Finite Group Databasebeta

← #10697474

Subgroups — #10697474

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#83335C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#79268C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic, complemented
5#69552C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic, complemented
8#10226—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5089—21no—
15#5089—21no—
16#5080—21no—
17#5079—21no—
18#5069—21no—
19#5067—21no—
20#5043—22no—
21#5020—22no—
22#5002—22no—
23C2×#810—22no—
24#4925C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4846—22no—
27#4736—22no—
28#4694—21no—
29#4682—21no—
30C3×#328—22no—
31#4451—22no—
32#3229—31no—
33#3226—31no—
34#3135—31no—
35#1960—31no—
36#1518D311yescharacteristic, complemented
37#1517—31no—
38#1281—31no—
39C2×#235—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C22×C3⋊C8D412yes—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D412yes—
65C3×C4⋊C8—21no—
66C3×C22⋊C8D412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95—21no—
70#835—24no—
71#830D412yes—
72#821—24no—
73#818—24no—
74#810—22no—
75#810—22no—
76#810—44no—
77#809—22no—
78#809—22no—
79#808—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#532—31no—
86C2×C4⋊Q8—31no—
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#431C2×D311yescharacteristic
101#416—62no—
102C2×#103—62no—
103C2×#97—31no—
104#382C2×D311yescharacteristic
105#328—22no—
106#327—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—41no—
115#235—42no—
116#235—42no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—42no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×#41—61no—
153C22×Q8—61no—
154C22×C8—32no—
155C4⋊Q8—32no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8D1211yescharacteristic
161C4⋊Q8D1211yescharacteristic
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—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#106—64no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182#103—62no—
183#103—62no—
184#103—64no—
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—21no—
196#95—21no—
197C23×C3#12711yescharacteristic
198C22×D3—41no—
199C3×Q8—42no—
200C3×Q8—81no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#13512yes—
215C2×C3×C4AGL(1,ℤ/8)12yes—
216#67—81no—
217#67—81no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222D12—42no—
223D12—42no—
224D12—81no—
225C4×D3—82no—
226Dic6—42no—
227Dic6—44no—
228Dic6—82no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232#41—122no—
233#41—122no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—61no—
237C2×Q8—62no—
238C2×Q8—121no—
239C2×Q8—122no—
240C2×Q8—122no—
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—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—41no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—41no—
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—81no—
290C3×C4—81no—
291C3×C4—161no—
292C3×C4#44912yes—
293C3×C4#46312yes—
294Dic3—82no—
295Dic3—81no—
296C23—121no—
297C23#94011yescharacteristic
298Q8—42no—
299Q8—81no—
300Q8—122no—
301Q8—124no—
302Q8—242no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307D4—241no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—242no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#88411yescharacteristic
324C2×C4#91412yes—
325C2×C4#91912yes—
326C2×C4#93011yescharacteristic
327C2×C4#93711yescharacteristic
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#517611yescharacteristic
344C22#519111yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—81no—
350C4—81no—
351C4—161no—
352C4—241no—
353C4—242no—
354C4#481012yes—
355C4#481912yes—
356C3#1022611yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#7005311yescharacteristic
361C2#7027711yescharacteristic
362C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1518D3
hypocenterC3#10226
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #83323C2semidirect
4. C3×#1518C2semidirect
5. #69552C2semidirect
7. #67741C2semidirect
36. #1518D3semidirect
356. C3#10226semidirect

Maximal subgroups

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

subgroupindex
#833352
#833232
#792682
C3×#15182
#695522
#695452
#677412
#102263