Finite Group Databasebeta

← #10692816

Subgroups — #10692816

Order 768.

576 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2372C211yesmaximal, characteristic, complemented
2#82943C211yesmaximal, characteristic
3#71719C211yesmaximal, characteristic, complemented
4#69432C212yesmaximal, complemented
5#67615C212yesmaximal, complemented
6#9441—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×#942—21no—
9C2×#932—21no—
10#5477C2211yescharacteristic
11C3×#391—21no—
12C3×#388—22no—
13C3×#386—21no—
14C3×C4×C22⋊C4—22no—
15C3×#376C2212yes—
16#5060C412yescomplemented
17#5059—22no—
18#5058C412yescomplemented
19#4926C2212yes—
20C4×#208C2211yescharacteristic
21#4540—22no—
22#4451—22no—
23#4451—22no—
24#3153—31no—
25#2372D311yescharacteristic, complemented
26#1533—31no—
27#1505—32no—
28#1300—32no—
29C22×C3×Q8D411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C23×C3×C4—21no—
32C2×#235—21no—
33C3×#124—22no—
34C3×#124C2×C412yes—
35C3×#124D412yes—
36C2×C3×C4⋊C4—21no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C42—22no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×#216—22no—
44#942—42no—
45#942—42no—
46C2×#215—22no—
47C2×#214—22no—
48#932—42no—
49#932—42no—
50C2×#213—22no—
51#930—22no—
52C2×#208—21no—
53#875—21no—
54#874C2×C411yescharacteristic, complemented
55C4×D12—22no—
56#869C2×C411yescharacteristic, complemented
57C3×#95—22no—
58C3×#95—22no—
59C3×#95D412yes—
60C3×#95D412yes—
61#830—22no—
62C4⋊(C3⋊C8)—22no—
63#803—22no—
64#803C2×C412yescomplemented
65C4×C3⋊C8—21no—
66C2×#137—61no—
67C2×AGL(1,ℤ/8)—61no—
68#532—31no—
69C2×#124C2×D311yescharacteristic
70#497—32no—
71#496—32no—
72#495—62no—
73#432—32no—
74C4×M16—31no—
75#391—21no—
76#388—22no—
77#386—21no—
78C4×C22⋊C4—22no—
79#376C2×D312yes—
80#329—62no—
81#327—62no—
82#327—62no—
83C24×C3—21no—
84C2×C3×Q8—21no—
85C2×C3×Q8—41no—
86C2×C3×Q8—41no—
87C2×C3×Q8C22⋊C412yes—
88C2×C3×D4—21no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4C22⋊C412yes—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—22no—
95C22×C3×C4—44no—
96C22×C3×C4—81no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D412yes—
99C2×#67—41no—
100#235—41no—
101#235—41no—
102#235—41no—
103#235—41no—
104C2×D12—21no—
105C2×D12—22no—
106C2×C4×D3—41no—
107C2×Dic6—21no—
108C3×C4⋊C4—81no—
109C3×C22⋊C4—24no—
110C3×C22⋊C4—24no—
111C3×C22⋊C4—44no—
112C3×C22⋊C4—81no—
113C3×C42—44no—
114C3×C42—81no—
115C3×C42C22×C411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4112yes—
118#216—44no—
119#215—44no—
120#214—44no—
121#213—44no—
122(C2×D3)⋊C4—42no—
123C4⋊Dic3—22no—
124#208—22no—
125#208—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—42no—
128C2×#41—61no—
129C22×Q8#6711yescharacteristic
130C22×D4#6711yescharacteristic
131C23×C4—21no—
132#137—122no—
133#137—122no—
134AGL(1,ℤ/8)—122no—
135AGL(1,ℤ/8)—122no—
136C2×Q16—62no—
137C2×SD16—62no—
138C2×SD16—62no—
139C2×D8—62no—
140C2×M16—61no—
141C4⋊Q8—31no—
142#127—31no—
143#124—22no—
144#124—61no—
145#124C4×D312yes—
146#124D1212yes—
147#121—62no—
148C4×D4—62no—
149C2×C4⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—21no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42C22×D311yescharacteristic
156C4⋊C8—62no—
157C22⋊C8—62no—
158#97—32no—
159#97—62no—
160C4×C8—61no—
161#95—22no—
162#95—22no—
163#95#6712yes—
164#95#6712yes—
165C23×C3—22no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3#12011yescharacteristic
171C22×D3—41no—
172C3×Q8—42no—
173C3×Q8—81no—
174C3×D4—42no—
175C3×D4—81no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—84no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D412yes—
191C2×C3×C4#12112yes—
192C2×C3×C4#12311yescharacteristic
193#67—81no—
194#67—81no—
195C2×Dic3—41no—
196D12—41no—
197D12—41no—
198D12—82no—
199C4×D3—81no—
200C4×D3—81no—
201Dic6—41no—
202Dic6—41no—
203C3⋊C8—24no—
204C3⋊C8—82no—
205C24—21no—
206#41—121no—
207#41—121no—
208#41—121no—
209#41—121no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—61no—
214C2×Q8(C2×D3)⋊C412yes—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—61no—
219C2×D4—62no—
220C2×D4—121no—
221C2×D4(C2×D3)⋊C412yes—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—44no—
226C22×C4—81no—
227C22×C4—121no—
228C22×C4C2×#6711yescharacteristic
229C22×C4C2×#6712yes—
230Q16—124no—
231SD16—124no—
232SD16—124no—
233D8—124no—
234M16—62no—
235M16—62no—
236C2×C8—62no—
237C2×C8—122no—
238C4⋊C4—62no—
239C4⋊C4—81no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—44no—
243C22⋊C4—81no—
244C22⋊C4—122no—
245C42—44no—
246C42—81no—
247C42C2×C4×D311yescharacteristic
248C42C2×D1211yescharacteristic
249C42#23512yes—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—82no—
256C22×C3#38611yescharacteristic
257C22×C3#45311yescharacteristic
258C22×C3#45311yescharacteristic
259C2×D3—81no—
260C2×D3—82no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—44no—
266C3×C4—42no—
267C3×C4—82no—
268Dic3—81no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—121no—
275C23#95311yescharacteristic
276Q8—42no—
277Q8—81no—
278Q8—121no—
279Q8—121no—
280D4—42no—
281D4—81no—
282D4—121no—
283D4—121no—
284D4—242no—
285D4—241no—
286D4—241no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—121no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4C2×(C2×D3)⋊C411yescharacteristic
304C2×C4C4×#6712yes—
305C2×C4#92911yescharacteristic
306C2×C4#93012yes—
307C8—64no—
308C8—242no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—82no—
312C2×C3#188711yescharacteristic
313C2×C3#190612yes—
314D3—161no—
315C22—22no—
316C22—44no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—242no—
322C22—241no—
323C22#521311yescharacteristic
324C22#524111yescharacteristic
325C22#524111yescharacteristic
326C4—22no—
327C4—22no—
328C4—21no—
329C4—21no—
330C4—44no—
331C4—42no—
332C4—82no—
333C4—241no—
334C3#944111yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—82no—
338C2—481no—
339C2#7164512yes—
340C2#7165011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2372C2
socleC22×C3#386
prime radical#82943C2
cosocleC2×C3×C42C23
hypercenter#2372D3
hypocenterC3#9441
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2372

Complements

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

NH ≅ G/Ntype
1. C3×#2372C2semidirect
3. #71719C2semidirect
4. #69432C2semidirect
5. #67615C2semidirect
16. #5060C4semidirect
18. #5058C4semidirect
25. #2372D3semidirect
54. #874C2×C4semidirect
56. #869C2×C4semidirect
64. #803C2×C4semidirect
334. C3#9441semidirect

Maximal subgroups

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

subgroupindex
C3×#23722
#829432
#717192
#694322
#676152
#94413