C411yescharacteristic, complemented
16#5076C411yescharacteristic, complemented
17#5073C411yescharacteristic, complemented
18#5037—22no—
19#5016—22no—
20#5012—21no—
21#5011—22no—
22D4×C3⋊C8—21no—
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×#327—22no—
27#4483—21no—
28#4483—21no—
29#4451—22no—
30#3207—31no—
31#2981—31no—
32#1505—31no—
33#1505D311yescharacteristic, complemented
34#1496—31no—
35#1287—31no—
36#1274—31no—
37C2×C3×#41—21no—
38C2×#235—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#127C2×C411yescharacteristic
41C3×#124—21no—
42C3×#121—21no—
43C3×#121D412yes—
44C3×C4×D4—21no—
45C3×C4×D4D412yes—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#214—22no—
49C2×#213—22no—
50#930—21no—
51#930—21no—
52C2×#208—22no—
53C22×C3⋊C8—22no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—21no—
62C3×#95—21no—
63C3×#95—21no—
64#830—22no—
65#830—22no—
66#830—21no—
67#830—21no—
68#830D412yes—
69#810—22no—
70#809—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75C4×C3⋊C8C2×C411yescharacteristic
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#532—31no—
80#532C2×D311yescharacteristic
81#493—31no—
82#492—31no—
83#487—31no—
84#486—31no—
85#463—62no—
86#449—62no—
87#436—61no—
88#435—62no—
89C8×D4—61no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431—31no—
93#431—31no—
94#408—62no—
95#376C2×D311yescharacteristic
96#327—22no—
97#327—62no—
98D4⋊C8—61no—
99D4⋊C8—61no—
100C3×#41—82no—
101C2×C3×Q8—21no—
102C2×C3×D4—22no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—41no—
112#235—82no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C42—22no—
124C3×C42—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#214—82no—
130#213—82no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135#208—82no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—82no—
142C2×#41—21no—
143C2×#41—61no—
144C2×SD16—62no—
145C2×D8—62no—
146C2×M16—62no—
147C22×C8—62no—
148C4⋊Q8—31no—
149C4⋊Q8C4×D311yescharacteristic
150#127—31no—
151#127C4×D311yescharacteristic
152#124—21no—
153#124—61no—
154#121—21no—
155#121—61no—
156#121—61no—
157#121#6712yes—
158C4×D4—21no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4D1212yes—
162C2×C42—21no—
163C2×C42C22×D311yescharacteristic
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—62no—
169#102—62no—
170C22⋊C8—21no—
171C22⋊C8—32no—
172C22⋊C8—62no—
173C22⋊C8—62no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#97—21no—
177#97—61no—
178C4×C8—31no—
179C4×C8—31no—
180C4×C8—31no—
181C4×C8—31no—
182#95—21no—
183#95—21no—
184C23×C3—22no—
185C23×C3#12111yescharacteristic
186C22×D3—41no—
187C3×Q8—81no—
188C3×D4—42no—
189C3×D4—41no—
190C3×D4—41no—
191C3×D4—81no—
192C3×D4—82no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4C2×C22⋊C411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12311yescharacteristic
209#67—82no—
210C2×Dic3—41no—
211D12—42no—
212D12—41no—
213D12—41no—
214D12—81no—
215C4×D3—82no—
216Dic6—42no—
217C3×C8—81no—
218C3⋊C8—41no—
219C3⋊C8—42no—
220C3⋊C8—41no—
221C3⋊C8—41no—
222C3⋊C8—41no—
223#41—82no—
224#41—242no—
225C2×Q8—21no—
226C2×Q8—61no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C2×D4(C2×D3)⋊C412yes—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—121no—
238C22×C4D3×D411yescharacteristic
239C22×C4#23711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241SD16—242no—
242D8—242no—
243M16—242no—
244C2×C8—41no—
245C2×C8—61no—
246C2×C8—61no—
247C2×C8—62no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—242no—
251C4⋊C4—21no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4(C2×D3)⋊C412yes—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—22no—
260C42—41no—
261C42C2×C4×D311yescharacteristic
262C42C2×D1211yescharacteristic
263C42D3×D411yescharacteristic
264C42#23711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3#38611yescharacteristic
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—42no—
281C3×C4—82no—
282C3×C4#44311yescharacteristic
283C3×C4#44411yescharacteristic
284Dic3—81no—
285C23—22no—
286C23—121no—
287C23#93911yescharacteristic
288Q8—81no—
289Q8—122no—
290D4—42no—
291D4—41no—
292D4—41no—
293D4—81no—
294D4—82no—
295D4—122no—
296D4—121no—
297D4—121no—
298D4—241no—
299D4—242no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—121no—
311C2×C4—242no—
312C2×C4#67⋊C411yescharacteristic
313C2×C4#67⋊C411yescharacteristic
314C2×C4#88211yescharacteristic
315C2×C4#88611yescharacteristic
316C2×C4C2×(C2×D3)⋊C411yescharacteristic
317C2×C4#93911yescharacteristic
318C8—81no—
319C8—121no—
320C8—122no—
321C8—121no—
322C8—121no—
323C8—121no—
324C2×C3—41no—
325C2×C3—82no—
326C2×C3#187611yescharacteristic
327C2×C3#190411yescharacteristic
328C2×C3#191011yescharacteristic
329D3—161no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#495811yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—82no—
347C4—241no—
348C4#482311yescharacteristic
349C4#482411yescharacteristic
350C3#928111yescharacteristic, complemented, Sylow
351C2—41no—
352C2—82no—
353C2—481no—
354C2#7004711yescharacteristic
355C2#7031411yescharacteristic
356C2#7037711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radical#83304C2
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9281
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. C3×#1505C2semidirect
4. #70801C2semidirect
5. #69432C2semidirect
6. #67756C2semidirect
7. #67614C2semidirect
14. #5093C4semidirect
15. #5088C4semidirect
16. #5076C4semidirect
17. #5073C4semidirect
33. #1505D3semidirect
350. C3#9281semidirect

Maximal subgroups

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

subgroupindex
#833042
#832802
C3×#15052
#708012
#694322
#677562
#676142
#92813