Finite Group Databasebeta

← #10697470

Subgroups — #10697470

Order 768.

488 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3229C211yesmaximal, characteristic, complemented
2#83335C211yesmaximal, characteristic
3#83215C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71786C211yesmaximal, characteristic, complemented
6#70790C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic, complemented
8#10226—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#498—21no—
13C3×#487—21no—
14C3×#483—22no—
15C3×#475—22no—
16C3×#431C2211yescharacteristic
17C2×#835—22no—
18#5089—21no—
19#5089—21no—
20#5088—21no—
21#5087—22no—
22#5078—21no—
23#5069—21no—
24#5067—21no—
25#5002—22no—
26#4967—22no—
27#4936C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4538—22no—
31#4470—22no—
32#3229D311yescharacteristic, complemented
33#3226—31no—
34#3135—31no—
35#1960—31no—
36#1518—31no—
37#1517—31no—
38#1281—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56#930—21no—
57#930—21no—
58C2×C4⋊Dic3—21no—
59C22×C3⋊C8D412yes—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C3×#106—24no—
66C3×C4⋊C8—21no—
67C3×#103—24no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70#835—44no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73#830D412yes—
74#810—22no—
75#810—22no—
76#809—24no—
77#809—22no—
78#809—22no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87#500—61no—
88#498—21no—
89#491—61no—
90#490—61no—
91#489—61no—
92#489—61no—
93#487—21no—
94#486—61no—
95#483—22no—
96#480—62no—
97#475—22no—
98#474—62no—
99#431—31no—
100#431C2×D311yescharacteristic
101#416—62no—
102C2×#103—62no—
103C2×#97—31no—
104#382—31no—
105#328—62no—
106#327—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118C22×Dic3—41no—
119#235—42no—
120#235—42no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×C4×D3—41no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—42no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×#41—61no—
153C22×Q8—21no—
154C22×C8—32no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8#6712yes—
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162#127—31no—
163#125—21no—
164#124—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167#121—61no—
168#121—61no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×D4—61no—
172C4×D4—61no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#106—24no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182#103—24no—
183#103—62no—
184#103—62no—
185#103—124no—
186#102—62no—
187#102—62no—
188#102—64no—
189C22⋊C8—32no—
190C22⋊C8#6712yes—
191#97—61no—
192#97—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—61no—
196#95—61no—
197C23×C3#12711yescharacteristic
198C22×D3—41no—
199C3×Q8—42no—
200C3×Q8—44no—
201C3×Q8—81no—
202C3×Q8—82no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#13512yes—
217C2×C3×C4AGL(1,ℤ/8)12yes—
218#67—81no—
219#67—81no—
220C2×Dic3—41no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224D12—42no—
225D12—42no—
226D12—81no—
227C4×D3—82no—
228Dic6—42no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232#41—122no—
233#41—122no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—121no—
245C22×C4—41no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4C2×#6711yescharacteristic
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—122no—
269C22⋊C4—41no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—41no—
273C42D3×D411yescharacteristic
274C42#23711yescharacteristic
275C42C2×#6711yescharacteristic
276C42C2×#6711yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3#39211yescharacteristic
280C22×C3#49511yescharacteristic
281C22×C3#49611yescharacteristic
282C2×D3—81no—
283C2×D3—81no—
284C2×D3—81no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4—82no—
292C3×C4#44912yes—
293C3×C4#46312yes—
294Dic3—81no—
295Dic3—161no—
296C23—121no—
297C23#94011yescharacteristic
298Q8—42no—
299Q8—44no—
300Q8—81no—
301Q8—82no—
302Q8—122no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307D4—241no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—121no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#91412yes—
321C2×C4#91612yes—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—122no—
330C8—122no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#200911yescharacteristic
334C2×C3#200911yescharacteristic
335C2×C3#201211yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#516811yescharacteristic
343C22#517611yescharacteristic
344C22#522311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—82no—
350C4—81no—
351C4—81no—
352C4—241no—
353C4—481no—
354C4#516512yes—
355C4#517212yes—
356C3#1022611yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#7186911yescharacteristic
361C2#7205511yescharacteristic
362C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3229D3
hypocenterC3#10226
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C3×#3229C2semidirect
4. #71838C2semidirect
5. #71786C2semidirect
7. #67741C2semidirect
32. #3229D3semidirect
356. C3#10226semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833352
#832152
#718382
#717862
#707902
#677412
#102263