Finite Group Databasebeta

← #10696553

Subgroups — #10696553

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
1#79318C211yesmaximal, characteristic
2#78538C211yesmaximal, characteristic
3#73410C211yesmaximal, characteristic, complemented
4#70824C211yesmaximal, characteristic, complemented
5#69863C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic
7C3×#1300C211yesmaximal, characteristic, complemented
8#10091—31nomaximal, Sylow
9C2×#909—22no—
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C3×#376C2211yescharacteristic
16#4936—21no—
17#4935—21no—
18#4926C2211yescharacteristic
19#4847—22no—
20#4736—21no—
21#4735—21no—
22#4715—22no—
23#4713—21no—
24#4702—21no—
25#4699—21no—
26#4693—21no—
27#4661—22no—
28#4656—22no—
29#4481—22no—
30#4465—22no—
31#4451—22no—
32#3234—31no—
33#3152—31no—
34#2660—31no—
35#1558—31no—
36#1528—31no—
37#1505—31no—
38#1300D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6D412yes—
41C2×C3×M16—21no—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—24no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53#909—48no—
54C2×Dic12—24no—
55C2×#204—24no—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D411yescharacteristic
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×C4×C8—21no—
74C3×#95—21no—
75C3×#95D412yes—
76C3⋊#95—24no—
77#830—21no—
78#821—22no—
79#821—22no—
80#818—22no—
81#817—22no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84C2×#137—62no—
85#532—31no—
86C2×C4⋊Q8—31no—
87C2×#125—31no—
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#376C2×D311yescharacteristic
104#333—62no—
105#330—62no—
106#327—62no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C2×#67—41no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—21no—
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—22no—
127C3×C42—24no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—84no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C2×C3⋊C8—41no—
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—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—64no—
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—61no—
185C4⋊C8D1212yes—
186#102—62no—
187#102—62no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190#97—21no—
191#97—61no—
192#97D1212yes—
193C4×C8—21no—
194#95—21no—
195#95—64no—
196#95D1212yes—
197C23×C3#12011yescharacteristic
198C22×D3—41no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12711yescharacteristic
211#67—82no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
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—42no—
228C3⋊C8—81no—
229#41—124no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×Q8—124no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—121no—
238C22×C4—22no—
239C22×C4—61no—
240C22×C4—62no—
241C22×C4—61no—
242C22×C4—121no—
243C22×C4C2×D1211yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246Q16—128no—
247SD16—128no—
248M16—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—121no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—244no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—24no—
270C42C2×D1211yescharacteristic
271C42C2×D1211yescharacteristic
272C42D3×D411yescharacteristic
273C42#23911yescharacteristic
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#45612yes—
288Dic3—82no—
289Dic3—81no—
290Dic3—81no—
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—24no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—122no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—124no—
314C2×C4—242no—
315C2×C4#87411yescharacteristic
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#89511yescharacteristic
319C2×C4#89511yescharacteristic
320C2×C4#89511yescharacteristic
321C8—24no—
322C8—42no—
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#467111yescharacteristic
335C22#482212yes—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—44no—
342C4—242no—
343C4—241no—
344C4—241no—
345C4—241no—
346C4#482612yes—
347C3#1009111yescharacteristic, complemented, Sylow
348C2—22no—
349C2—481no—
350C2#6876111yescharacteristic
351C2#6876811yescharacteristic
352C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1300D3
hypocenterC3#10091
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #73410C2semidirect
4. #70824C2semidirect
5. #69863C2semidirect
7. C3×#1300C2semidirect
38. #1300D3semidirect
347. C3#10091semidirect

Maximal subgroups

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

subgroupindex
#793182
#785382
#734102
#708242
#698632
#694322
C3×#13002
#100913