Finite Group Databasebeta

← #10692678

Subgroups — #10692678

Order 768.

442 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3184C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic, complemented
3#71813C211yesmaximal, characteristic, complemented
4#71813C211yesmaximal, characteristic, complemented
5#71799C211yesmaximal, characteristic, complemented
6#67670C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9417—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#498C411yescharacteristic, complemented
12C3×#493C411yescharacteristic, complemented
13C3×#486C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#469—22no—
16C3×#442—22no—
17C3×C4×M16C2211yescharacteristic
18#5234—22no—
19#5011—22no—
20#5011—22no—
21#4937C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4540—22no—
26#4537—22no—
27#4537—22no—
28#3184D311yescharacteristic
29#2980—31no—
30#1536—31no—
31#1514—31no—
32#1514—31no—
33#1291—31no—
34#1288—31no—
35C2×C3×#41—21no—
36C2×C3×#41—21no—
37C2×C3×D8—22no—
38C2×C3×M16—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#127C2×C411yescharacteristic
43C3×#124—21no—
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121D412yes—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×#208—22no—
55C22×C3⋊C8—22no—
56C3×#106—22no—
57C3×#102—22no—
58C3×#102—22no—
59C3×#97—21no—
60C3×C4×C8C2×C411yescharacteristic
61C3×C4×C8C2×C411yescharacteristic
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—24no—
66#830—21no—
67#830—21no—
68#830D412yes—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73#803—21no—
74#803D412yes—
75#532C2×D311yescharacteristic
76#532C2×D311yescharacteristic
77#498Dic311yescharacteristic, complemented
78#493Dic311yescharacteristic, complemented
79#486Dic311yescharacteristic, complemented
80#486Dic311yescharacteristic, complemented
81#469—22no—
82#442—22no—
83#435—62no—
84#435—62no—
85#432—31no—
86#432—31no—
87#432—31no—
88#408—62no—
89C4×M16C2×D311yescharacteristic
90#383—31no—
91#329—62no—
92#327—62no—
93#327—62no—
94C3×#41—81no—
95C3×#41—81no—
96C3×#41—82no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×D4—22no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—41no—
113C3×D8—44no—
114C3×M16—42no—
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#208—82no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—82no—
134C2×#41—21no—
135C2×#41—21no—
136C2×D8—22no—
137C2×M16—21no—
138C2×M16—62no—
139C22×C8—62no—
140C4⋊Q8C2×Dic311yescharacteristic
141C4⋊Q8C2×Dic311yescharacteristic
142#127C2×Dic311yescharacteristic
143#127C2×Dic311yescharacteristic
144#124—21no—
145#124—21no—
146#121—21no—
147#121—21no—
148#121—21no—
149#121#6712yes—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4#6712yes—
154C2×C42C22×D311yescharacteristic
155#106—22no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159#102—22no—
160#102—22no—
161C22⋊C8—32no—
162C22⋊C8—64no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165#97—21no—
166#97—32no—
167#97—61no—
168#97—61no—
169C4×C8C2×Dic311yescharacteristic
170C4×C8C2×Dic311yescharacteristic
171#95—61no—
172#95—61no—
173#95—61no—
174C23×C3—22no—
175C23×C3—41no—
176C23×C3#12111yescharacteristic
177C3×Q8—81no—
178C3×Q8—81no—
179C3×D4—42no—
180C3×D4—42no—
181C3×D4—81no—
182C3×D4—81no—
183C3×D4—81no—
184C3×D4—81no—
185C3×D4—82no—
186C3×D4—82no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—41no—
194C2×C3×C4—82no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C3×C8—22no—
205C3×C8—41no—
206C3⋊C8—42no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209#41—81no—
210#41—81no—
211#41—82no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4#21712yes—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—121no—
225C22×C4D3×D411yescharacteristic
226C22×C4#23711yescharacteristic
227C22×C4C2×#6711yescharacteristic
228D8—44no—
229M16—42no—
230M16—242no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—121no—
236C2×C8—242no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4#21712yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C42D3×D411yescharacteristic
246C42#23711yescharacteristic
247C42C22×Dic311yescharacteristic
248C42C2×#6711yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3#38611yescharacteristic
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—81no—
267Dic3—161no—
268C23—22no—
269C23—41no—
270C23#93911yescharacteristic
271Q8—81no—
272Q8—81no—
273D4—42no—
274D4—42no—
275D4—81no—
276D4—81no—
277D4—81no—
278D4—81no—
279D4—82no—
280D4—82no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—241no—
290C2×C4—241no—
291C2×C4—241no—
292C2×C4D4×Dic311yescharacteristic
293C2×C4D4×Dic311yescharacteristic
294C2×C4#93511yescharacteristic
295C2×C4#93711yescharacteristic
296C2×C4#93911yescharacteristic
297C2×C4C2×#21711yescharacteristic
298C8—22no—
299C8—41no—
300C8—122no—
301C8—122no—
302C8—241no—
303C2×C3—41no—
304C2×C3—82no—
305C2×C3—161no—
306C2×C3#188411yescharacteristic
307C2×C3#190311yescharacteristic
308C2×C3#190411yescharacteristic
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#522011yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—481no—
328C3#941711yescharacteristic, complemented, Sylow
329C2—41no—
330C2—82no—
331C2—161no—
332C2#7181711yescharacteristic
333C2#7208911yescharacteristic
334C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3184C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3184D3
hypocenterC3#9417
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3184

Complements

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

NH ≅ G/Ntype
2. #83279C2semidirect
3. #71813C2semidirect
4. #71813C2semidirect
5. #71799C2semidirect
6. #67670C2semidirect
7. #67666C2semidirect
11. C3×#498C4semidirect
12. C3×#493C4semidirect
13. C3×#486C4semidirect
14. C3×#486C4semidirect
77. #498Dic3semidirect
78. #493Dic3semidirect
79. #486Dic3semidirect
80. #486Dic3semidirect
328. C3#9417semidirect

Maximal subgroups

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

subgroupindex
C3×#31842
#832792
#718132
#717992
#676702
#676662
#94173