Finite Group Databasebeta

← #10696774

Subgroups — #10696774

Order 768.

442 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3236C211yesmaximal, characteristic, complemented
2#83361C211yesmaximal, characteristic
3#83012C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic, complemented
5#71737C211yesmaximal, characteristic, complemented
6#70801C211yesmaximal, characteristic
7#67753C211yesmaximal, characteristic, complemented
8#10123—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#501—21no—
13C3×#491—21no—
14C3×#484—21no—
15C3×#483—21no—
16C3×#479—22no—
17C3×#432C2211yescharacteristic
18#5087—21no—
19#5084—21no—
20#5069—21no—
21#5068—22no—
22#5067—21no—
23#5067—21no—
24#5064—21no—
25#5003—22no—
26#4966—21no—
27#4966—21no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32#4528—22no—
33#4483—22no—
34#3236D311yescharacteristic, complemented
35#3235—31no—
36#3142—31no—
37#1496—31no—
38#1485—31no—
39#1470—31no—
40#1291—31no—
41C2×#235—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59#930—21no—
60#930—21no—
61C2×#208—21no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67C3×#107—22no—
68C3×#106—22no—
69C3×C4⋊C8D412yes—
70C3×#103—22no—
71C3×#103—22no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74#830—21no—
75#830D412yes—
76#810—22no—
77#810—22no—
78#809—22no—
79#809—22no—
80#809—22no—
81#809—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91#532—31no—
92C2×#125C2×D311yescharacteristic
93#501—21no—
94#501—61no—
95#491—21no—
96#491—61no—
97#490—61no—
98#490—61no—
99#488—62no—
100#484—21no—
101#483—21no—
102#482—62no—
103#479—22no—
104#476—61no—
105#474—61no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431—31no—
109#417—61no—
110#417—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—41no—
123#235—44no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154#208—42no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×#41—61no—
159C2×M16—61no—
160C4⋊Q8—31no—
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163#127—31no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#124—61no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172#121—61no—
173#121—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×D4—61no—
177C4×D4—61no—
178#117—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6711yescharacteristic
183C2×C42C22×D311yescharacteristic
184#107—22no—
185#107—62no—
186#106—22no—
187#106—62no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—22no—
192#103—22no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—32no—
201C22⋊C8—61no—
202#97—21no—
203#97—61no—
204#97—61no—
205C4×C8—32no—
206C4×C8—61no—
207C23×C3#12111yescharacteristic
208C22×D3—41no—
209C3×Q8—42no—
210C3×Q8—42no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—81no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226#67—82no—
227C2×Dic3—41no—
228D12—42no—
229D12—42no—
230D12—42no—
231C4×D3—82no—
232Dic6—42no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237#41—124no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—121no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4#23711yescharacteristic
250C22×C4C2×#6711yescharacteristic
251M16—122no—
252C2×C8—22no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C22⋊C4—41no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42—41no—
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42C2×#6711yescharacteristic
279C42C2×#6711yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#39211yescharacteristic
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4#45412yes—
297Dic3—81no—
298C23—121no—
299C23#93911yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—122no—
303D4—122no—
304D4—122no—
305D4—122no—
306D4—242no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—121no—
317C2×C4—242no—
318C2×C4#93711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#94011yescharacteristic
323C2×C4#95311yescharacteristic
324C8—42no—
325C8—122no—
326C8—122no—
327C8—241no—
328C2×C3—41no—
329C2×C3#200311yescharacteristic
330C2×C3#200911yescharacteristic
331C2×C3#201711yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—241no—
350C4#524512yes—
351C3#1012311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—481no—
354C2#7186711yescharacteristic
355C2#7211111yescharacteristic
356C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3236C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3236D3
hypocenterC3#10123
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

Complements

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

NH ≅ G/Ntype
1. C3×#3236C2semidirect
4. #71825C2semidirect
5. #71737C2semidirect
7. #67753C2semidirect
34. #3236D3semidirect
351. C3#10123semidirect

Maximal subgroups

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

subgroupindex
C3×#32362
#833612
#830122
#718252
#717372
#708012
#677532
#101233