Finite Group Databasebeta

← #10696769

Subgroups — #10696769

Order 768.

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

normal subgroups only

This group has 476 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#83307C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic, complemented
3#78529C211yesmaximal, characteristic, complemented
4C3×#1484C211yesmaximal, characteristic, complemented
5#70787C211yesmaximal, characteristic, complemented
6#70765C211yesmaximal, characteristic, complemented
7#67670C211yesmaximal, characteristic, complemented
8#10122—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5238—22no—
15#5133—21no—
16#5133—21no—
17#5083—21no—
18#5081—21no—
19#5072—21no—
20#5071—21no—
21#5058—21no—
22#5055—21no—
23#5034—21no—
24#5033—21no—
25#5009—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4713—21no—
29#4703—21no—
30#4697—22no—
31C3×D4⋊C8—22no—
32#4469—22no—
33#4459—22no—
34#3234—31no—
35#3214—31no—
36#3142—31no—
37#1496—31no—
38#1484D311yescharacteristic, complemented
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×#235—21no—
43C2×C3×M16—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#215—22no—
53C2×#213—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68#869D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8D412yes—
74#835—22no—
75#832—22no—
76#832—22no—
77#832—22no—
78#830—21no—
79#830D412yes—
80#821—22no—
81#821—22no—
82#816—22no—
83#816—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803—21no—
90#532—31no—
91#532C2×D311yescharacteristic
92C2×#125—31no—
93#501—61no—
94#496—61no—
95#491—61no—
96#490—61no—
97#490—61no—
98#489—61no—
99#488—62no—
100#481—61no—
101#480—61no—
102#478—62no—
103#461—61no—
104#459—61no—
105#449—62no—
106#432—31no—
107#432—31no—
108#431C2×D311yescharacteristic
109#417—61no—
110#417—61no—
111C4×M16C2×D311yescharacteristic
112D4⋊C8—22no—
113D4⋊C8—62no—
114#324—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128#235—44no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C3×M16—42no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#215—82no—
146#213—82no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×#41—21no—
162C2×#41—61no—
163C2×SD16—62no—
164C2×D8—62no—
165C2×M16—21no—
166C4⋊Q8—31no—
167C4⋊Q8D1211yescharacteristic
168#127—31no—
169#127D1211yescharacteristic
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174#124—21no—
175#124—61no—
176#121—21no—
177#121—21no—
178#121—61no—
179#121—61no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194#103—62no—
195#103—62no—
196#103—62no—
197#102—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205#97—21no—
206#97—61no—
207#97—61no—
208C4×C8—21no—
209C4×C8#6712yes—
210C23×C3—41no—
211C23×C3#12111yescharacteristic
212C22×D3—41no—
213C3×Q8—42no—
214C3×D4—42no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—82no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—82no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229#67—82no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—81no—
235D12—41no—
236D12—41no—
237D12—42no—
238D12—42no—
239C4×D3—82no—
240Dic6—42no—
241C3×C8—42no—
242C3×C8—81no—
243C3⋊C8—42no—
244C3⋊C8—42no—
245#41—82no—
246#41—124no—
247C2×Q8—21no—
248C2×Q8—61no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—121no—
257C22×C4—41no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4#23511yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264SD16—242no—
265D8—242no—
266M16—42no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—62no—
270C2×C8—62no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42C2×D1211yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—81no—
307C3×C4#45412yes—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23—41no—
312C23—121no—
313C23#88411yescharacteristic
314Q8—42no—
315Q8—122no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—82no—
320D4—121no—
321D4—121no—
322D4—122no—
323D4—122no—
324D4—242no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—82no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4—241no—
336C2×C4#88211yescharacteristic
337C2×C4#88211yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4#93011yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#94011yescharacteristic
342C8—42no—
343C8—81no—
344C8—122no—
345C8—122no—
346C2×C3—41no—
347C2×C3—161no—
348C2×C3#200311yescharacteristic
349C2×C3#200811yescharacteristic
350C2×C3#201711yescharacteristic
351D3—161no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22#496011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—241no—
369C4—241no—
370C4—241no—
371C4#482712yes—
372C3#1012211yescharacteristic, complemented, Sylow
373C2—41no—
374C2—161no—
375C2—481no—
376C2#7001311yescharacteristic
377C2#7033111yescharacteristic
378C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1484C2
socleC22×C3#392
prime radical#83307C2
hypercenter#1484D3
hypocenterC3#10122
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
2. #83273C2semidirect
3. #78529C2semidirect
4. C3×#1484C2semidirect
5. #70787C2semidirect
6. #70765C2semidirect
7. #67670C2semidirect
38. #1484D3semidirect
372. C3#10122semidirect

Maximal subgroups

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

subgroupindex
#833072
#832732
#785292
C3×#14842
#707872
#707652
#676702
#101223