Finite Group Databasebeta

← #10692674

Subgroups — #10692674

Order 768.

442 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 91 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#83295C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic, complemented
3C3×#1514C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic, complemented
5#69564C211yesmaximal, characteristic, complemented
6#67753C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9417—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5234—22no—
14#5088C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5073C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5039—22no—
19#5014—22no—
20#5011—22no—
21#5011—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C3×#327—22no—
26#4464—22no—
27#4451—22no—
28#3184—31no—
29#2980—31no—
30#1536—31no—
31#1514—31no—
32#1514D311yescharacteristic, complemented
33#1291—31no—
34#1288—31no—
35C2×C3×#41—21no—
36C2×#235—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#127C2×C411yescharacteristic
39C3×#124—21no—
40C3×#121—21no—
41C3×#121D412yes—
42C3×C4×D4—21no—
43C3×C4×D4D412yes—
44C2×C3×C42C2311yescharacteristic
45C2×#213—22no—
46#930—21no—
47#930—21no—
48C2×#208—21no—
49C2×#208—22no—
50C22×C3⋊C8—22no—
51#875—21no—
52#874C2×C411yescharacteristic
53C4×D12—21no—
54C4×D12—21no—
55#869C2×C411yescharacteristic
56C3×C4⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#97—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62#832—22no—
63#830—24no—
64#830—21no—
65#830D412yes—
66#809—22no—
67#808—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70#803—21no—
71#803—21no—
72#803D412yes—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#532—31no—
76#532C2×D311yescharacteristic
77#498—31no—
78#493—31no—
79#486—31no—
80#486—31no—
81#469—62no—
82#442—62no—
83#435—62no—
84#435—62no—
85#432—31no—
86#432—31no—
87#432C2×D311yescharacteristic
88#408—62no—
89C4×M16—31no—
90#383C2×D311yescharacteristic
91#329—62no—
92#327—22no—
93#327—62no—
94C3×#41—81no—
95C3×#41—81no—
96C2×C3×Q8—21no—
97C2×C3×D4—22no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4C22⋊C412yes—
101C22×C3×C4—22no—
102C22×C3×C4—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—41no—
107#235—82no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4C22⋊C412yes—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#213—44no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127#208—42no—
128#208—82no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—82no—
134C2×#41—21no—
135C2×#41—61no—
136C2×D8—62no—
137C2×M16—61no—
138C2×M16—62no—
139C22×C8—62no—
140C4⋊Q8—31no—
141C4⋊Q8C4×D311yescharacteristic
142#127—31no—
143#127C4×D311yescharacteristic
144#124—21no—
145#124—61no—
146#121—21no—
147#121—61no—
148#121—61no—
149#121#6712yes—
150C4×D4—21no—
151C4×D4—61no—
152C4×D4—61no—
153C4×D4D1212yes—
154C2×C42C22×D311yescharacteristic
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159#102—62no—
160#102—62no—
161C22⋊C8—21no—
162C22⋊C8—32no—
163C22⋊C8—64no—
164C22⋊C8—61no—
165#97—21no—
166#97—32no—
167#97—61no—
168#97—61no—
169C4×C8—31no—
170C4×C8—31no—
171#95—21no—
172#95—21no—
173#95—21no—
174C23×C3—22no—
175C23×C3#12111yescharacteristic
176C22×D3—41no—
177C3×Q8—81no—
178C3×D4—42no—
179C3×D4—81no—
180C3×D4—81no—
181C3×D4—82no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197#67—82no—
198C2×Dic3—41no—
199D12—42no—
200D12—81no—
201D12—81no—
202C4×D3—82no—
203Dic6—81no—
204C3×C8—81no—
205C3⋊C8—22no—
206C3⋊C8—42no—
207C3⋊C8—42no—
208C3⋊C8—41no—
209#41—81no—
210#41—81no—
211#41—242no—
212C2×Q8—21no—
213C2×Q8—61no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—61no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4—121no—
221C2×D4(C2×D3)⋊C412yes—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—121no—
225C22×C4D3×D411yescharacteristic
226C22×C4#23711yescharacteristic
227C22×C4C2×#6711yescharacteristic
228D8—124no—
229M16—122no—
230M16—242no—
231C2×C8—41no—
232C2×C8—61no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—61no—
236C2×C8—242no—
237C4⋊C4—21no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4(C2×D3)⋊C412yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C42C2×C4×D311yescharacteristic
246C42C2×D1211yescharacteristic
247C42D3×D411yescharacteristic
248C42#23711yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3#38611yescharacteristic
256C2×D3—81no—
257C2×D3—81no—
258C2×D3—81no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—161no—
267Dic3—81no—
268C23—22no—
269C23—121no—
270C23#93911yescharacteristic
271Q8—81no—
272Q8—241no—
273D4—42no—
274D4—81no—
275D4—81no—
276D4—82no—
277D4—122no—
278D4—241no—
279D4—241no—
280D4—242no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—121no—
291C2×C4—242no—
292C2×C4#67⋊C411yescharacteristic
293C2×C4#67⋊C411yescharacteristic
294C2×C4#88211yescharacteristic
295C2×C4#88611yescharacteristic
296C2×C4C2×(C2×D3)⋊C411yescharacteristic
297C2×C4#93911yescharacteristic
298C8—62no—
299C8—81no—
300C8—122no—
301C8—122no—
302C8—121no—
303C2×C3—41no—
304C2×C3—82no—
305C2×C3#188411yescharacteristic
306C2×C3#190311yescharacteristic
307C2×C3#190411yescharacteristic
308D3—161no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—241no—
316C22—241no—
317C22—241no—
318C22#495811yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—161no—
327C4—241no—
328C3#941711yescharacteristic, complemented, Sylow
329C2—41no—
330C2—82no—
331C2—481no—
332C2#7004511yescharacteristic
333C2#7031411yescharacteristic
334C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#386
prime radical#83295C2
cosocleC2×C3×C42C23
hypercenter#1514D3
hypocenterC3#9417
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
2. #83279C2semidirect
3. C3×#1514C2semidirect
4. #70811C2semidirect
5. #69564C2semidirect
6. #67753C2semidirect
7. #67633C2semidirect
14. #5088C4semidirect
15. #5080C4semidirect
16. #5073C4semidirect
17. #5070C4semidirect
32. #1514D3semidirect
328. C3#9417semidirect

Maximal subgroups

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

subgroupindex
#832952
#832792
C3×#15142
#708112
#695642
#677532
#676332
#94173