Finite Group Databasebeta

← #10707772

Subgroups — #10707772

Order 768.

382 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 57 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#78207C211yesmaximal, characteristic, complemented
2#69718C211yesmaximal, characteristic
3#68977C211yesmaximal, characteristic
4#68133C211yesmaximal, characteristic
5C3×#1353C211yesmaximal, characteristic, complemented
6#67494C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#12182—31nomaximal, Sylow
9#5510C2211yescharacteristic
10C3×#400C2211yescharacteristic
11C2×#825—21no—
12#5133—21no—
13#4979C2211yescharacteristic
14#4977C2211yescharacteristic
15#4976—21no—
16#4968—21no—
17#4738—21no—
18#4657—21no—
19#4652—21no—
20#4652—21no—
21#4648—21no—
22#4646—21no—
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4528C2211yescharacteristic
26#4470—21no—
27#4451—21no—
28#2612—31no—
29#2040—31no—
30#1616—31no—
31#1604—31no—
32#1354—31no—
33#1353D311yescharacteristic, complemented
34#1324—31no—
35C2×#235—21no—
36C22×Dic6D411yescharacteristic
37C2×C3×M16—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40#930—21no—
41#930—21no—
42C2×C4⋊Dic3D411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48C2×C4×Dic3—21no—
49C2×C4×Dic3D411yescharacteristic
50C2×#204—41no—
51#897—41no—
52#895—41no—
53C3×C22⋊C8—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95D411yescharacteristic
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D411yescharacteristic
64#830—21no—
65#825—42no—
66#821—41no—
67#816—41no—
68#816—41no—
69#809—41no—
70#808—41no—
71#537—31no—
72#481—61no—
73#448—61no—
74C2×#104—61no—
75#417—61no—
76#400C2×D311yescharacteristic
77#398—61no—
78#395—31no—
79#391—61no—
80#389—61no—
81#387—61no—
82#387—61no—
83#382—31no—
84#376—61no—
85#343C2×D311yescharacteristic
86#327—61no—
87#327—61no—
88#324—31no—
89#324C2×D311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C2×#67—41no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100#235—41no—
101#235—41no—
102#235—41no—
103#235—41no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×C4×D3—41no—
108C2×Dic6—21no—
109C2×Dic6—41no—
110C2×Dic6—42no—
111C3×M16—41no—
112C3×M16—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—41no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124Dic3⋊C4—42no—
125Dic3⋊C4—81no—
126C4×Dic3—21no—
127C4×Dic3—42no—
128C4×Dic3—42no—
129C4×Dic3SD1612yescomplemented
130C2×C3⋊C8—41no—
131#204—82no—
132C2×#41—61no—
133C22×Q8—31no—
134C2×SD16—121no—
135C2×M16—21no—
136C22⋊Q8—121no—
137#121—61no—
138#121—61no—
139#121—121no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4D1211yescharacteristic
146C2×C4⋊C4C22×D311yescharacteristic
147C2×C42—31no—
148C2×C42—61no—
149C2×C42—61no—
150#106—121no—
151#104—122no—
152#103—121no—
153#103—121no—
154#102—121no—
155#102—121no—
156C22⋊C8—21no—
157C22⋊C8—61no—
158#95—21no—
159#95—21no—
160#95—31no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95D1211yescharacteristic
168C23×C3#12011yescharacteristic
169C22×D3—41no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×SD1611yescharacteristic
183C2×C3×C4#13711yescharacteristic
184#67—81no—
185#67—81no—
186C2×Dic3—22no—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199D12—41no—
200D12—41no—
201D12—81no—
202D12—81no—
203C4×D3—81no—
204C4×D3—81no—
205Dic6—42no—
206Dic6—41no—
207Dic6—41no—
208Dic6—81no—
209C3×C8—81no—
210C3×C8—81no—
211C3⋊C8—81no—
212#41—121no—
213#41—121no—
214#41—121no—
215#41—121no—
216C2×Q8—61no—
217C2×Q8—121no—
218C2×Q8—122no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—121no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—121no—
230C22×C4C2×D1211yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233SD16—242no—
234M16—41no—
235M16—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—121no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—122no—
245C4⋊C4—122no—
246C4⋊C4—122no—
247C4⋊C4—241no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C42—32no—
251C42—61no—
252C42—122no—
253C42—122no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#44811yescharacteristic
257C22×C3#45411yescharacteristic
258C22×C3B(4,𝔽2)11yescharacteristic
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—81no—
268Dic3—42no—
269Dic3—42no—
270Dic3—42no—
271Dic3—81no—
272Dic3—81no—
273C23—121no—
274C23#88211yescharacteristic
275Q8—121no—
276Q8—121no—
277Q8—122no—
278Q8—241no—
279D4—121no—
280D4—121no—
281D4—241no—
282D4—241no—
283D4—241no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—241no—
311C2×C4—241no—
312C2×C4D3×SD1611yescharacteristic
313C2×C4#91811yescharacteristic
314C8—81no—
315C8—81no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3#159911yescharacteristic
320C2×C3#219711yescharacteristic
321C2×C3#220411yescharacteristic
322D3—161no—
323C22—21no—
324C22—21no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#474111yescharacteristic
329C22#480611yescharacteristic
330C22#482711yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—122no—
338C4—122no—
339C4—122no—
340C4—241no—
341C4—241no—
342C3#1218211yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—481no—
346C2#6832511yescharacteristic
347C2#6913811yescharacteristic
348C2#6916111yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3B(4,𝔽2)
prime radical#5510C22
hypercenter#1353D3
hypocenterC3#12182
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
1. #78207C2semidirect
5. C3×#1353C2semidirect
6. #67494C2semidirect
7. #67452C2semidirect
33. #1353D3semidirect
129. C4×Dic3SD16semidirect
342. C3#12182semidirect

Maximal subgroups

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

subgroupindex
#782072
#697182
#689772
#681332
C3×#13532
#674942
#674522
#121823