Finite Group Databasebeta

← #10697472

Subgroups — #10697472

Order 768.

488 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 95 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#83223C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#78497C211yesmaximal, characteristic
4C3×#1960C211yesmaximal, characteristic, complemented
5#69546C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic
7#67661C211yesmaximal, characteristic, complemented
8#10226—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×#103—22no—
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5113—22no—
16#5094—22no—
17#5093—21no—
18#5088—21no—
19#5087—22no—
20#5080—21no—
21#5078—21no—
22#4971—22no—
23#4967—22no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4714—21no—
27#4714—21no—
28#4713—21no—
29#4703—21no—
30#4452—22no—
31#4451—22no—
32#3229—31no—
33#3226—31no—
34#3135—31no—
35#1960D311yescharacteristic, complemented
36#1518—31no—
37#1517—31no—
38#1281—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×C3×C8D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#930—21no—
51#930—21no—
52#926—21no—
53#924—21no—
54#924—21no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#103—44no—
66C3×#97—21no—
67C3×#97—21no—
68C3×#95—21no—
69C3×#95—21no—
70#830D412yes—
71#830D412yes—
72#821—22no—
73#821—22no—
74#816—22no—
75#816—22no—
76#810—24no—
77#809—24no—
78#808—24no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87#500—61no—
88#498—61no—
89#491—61no—
90#490—61no—
91#489—61no—
92#489—61no—
93#487—61no—
94#486—61no—
95#483—62no—
96#480—62no—
97#475—62no—
98#474—62no—
99#431—31no—
100#431—31no—
101#416—62no—
102C2×#103—22no—
103C2×#97C2×D311yescharacteristic
104#382C2×D311yescharacteristic
105#328—62no—
106#327—62no—
107C2×C3×Q8—22no—
108C2×C3×Q8—41no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117#235—42no—
118#235—42no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—42no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×#41—61no—
153C22×Q8—21no—
154C22×C8#6712yes—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8D1212yes—
162#127—31no—
163#125—61no—
164#124—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167#121—61no—
168#121—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—61no—
172C4×D4—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#106—64no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#103—44no—
183#103—64no—
184#103—62no—
185#103—62no—
186#102—64no—
187#102—62no—
188#102—62no—
189C22⋊C8—32no—
190C22⋊C8—32no—
191#97—21no—
192#97—21no—
193C4×C8—61no—
194C4×C8—61no—
195#95—21no—
196#95—21no—
197C23×C3#12711yescharacteristic
198C22×D3—41no—
199C3×Q8—44no—
200C3×Q8—82no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#13512yes—
216C2×C3×C4AGL(1,ℤ/8)12yes—
217#67—81no—
218#67—81no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222D12—42no—
223D12—42no—
224D12—81no—
225C4×D3—82no—
226Dic6—42no—
227Dic6—42no—
228Dic6—81no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232#41—122no—
233#41—122no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—121no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4C2×#6711yescharacteristic
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—121no—
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
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—82no—
290C3×C4—161no—
291C3×C4#44912yes—
292C3×C4#46312yes—
293Dic3—81no—
294Dic3—81no—
295Dic3—81no—
296C23—121no—
297C23#94011yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—122no—
301Q8—122no—
302Q8—241no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307D4—241no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—242no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#88411yescharacteristic
324C2×C4#93011yescharacteristic
325C2×C4#93212yes—
326C2×C4#93711yescharacteristic
327C2×C4#95012yes—
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#496011yescharacteristic
343C22#505911yescharacteristic
344C22#507411yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—82no—
350C4—161no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4#501612yes—
355C4#503712yes—
356C3#1022611yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#6965911yescharacteristic
361C2#7035111yescharacteristic
362C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1960D3
hypocenterC3#10226
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. #83223C2semidirect
4. C3×#1960C2semidirect
5. #69546C2semidirect
7. #67661C2semidirect
35. #1960D3semidirect
356. C3#10226semidirect

Maximal subgroups

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

subgroupindex
#832232
#832152
#784972
C3×#19602
#695462
#695452
#676612
#102263