Finite Group Databasebeta

← #10691776

Subgroups — #10691776

Order 768.

422 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3228C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic, complemented
4#71827C211yesmaximal, characteristic, complemented
5#71812C211yesmaximal, characteristic
6#67747C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9279—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#500C411yescharacteristic, complemented
12C3×#498C411yescharacteristic, complemented
13C3×#486C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#482—22no—
16C3×#478—22no—
17C3×#431C2211yescharacteristic
18#5234—22no—
19#5012—21no—
20#5011—22no—
21D4×C3⋊C8—21no—
22#4932C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—22no—
27#4537—22no—
28#4535—21no—
29#4535—21no—
30#3228D311yescharacteristic
31#2981—31no—
32#1506—31no—
33#1505—31no—
34#1497—31no—
35#1287—31no—
36#1274—31no—
37C2×C3×#41—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#125—21no—
43C3×#124—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×#121—21no—
47C3×#121D412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4D412yes—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C2×C4×Dic3—21no—
55C2×#208—22no—
56C22×C3⋊C8—22no—
57C3×#107—22no—
58C3×#106—22no—
59C3×C4⋊C8—21no—
60C3×#102—22no—
61C3×#102—22no—
62C3×C22⋊C8—21no—
63C3×C4×C8C2×C411yescharacteristic
64C3×C4×C8C2×C411yescharacteristic
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—22no—
68#830—22no—
69#830—21no—
70#830—21no—
71#830D412yes—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533C2×D311yescharacteristic
80#532C2×D311yescharacteristic
81#500Dic311yescharacteristic, complemented
82#498Dic311yescharacteristic, complemented
83#486Dic311yescharacteristic, complemented
84#486Dic311yescharacteristic, complemented
85#482—22no—
86#478—22no—
87#436—61no—
88#435—62no—
89C8×D4—61no—
90#432—31no—
91#432—31no—
92#431—31no—
93#431C2×D311yescharacteristic
94#408—62no—
95#376—31no—
96#328—62no—
97#327—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C3×#41—82no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×D4—22no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—41no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4C22⋊C412yes—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—41no—
126C3×C42—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C4×Dic3—22no—
132C4×Dic3—41no—
133#208—82no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—82no—
139C2×#41—21no—
140C2×M16—62no—
141C22×C8—62no—
142C4⋊Q8C2×Dic311yescharacteristic
143C4⋊Q8C2×Dic311yescharacteristic
144C4⋊Q8C2×Dic311yescharacteristic
145#127C2×Dic311yescharacteristic
146#125—21no—
147#124—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150#121—21no—
151#121#6712yes—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—21no—
155C4×D4#6712yes—
156#117—21no—
157C2×C42—61no—
158C2×C42C22×D311yescharacteristic
159#107—22no—
160#106—22no—
161C4⋊C8—21no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#102—22no—
166#102—22no—
167C22⋊C8—21no—
168C22⋊C8—32no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#97—61no—
174#97—61no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8C2×Dic311yescharacteristic
178C4×C8C2×Dic311yescharacteristic
179#95—61no—
180#95—61no—
181C23×C3—22no—
182C23×C3#12111yescharacteristic
183C3×Q8—41no—
184C3×Q8—41no—
185C3×Q8—81no—
186C3×Q8—81no—
187C3×D4—42no—
188C3×D4—41no—
189C3×D4—41no—
190C3×D4—81no—
191C3×D4—82no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12311yescharacteristic
207C2×Dic3—42no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C3×C8—41no—
211C3×C8—41no—
212C3⋊C8—41no—
213C3⋊C8—42no—
214C3⋊C8—41no—
215C3⋊C8—81no—
216#41—82no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4#21712yes—
224C22×C4—22no—
225C22×C4—121no—
226C22×C4D3×D411yescharacteristic
227C22×C4#23711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—242no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—62no—
235C2×C8—121no—
236C2×C8—242no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4#21712yes—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—41no—
248C42—41no—
249C42—62no—
250C42—121no—
251C42D3×D411yescharacteristic
252C42#23711yescharacteristic
253C42C22×Dic311yescharacteristic
254C42C2×#6711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3#38611yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—42no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4#44311yescharacteristic
271C3×C4#44411yescharacteristic
272Dic3—82no—
273C23—22no—
274C23#93911yescharacteristic
275Q8—41no—
276Q8—41no—
277Q8—81no—
278Q8—81no—
279D4—42no—
280D4—41no—
281D4—41no—
282D4—81no—
283D4—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—122no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4D4×Dic311yescharacteristic
297C2×C4D4×Dic311yescharacteristic
298C2×C4#93511yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#93911yescharacteristic
301C2×C4C2×#21711yescharacteristic
302C8—41no—
303C8—41no—
304C8—121no—
305C8—122no—
306C8—121no—
307C8—241no—
308C2×C3—41no—
309C2×C3—82no—
310C2×C3#187611yescharacteristic
311C2×C3#190411yescharacteristic
312C2×C3#190911yescharacteristic
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22#522011yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—81no—
328C4—242no—
329C4#519611yescharacteristic
330C4#519711yescharacteristic
331C3#927911yescharacteristic, complemented, Sylow
332C2—41no—
333C2—82no—
334C2#7186511yescharacteristic
335C2#7208711yescharacteristic
336C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3228C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3228D3
hypocenterC3#9279
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
3. #71836C2semidirect
4. #71827C2semidirect
6. #67747C2semidirect
7. #67737C2semidirect
11. C3×#500C4semidirect
12. C3×#498C4semidirect
13. C3×#486C4semidirect
14. C3×#486C4semidirect
81. #500Dic3semidirect
82. #498Dic3semidirect
83. #486Dic3semidirect
84. #486Dic3semidirect
331. C3#9279semidirect

Maximal subgroups

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

subgroupindex
C3×#32282
#832802
#718362
#718272
#718122
#677472
#677372
#92793