Finite Group Databasebeta

← #10696552

Subgroups — #10696552

Order 768.

560 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 83 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#83044C211yesmaximal, characteristic, complemented
2#83019C211yesmaximal, characteristic
3#78538C211yesmaximal, characteristic
4C3×#1528C211yesmaximal, characteristic, complemented
5#70832C211yesmaximal, characteristic
6#70824C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10091—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#909—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5067—21no—
16#5066—22no—
17#5064—21no—
18#4981—22no—
19#4979—22no—
20#4977—21no—
21#4975—21no—
22#4968—21no—
23#4967—21no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26#4715—22no—
27#4699—21no—
28#4693—21no—
29C3×#330—22no—
30#4481—22no—
31#4470—22no—
32#3234—31no—
33#3152—31no—
34#2660—31no—
35#1558—31no—
36#1528D311yescharacteristic, complemented
37#1505—31no—
38#1300—31no—
39C2×#235—21no—
40C22×Dic6D412yes—
41C2×C3×M16—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×Dic3⋊C4—24no—
56C2×C4×Dic3—21no—
57#909—48no—
58C2×Dic12—24no—
59C2×#204—24no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×C4×C8—21no—
74C3⋊#95—24no—
75C3⋊#95—21no—
76C3⋊#95D412yes—
77#830—21no—
78#809—22no—
79#809—22no—
80#808—22no—
81#807—22no—
82C4⋊(C3⋊C8)D412yes—
83#803—21no—
84C2×#137—62no—
85#532—31no—
86C2×C4⋊Q8—31no—
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#432C2×D311yescharacteristic
98C4×M16C2×D311yescharacteristic
99#398—62no—
100#395—62no—
101#384—61no—
102#382—61no—
103#376—31no—
104#333—62no—
105#330—22no—
106#327—62no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C22×Dic3—22no—
114C22×Dic3—22no—
115#235—44no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—44no—
124C3×M16—24no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
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—41no—
134C3×C4⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147Dic3⋊C4—84no—
148C4×Dic3—24no—
149C2×C3⋊C8—22no—
150Dic12—48no—
151#204—48no—
152C2×#41—61no—
153C22×Q8—32no—
154#137—128no—
155C2×Q16—64no—
156C2×SD16—64no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
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—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—64no—
178C2×C4⋊C4D1211yescharacteristic
179C2×C4⋊C4D1211yescharacteristic
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—32no—
186#102—62no—
187#102—62no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190#97—21no—
191#97—61no—
192#97#6712yes—
193C4×C8—21no—
194#95—32no—
195#95—64no—
196#95—61no—
197C23×C3#12011yescharacteristic
198C22×D3—41no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12711yescharacteristic
212#67—82no—
213C2×Dic3—24no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—44no—
219D12—42no—
220D12—42no—
221D12—81no—
222C4×D3—82no—
223Dic6—44no—
224Dic6—42no—
225Dic6—82no—
226C3×C8—24no—
227C3×C8—81no—
228C3⋊C8—42no—
229#41—124no—
230C2×Q8—62no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—124no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—121no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—121no—
243C22×C4C2×D1211yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246Q16—128no—
247SD16—128no—
248M16—24no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—62no—
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—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—244no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—64no—
270C42#23511yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42C2×#6711yescharacteristic
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—81no—
287C3×C4—81no—
288C3×C4#45612yes—
289Dic3—44no—
290Dic3—82no—
291Dic3—81no—
292C23—121no—
293C23#88211yescharacteristic
294Q8—124no—
295Q8—122no—
296Q8—242no—
297D4—122no—
298D4—122no—
299D4—241no—
300D4—242no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—64no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—122no—
313C2×C4—124no—
314C2×C4—242no—
315C2×C4#88211yescharacteristic
316C2×C4#88411yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4#93011yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#94011yescharacteristic
321C8—24no—
322C8—81no—
323C8—122no—
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#482212yes—
335C22#496011yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—124no—
344C4—242no—
345C4—241no—
346C4#475212yes—
347C3#1009111yescharacteristic, complemented, Sylow
348C2—22no—
349C2—481no—
350C2#7011911yescharacteristic
351C2#7038511yescharacteristic
352C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1528C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1528D3
hypocenterC3#10091
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
1. #83044C2semidirect
4. C3×#1528C2semidirect
6. #70824C2semidirect
7. #67665C2semidirect
36. #1528D3semidirect
347. C3#10091semidirect

Maximal subgroups

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

subgroupindex
#830442
#830192
#785382
C3×#15282
#708322
#708242
#676652
#100913