Finite Group Databasebeta

← #10696550

Subgroups — #10696550

Order 768.

560 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2660C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic
3#83019C211yesmaximal, characteristic
4#71777C211yesmaximal, characteristic, complemented
5#71735C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic, complemented
8#10091—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#942—22no—
12#5477C2211yescharacteristic
13C3×#398—22no—
14C3×#395—22no—
15C3×#384—21no—
16C3×#382—21no—
17C3×#376C2211yescharacteristic
18#5092—21no—
19#5086—22no—
20#5085—21no—
21#5067—21no—
22#5066—22no—
23#5064—21no—
24#4968—21no—
25#4967—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4546—22no—
30#4543—22no—
31#4451—22no—
32#3234—31no—
33#3152—31no—
34#2660D311yescharacteristic, complemented
35#1558—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8D412yes—
40C2×#235—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59#942—48no—
60C2×#215—24no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×#95—24no—
70C3×#95—21no—
71C3×#95D412yes—
72#830—21no—
73#830—21no—
74#809—22no—
75#809—22no—
76#808—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×#125C2×D311yescharacteristic
88#501—61no—
89#497—62no—
90#494—61no—
91#492—61no—
92#490—61no—
93#481—61no—
94#480—61no—
95#478—62no—
96#432—31no—
97#432—31no—
98C4×M16—31no—
99#398—22no—
100#395—22no—
101#384—21no—
102#382—21no—
103#376C2×D311yescharacteristic
104#333—62no—
105#330—62no—
106#327—62no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—44no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118#235—44no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—84no—
137C3×C42—24no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#216—48no—
143#215—48no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148#208—24no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×#41—61no—
153C22×Q8#6712yes—
154#137—128no—
155C2×Q16—64no—
156C2×SD16—64no—
157C2×M16—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163#127—31no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186#102—62no—
187#102—62no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—32no—
191#97—61no—
192#97—61no—
193C4×C8—61no—
194#95—24no—
195#95—21no—
196#95#6712yes—
197C23×C3#12011yescharacteristic
198C22×D3—41no—
199C3×Q8—44no—
200C3×Q8—82no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—24no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—44no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219#67—82no—
220C2×Dic3—41no—
221D12—42no—
222D12—42no—
223D12—81no—
224C4×D3—82no—
225Dic6—42no—
226C3⋊C8—24no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229#41—124no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—44no—
233C2×Q8—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—121no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—121no—
243C22×C4C2×#6711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245C22×C4C2×#6711yescharacteristic
246Q16—128no—
247SD16—128no—
248M16—64no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—84no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—24no—
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#45312yes—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—44no—
287C3×C4—81no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4#45612yes—
291Dic3—81no—
292C23—121no—
293C23#95311yescharacteristic
294Q8—44no—
295Q8—82no—
296Q8—122no—
297D4—122no—
298D4—122no—
299D4—241no—
300D4—242no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—121no—
314C2×C4—242no—
315C2×C4#93711yescharacteristic
316C2×C4#93711yescharacteristic
317C2×C4#93811yescharacteristic
318C2×C4#93911yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C8—64no—
322C8—122no—
323C8—241no—
324C2×C3—22no—
325C2×C3#200211yescharacteristic
326C2×C3#200511yescharacteristic
327C2×C3#200911yescharacteristic
328D3—161no—
329C22—21no—
330C22—21no—
331C22—241no—
332C22—241no—
333C22—241no—
334C22#522311yescharacteristic
335C22#524112yes—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—44no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—241no—
346C4#509512yes—
347C3#1009111yescharacteristic, complemented, Sylow
348C2—22no—
349C2—481no—
350C2#7179511yescharacteristic
351C2#7185611yescharacteristic
352C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2660C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2660D3
hypocenterC3#10091
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2660

Complements

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

NH ≅ G/Ntype
1. C3×#2660C2semidirect
4. #71777C2semidirect
5. #71735C2semidirect
7. #67615C2semidirect
34. #2660D3semidirect
347. C3#10091semidirect

Maximal subgroups

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

subgroupindex
C3×#26602
#832032
#830192
#717772
#717352
#694322
#676152
#100913