Finite Group Databasebeta

← #10696775

Subgroups — #10696775

Order 768.

442 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 79 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#83021C211yesmaximal, characteristic, complemented
2#83012C211yesmaximal, characteristic
3#79319C211yesmaximal, characteristic
4C3×#1470C211yesmaximal, characteristic, complemented
5#70769C211yesmaximal, characteristic, complemented
6#70765C211yesmaximal, characteristic
7#67754C211yesmaximal, characteristic, complemented
8#10123—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5069—21no—
15#5069—21no—
16#5068—22no—
17#5067—21no—
18#5064—21no—
19#5063—22no—
20#4971—21no—
21#4970—21no—
22#4966—21no—
23#4966—21no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4846—21no—
27#4842—21no—
28#4737—22no—
29#4713—21no—
30#4702—21no—
31C3×#324—22no—
32#4484—22no—
33#4459—22no—
34#3236—31no—
35#3235—31no—
36#3142—31no—
37#1496—31no—
38#1485—31no—
39#1470D311yescharacteristic, complemented
40#1291—31no—
41C2×#235—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57#875—21no—
58#874D411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#97—21no—
72C3×C4×C8—21no—
73#830—21no—
74#821—22no—
75#821—22no—
76#818—22no—
77#817—22no—
78#810—22no—
79#810—22no—
80#810—22no—
81#810—22no—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#532—31no—
92C2×#125C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#491—61no—
96#491—61no—
97#490—61no—
98#490—61no—
99#488—62no—
100#484—61no—
101#483—61no—
102#482—62no—
103#479—62no—
104#476—61no—
105#474—61no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#417—61no—
110#417—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—22no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—41no—
121#235—44no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155#208—42no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×#41—61no—
159C2×M16—61no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163#127—31no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—61no—
169#124—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—61no—
173#121—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—61no—
177C4×D4—61no—
178#117—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4D1211yescharacteristic
182C2×C4⋊C4D1211yescharacteristic
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—62no—
186#106—62no—
187#106—62no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—32no—
191#103—62no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8#6712yes—
202#97—21no—
203#97—61no—
204#97—61no—
205C4×C8—21no—
206C4×C8—32no—
207C23×C3#12111yescharacteristic
208C22×D3—41no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222#67—82no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226D12—42no—
227D12—42no—
228D12—42no—
229C4×D3—82no—
230Dic6—42no—
231Dic6—42no—
232Dic6—42no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237#41—124no—
238C2×Q8—61no—
239C2×Q8—61no—
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×C4C2×D1211yescharacteristic
249C22×C4#23511yescharacteristic
250C22×C4C2×#6711yescharacteristic
251M16—122no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42—121no—
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
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#45412yes—
295Dic3—81no—
296Dic3—81no—
297Dic3—81no—
298C23—121no—
299C23#93011yescharacteristic
300Q8—122no—
301Q8—122no—
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—81no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—242no—
318C2×C4#88211yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#93711yescharacteristic
323C2×C4#94011yescharacteristic
324C8—42no—
325C8—81no—
326C8—122no—
327C8—122no—
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#496011yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4#475312yes—
351C3#1012311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—481no—
354C2#7001311yescharacteristic
355C2#7005311yescharacteristic
356C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1470C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1470D3
hypocenterC3#10123
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

Complements

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

NH ≅ G/Ntype
1. #83021C2semidirect
4. C3×#1470C2semidirect
5. #70769C2semidirect
7. #67754C2semidirect
39. #1470D3semidirect
351. C3#10123semidirect

Maximal subgroups

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

subgroupindex
#830212
#830122
#793192
C3×#14702
#707692
#707652
#677542
#101233