Finite Group Databasebeta

← #10692777

Subgroups — #10692777

Order 768.

400 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 103 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#83341C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic
3C3×#1561C211yesmaximal, characteristic
4#70507C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#67710C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#9434—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5080—21no—
14#5078—21no—
15#5064C414yes—
16#5024—21no—
17#5024—21no—
18Q8×C3⋊C8—21no—
19Q8×C3⋊C8—21no—
20#5000—22no—
21#4995—22no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C3×#332—21no—
26C3×#332—21no—
27#4482—21no—
28#4482—21no—
29#4481—21no—
30#4481—21no—
31#3165—31no—
32#2986—31no—
33#1561D311yescharacteristic
34#1548—31no—
35#1546—31no—
36#1299—31no—
37#1299—31no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125C2×C412yes—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117D412yes—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×#208—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869C2×C412yes—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61#830—21no—
62#830—21no—
63#808—24no—
64#807—24no—
65C4⋊(C3⋊C8)—22no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)D411yescharacteristic
68C4⋊(C3⋊C8)D411yescharacteristic
69C4⋊(C3⋊C8)D411yescharacteristic
70C4⋊(C3⋊C8)D411yescharacteristic
71#803—22no—
72#803C2×C412yes—
73C4×C3⋊C8—22no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79#501—34no—
80#500—61no—
81#498—61no—
82#446—61no—
83#446—61no—
84C8×Q8—61no—
85C8×Q8—61no—
86#433—62no—
87#431—31no—
88#431—31no—
89#428—62no—
90#423C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92C4×M16—31no—
93#333—61no—
94#333—61no—
95#332—21no—
96#332—21no—
97#332—61no—
98#332—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—41no—
105C2×Dic6—22no—
106C2×Dic6—41no—
107C2×Dic6—81no—
108C2×Dic6—81no—
109C3×M16—81no—
110C3×M16—81no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4C22⋊C412yes—
118C3×C4⋊C4C22⋊C412yes—
119C3×C22⋊C4—22no—
120C3×C42—22no—
121C3×C42C22×C411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129#208—22no—
130#208—22no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C22×Q8—61no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—61no—
140C4⋊Q8—32no—
141C4⋊Q8—61no—
142C4⋊Q8—61no—
143C4⋊Q8—61no—
144C4⋊Q8#6711yescharacteristic
145C4⋊Q8#6711yescharacteristic
146#125C4×D312yes—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151#117D1212yes—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C42C22×D311yescharacteristic
155#107—64no—
156#106—64no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—31no—
162C4⋊C8—31no—
163C4⋊C8—31no—
164C4⋊C8—31no—
165C4⋊C8—62no—
166C4⋊C8—62no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—32no—
170#97—62no—
171C4×C8—62no—
172C4×C8—61no—
173C4×C8—61no—
174C4×C8—61no—
175C23×C3#12311yescharacteristic
176C3×Q8—41no—
177C3×Q8—41no—
178C3×Q8—41no—
179C3×Q8—41no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×Dic3—42no—
195C2×Dic3—82no—
196Dic6—81no—
197Dic6—81no—
198Dic6—82no—
199C3×C8—81no—
200C3×C8—81no—
201C3⋊C8—24no—
202C3⋊C8—41no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—41no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—62no—
209C2×Q8—121no—
210C2×Q8—241no—
211C2×Q8—241no—
212C22×C4—121no—
213C22×C4C2×D1211yescharacteristic
214C22×C4#23711yescharacteristic
215C22×C4#23711yescharacteristic
216M16—62no—
217M16—62no—
218M16—81no—
219M16—81no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—62no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4(C2×D3)⋊C412yes—
236C4⋊C4(C2×D3)⋊C412yes—
237C22⋊C4—22no—
238C42—22no—
239C42C2×C4×D311yescharacteristic
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
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#45611yescharacteristic
253C3×C4#45611yescharacteristic
254C3×C4#47111yescharacteristic
255C3×C4#47111yescharacteristic
256C3×C4#47211yescharacteristic
257C3×C4#47311yescharacteristic
258Dic3—82no—
259C23#88611yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—242no—
265Q8—241no—
266Q8—241no—
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—122no—
276C2×C4—242no—
277C2×C4#67⋊C411yescharacteristic
278C2×C4#67⋊C411yescharacteristic
279C2×C4#88211yescharacteristic
280C2×C4C2×(C2×D3)⋊C411yescharacteristic
281C2×C4#93911yescharacteristic
282C2×C4#93911yescharacteristic
283C8—64no—
284C8—81no—
285C8—81no—
286C8—121no—
287C8—121no—
288C8—121no—
289C8—121no—
290C2×C3—41no—
291C2×C3#188611yescharacteristic
292C2×C3#190711yescharacteristic
293C2×C3#190811yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#495811yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—242no—
304C4#475211yescharacteristic
305C4#475211yescharacteristic
306C4#519911yescharacteristic
307C4#520011yescharacteristic
308C4#520311yescharacteristic
309C4#520411yescharacteristic
310C3#943411yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7010311yescharacteristic
313C2#7030311yescharacteristic
314C2#7030411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1561C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1561D3
hypocenterC3#9434
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

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
#833412
#833322
C3×#15612
#705072
#705062
#677102
#94343