Finite Group Databasebeta

← #11711873

Subgroups — #11711873

Order 768.

318 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 51 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#85322C211yesmaximal, characteristic
2#2982⋊C3C211yesmaximal, characteristic
3#73057C211yesmaximal, characteristic
4#73010C211yesmaximal, characteristic, complemented
5#73009C211yesmaximal, characteristic, complemented
6#226⋊C8C211yesmaximal, characteristic, complemented
7C8×#226C211yesmaximal, characteristic, complemented
8#17925—31nomaximal, Sylow
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10#5419C411yescharacteristic
11#5419C411yescharacteristic
12#5391C411yescharacteristic
13C4⋊#226C411yescharacteristic
14C4×#226C2211yescharacteristic
15C4×#226C2211yescharacteristic
16#4720—41nomaximal
17C8×SL(2,𝔽3)C2211yescharacteristic
18C8×SL(2,𝔽3)C2211yescharacteristic
19SL(2,𝔽3)⋊C8C2211yescharacteristic
20SL(2,𝔽3)⋊C8C2211yescharacteristic
21#3104—31no—
22#2985—31no—
23#2982D311yescharacteristic
24#1587—31no—
25#1587—31no—
26#1584—31no—
27#1584—31no—
28#1579—31no—
29#1579—31no—
30C8×Q16—31no—
31C8×Q16—31no—
32#1572—31no—
33#1491—31no—
34#1485—31no—
35#1453—31no—
36C22×SL(2,𝔽3)C2×C411yescharacteristic
37C2×#226C2×C411yescharacteristic
38C2×#226C2×C411yescharacteristic
39#926—41no—
40C4×SL(2,𝔽3)C2×C411yescharacteristic
41C4×SL(2,𝔽3)C2311yescharacteristic
42#859C2×C411yescharacteristic
43#859C2×C411yescharacteristic
44C3×C22⋊C8—41no—
45#830—41no—
46#815—41no—
47Dic3⋊C8—41no—
48Dic3⋊C8—41no—
49C8×Dic3—41no—
50#533—31no—
51C2×C4×Q8C2×D311yescharacteristic
52#484—31no—
53#484—31no—
54#477—31no—
55#477—31no—
56#462—31no—
57#462—31no—
58#451—31no—
59#451—31no—
60#446—31no—
61#446—31no—
62#446—61no—
63#446—61no—
64C8×Q8—31no—
65C8×Q8—31no—
66C8×Q8—31no—
67C8×Q8—31no—
68C8×Q8C2×D311yescharacteristic
69C8×Q8C2×D311yescharacteristic
70C4×Q16—31no—
71C4×Q16—31no—
72C4×Q16—31no—
73C4×Q16—31no—
74#433—61no—
75#433—61no—
76#432—31no—
77#431—31no—
78#431—31no—
79#431—31no—
80C4×M16—31no—
81#335—31no—
82#335—31no—
83#334—31no—
84#334—31no—
85Q8⋊C8—31no—
86Q8⋊C8—31no—
87Q8⋊C8—31no—
88Q8⋊C8—31no—
89Q8⋊C8—31no—
90Q8⋊C8—31no—
91Q8⋊C8—31no—
92Q8⋊C8—31no—
93#322—31no—
94#322—31no—
95#322—31no—
96C82—31no—
97C22×C3×C4—41no—
98C2×SL(2,𝔽3)—21no—
99C2×SL(2,𝔽3)C22×C411yescharacteristic
100#226—21no—
101#226—21no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104#217—41no—
105#217—41no—
106C4⋊Dic3—41no—
107C4⋊Dic3—41no—
108C4×Dic3—41no—
109C4×Dic3—41no—
110C2×C3⋊C8—41no—
111C2×C3⋊C8—41no—
112C22×Q8C4×D311yescharacteristic
113C2×Q16—31no—
114C2×Q16—31no—
115C2×Q16—31no—
116C2×Q16—31no—
117C2×M16—61no—
118C4⋊Q8—31no—
119C4⋊Q8—31no—
120#125—61no—
121C22⋊Q8—31no—
122C22⋊Q8—31no—
123C22⋊Q8—61no—
124C4×Q8—31no—
125C4×Q8—31no—
126C4×Q8—31no—
127C4×Q8—31no—
128C4×Q8—61no—
129C4×Q8C4×D311yescharacteristic
130C4×Q8C22×D311yescharacteristic
131#117—61no—
132C2×C4⋊C4—31no—
133C2×C42—31no—
134#107—31no—
135#107—31no—
136#107—31no—
137#107—31no—
138C4⋊C8—31no—
139C4⋊C8—31no—
140C4⋊C8—31no—
141C4⋊C8—31no—
142C4⋊C8—31no—
143C4⋊C8—31no—
144C4⋊C8—31no—
145C4⋊C8—31no—
146C4⋊C8—61no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150#103—31no—
151#103—31no—
152#103—31no—
153#103—31no—
154#103—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158C22⋊C8—31no—
159C22⋊C8—31no—
160C22⋊C8—31no—
161C22⋊C8—61no—
162C22⋊C8S411yescharacteristic
163#97—61no—
164#97—61no—
165#97—61no—
166#97—61no—
167C4×C8—31no—
168C4×C8—31no—
169C4×C8—31no—
170C4×C8—31no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—61no—
174C4×C8—61no—
175C4×C8—61no—
176C23×C3—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185SL(2,𝔽3)#13111yescharacteristic
186C3×C8—81no—
187C3×C8—81no—
188C3⋊C8—81no—
189C3⋊C8—81no—
190C2×Q8—21no—
191C2×Q8—31no—
192C2×Q8—31no—
193C2×Q8—31no—
194C2×Q8—61no—
195C2×Q8—61no—
196C2×Q8C2×C4×D311yescharacteristic
197C22×C4—31no—
198C22×C4—31no—
199C22×C4C2×S411yescharacteristic
200Q16—61no—
201Q16—61no—
202Q16—62no—
203Q16—62no—
204Q16—61no—
205Q16—61no—
206M16—122no—
207C2×C8—31no—
208C2×C8—31no—
209C2×C8—31no—
210C2×C8—31no—
211C2×C8—31no—
212C2×C8—31no—
213C2×C8—61no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—61no—
217C2×C8C2×S411yescharacteristic
218C2×C8C2×S411yescharacteristic
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C42—31no—
234C42—31no—
235C42—31no—
236C42—31no—
237C42—61no—
238C42—61no—
239C22×C3—41no—
240C22×C3—81no—
241C22×C3—81no—
242C22×C3—81no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—81no—
246Dic3—81no—
247Dic3—81no—
248Dic3—81no—
249Dic3—81no—
250C23C4×S411yescharacteristic
251Q8—31no—
252Q8—32no—
253Q8—61no—
254Q8—61no—
255Q8—61no—
256Q8—61no—
257Q8—61no—
258Q8—241no—
259Q8#90111yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—121no—
273C2×C4C4×S411yescharacteristic
274C2×C4C22×S411yescharacteristic
275C8—21no—
276C8—21no—
277C8—61no—
278C8—61no—
279C8—61no—
280C8—61no—
281C8—62no—
282C8—61no—
283C8—61no—
284C8—121no—
285C8—121no—
286C8—121no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—161no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22C2×C4×S411yescharacteristic
295C4—21no—
296C4—31no—
297C4—31no—
298C4—61no—
299C4—61no—
300C4—61no—
301C4—121no—
302C4—121no—
303C4—121no—
304C4—121no—
305C4#592711yescharacteristic
306C4#592711yescharacteristic
307C3—41noSylow
308C2—41no—
309C2#8535011yescharacteristic
310C2#8543711yescharacteristic
311C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2982D3
socleC22C2×C4×S4
prime radicalC22×SL(2,𝔽3)C2×C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC22⋊C8S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #73010C2semidirect
5. #73009C2semidirect
6. #226⋊C8C2semidirect
7. C8×#226C2semidirect

Maximal subgroups

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

subgroupindex
#853222
#2982⋊C32
#730572
#730102
#730092
#226⋊C82
C8×#2262
#179253
#47204