Finite Group Databasebeta

← #10692781

Subgroups — #10692781

Order 768.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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×#3165C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic
3#71834C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#67669C211yesmaximal, characteristic
7#67669C211yesmaximal, characteristic
8#9434—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#501C414yes—
12C3×#500—21no—
13C3×#498—21no—
14C3×#428—22no—
15C3×C4×M16C2211yescharacteristic
16#5024—21no—
17#5024—21no—
18Q8×C3⋊C8—21no—
19Q8×C3⋊C8—21no—
20#5000—22no—
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4924C2211yescharacteristic
25#4547—21no—
26#4547—21no—
27#4547—21no—
28#4547—21no—
29#4546—21no—
30#4546—21no—
31#3165D311yescharacteristic
32#2986—31no—
33#1561—31no—
34#1548—31no—
35#1546—31no—
36#1299—31no—
37#1299—31no—
38C22×C3×Q8—21no—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125C2×C412yes—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117D412yes—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#208—21no—
56C2×#208—21no—
57C3×#107—24no—
58C3×#106—24no—
59C3×#97C2×C412yes—
60C3×C4×C8—21no—
61#830—21no—
62#830—21no—
63C4⋊(C3⋊C8)—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D411yescharacteristic
70C4⋊(C3⋊C8)D411yescharacteristic
71C4⋊(C3⋊C8)D411yescharacteristic
72C4⋊(C3⋊C8)D411yescharacteristic
73#803—22no—
74C4×C3⋊C8—22no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8C2×D311yescharacteristic
79#501Dic314yes—
80#500—21no—
81#498—21no—
82#446—61no—
83#446—61no—
84C8×Q8—61no—
85C8×Q8—61no—
86#433—62no—
87#431—31no—
88#431—31no—
89#428—22no—
90#423—31no—
91#423—31no—
92C4×M16C2×D311yescharacteristic
93#333—61no—
94#333—61no—
95#332—61no—
96#332—61no—
97#332—61no—
98#332—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—81no—
104C2×C3×Q8—81no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C3×M16—22no—
110C3×M16—22no—
111C2×C3×C8—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C42—22no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128#208—81no—
129#208—81no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×Q8—21no—
137C2×M16—21no—
138C2×M16—61no—
139C2×M16—61no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8C2×Dic312yes—
144C4⋊Q8#6711yescharacteristic
145C4⋊Q8#6711yescharacteristic
146#125C2×Dic312yes—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151#117#6712yes—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C42C22×D311yescharacteristic
155#107—24no—
156#106—24no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8—62no—
162C4⋊C8—62no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—62no—
170#97C2×Dic312yes—
171C4×C8—21no—
172C4×C8—62no—
173C4×C8—61no—
174C4×C8—61no—
175C23×C3#12311yescharacteristic
176C3×Q8—41no—
177C3×Q8—41no—
178C3×Q8—41no—
179C3×Q8—41no—
180C3×Q8—82no—
181C3×Q8—81no—
182C3×Q8—81no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—42no—
192C2×C3×C4—82no—
193C2×C3×C4C2×C22⋊C411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C3×C8—24no—
200C3⋊C8—41no—
201C3⋊C8—41no—
202C3⋊C8—41no—
203C3⋊C8—41no—
204C3⋊C8—81no—
205C3⋊C8—81no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—81no—
211C2×Q8—81no—
212C22×C4—41no—
213C22×C4#23711yescharacteristic
214C22×C4#23711yescharacteristic
215C22×C4C2×#6711yescharacteristic
216M16—22no—
217M16—22no—
218M16—241no—
219M16—241no—
220C2×C8—22no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—121no—
226C2×C8—121no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4#21712yes—
236C4⋊C4#21712yes—
237C22⋊C4—22no—
238C42—22no—
239C42D3×D411yescharacteristic
240C42D3×D411yescharacteristic
241C42C22×Dic311yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—42no—
251C3×C4—42no—
252C3×C4—82no—
253C3×C4#45611yescharacteristic
254C3×C4#45611yescharacteristic
255C3×C4#47111yescharacteristic
256C3×C4#47111yescharacteristic
257C3×C4#47211yescharacteristic
258C3×C4#47311yescharacteristic
259C23#93511yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—82no—
265Q8—81no—
266Q8—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4D4×Dic311yescharacteristic
278C2×C4D4×Dic311yescharacteristic
279C2×C4#93711yescharacteristic
280C2×C4#93911yescharacteristic
281C2×C4#93911yescharacteristic
282C2×C4C2×#21711yescharacteristic
283C8—24no—
284C8—121no—
285C8—121no—
286C8—121no—
287C8—121no—
288C8—241no—
289C8—241no—
290C2×C3—41no—
291C2×C3#188611yescharacteristic
292C2×C3#190711yescharacteristic
293C2×C3#190811yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#522011yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4#509511yescharacteristic
305C4#509511yescharacteristic
306C4#519911yescharacteristic
307C4#520011yescharacteristic
308C4#520311yescharacteristic
309C4#520411yescharacteristic
310C3#943411yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7178411yescharacteristic
313C2#7209111yescharacteristic
314C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3165D3
hypocenterC3#9434
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
310. C3#9434semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833322
#718342
#717742
#717732
#676692
#94343