Finite Group Databasebeta

← #11715628

Subgroups — #11715628

Order 768.

341 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 25 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
1C2×Q8⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#85553C211yesmaximal, characteristic, complemented
3#85552C211yesmaximal, characteristic, complemented
4#18769—31nomaximal, Sylow
5Q8⋊SL(2,𝔽3)C2211yescharacteristic
6#5420—41nomaximal
7#5419—41nomaximal
8#5419—41nomaximal
9C2×Q82D311yescharacteristic
10#3209—31no—
11#3104—31no—
12#3078—31no—
13#1645—31no—
14#1645—31no—
15#1645—31no—
16#1645—31no—
17#1633—31no—
18#1633—31no—
19#1633—31no—
20#1633—31no—
21#1548—31no—
22#1487—31no—
23#1479—31no—
24C22×SL(2,𝔽3)—41no—
25C22×SL(2,𝔽3)—41no—
26C22×SL(2,𝔽3)—41no—
27C22×SL(2,𝔽3)—81no—
28C2×#226—41no—
29C2×#226—41no—
30#859—41no—
31#859—41no—
32#859—41no—
33#859—41no—
34Q82—31no—
35Q82—31no—
36Q82—31no—
37Q82—61no—
38Q82C2×D311yescharacteristic
39#533—31no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—61no—
42C2×C4×Q8—31no—
43C2×C4×Q8—31no—
44#494—31no—
45#494—31no—
46#487—31no—
47#487—31no—
48#484—61no—
49#479—61no—
50#479—61no—
51#477—31no—
52#477—31no—
53#477—31no—
54#477—31no—
55#475—31no—
56#475—31no—
57#462—31no—
58#462—31no—
59#462—61no—
60#462—61no—
61#451—61no—
62#451—61no—
63#451—61no—
64C4×Q16—61no—
65C4×Q16—61no—
66#431—31no—
67#431—31no—
68#428—61no—
69#423—31no—
70#419—61no—
71#333—31no—
72#333—31no—
73#333—31no—
74#333—31no—
75Q8⋊C8—31no—
76Q8⋊C8—31no—
77Q8⋊C8—31no—
78Q8⋊C8—31no—
79Q8⋊C8—31no—
80Q8⋊C8—31no—
81Q8⋊C8—31no—
82Q8⋊C8—31no—
83C2×SL(2,𝔽3)—42no—
84C2×SL(2,𝔽3)—41no—
85C2×SL(2,𝔽3)—41no—
86C2×SL(2,𝔽3)—41no—
87C2×SL(2,𝔽3)—81no—
88C2×SL(2,𝔽3)—81no—
89C2×SL(2,𝔽3)—81no—
90C2×SL(2,𝔽3)—82no—
91#226—81no—
92#226—81no—
93#217—161no—
94C22×Q8—21no—
95C22×Q8—31no—
96C22×Q8S411yescharacteristic
97C22×Q8S411yescharacteristic
98C22×Q8S411yescharacteristic
99C2×Q16—61no—
100C2×Q16—61no—
101C2×Q16—61no—
102C2×Q16—61no—
103C2×Q16—61no—
104C2×Q16—61no—
105C2×M16—61no—
106C4⋊Q8—31no—
107C4⋊Q8—31no—
108C4⋊Q8—31no—
109C4⋊Q8—31no—
110C4⋊Q8—31no—
111C4⋊Q8—31no—
112C4⋊Q8—61no—
113C4⋊Q8—61no—
114C4⋊Q8—61no—
115C4⋊Q8—61no—
116C4⋊Q8—61no—
117C4⋊Q8—61no—
118C4⋊Q8—61no—
119C4⋊Q8—61no—
120C4⋊Q8—61no—
121C4⋊Q8—61no—
122#125—61no—
123C22⋊Q8—61no—
124C22⋊Q8—61no—
125C4×Q8—31no—
126C4×Q8—31no—
127C4×Q8—31no—
128C4×Q8—31no—
129C4×Q8—31no—
130C4×Q8—31no—
131C4×Q8—31no—
132C4×Q8—31no—
133C4×Q8—61no—
134C4×Q8—61no—
135C4×Q8—61no—
136C4×Q8—61no—
137C4×Q8—61no—
138C4×Q8—61no—
139#117—61no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C42—31no—
145C2×C42—61no—
146#107—61no—
147#107—61no—
148#107—61no—
149#107—61no—
150#106—61no—
151#106—61no—
152C4⋊C8—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155C4⋊C8—31no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—61no—
159#103—61no—
160#103—61no—
161#103—61no—
162#103—61no—
163#103—61no—
164#103—61no—
165#103—61no—
166#103—61no—
167#103—61no—
168#103—61no—
169#103—61no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172C4×C8—31no—
173C4×C8—31no—
174C4×C8—61no—
175C23×C3—161no—
176C2×Dic3—161no—
177C2×Dic3—161no—
178SL(2,𝔽3)—41no—
179SL(2,𝔽3)—41no—
180SL(2,𝔽3)—41no—
181SL(2,𝔽3)—81no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—31no—
185C2×Q8—31no—
186C2×Q8—32no—
187C2×Q8—31no—
188C2×Q8—31no—
189C2×Q8—31no—
190C2×Q8—61no—
191C2×Q8—61no—
192C2×Q8—61no—
193C2×Q8—61no—
194C2×Q8—61no—
195C2×Q8—81no—
196C2×Q8—82no—
197C2×Q8—241no—
198C2×Q8#22612yescomplemented
199C2×Q8C2×S411yescharacteristic
200C2×Q8C2×S411yescharacteristic
201C2×Q8C2×S411yescharacteristic
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—61no—
206Q16—121no—
207Q16—122no—
208Q16—121no—
209Q16—241no—
210Q16—241no—
211Q16—241no—
212Q16—241no—
213M16—122no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—61no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C22⋊C4—121no—
241C42—31no—
242C42—31no—
243C42—31no—
244C42—31no—
245C42—61no—
246C42—61no—
247C42—61no—
248C42—61no—
249C42—121no—
250C22×C3—162no—
251C22×C3—161no—
252C22×C3—321no—
253C22×C3—321no—
254Dic3—321no—
255Dic3—321no—
256C23#102011yescharacteristic
257Q8—31no—
258Q8—32no—
259Q8—31no—
260Q8—32no—
261Q8—41no—
262Q8—61no—
263Q8—61no—
264Q8—81no—
265Q8—121no—
266Q8—121no—
267Q8—121no—
268Q8—121no—
269Q8—121no—
270Q8—121no—
271Q8—121no—
272Q8—241no—
273Q8—241no—
274Q8C2×#22611yescharacteristic
275Q8#98311yescharacteristic
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—121no—
291C2×C4—121no—
292C8—121no—
293C8—121no—
294C8—121no—
295C8—121no—
296C8—121no—
297C8—121no—
298C2×C3—161no—
299C2×C3—161no—
300C2×C3—161no—
301C2×C3—322no—
302C22—21no—
303C22—21no—
304C22#593012yes—
305C22C2×#102011yescharacteristic
306C4—31no—
307C4—31no—
308C4—32no—
309C4—31no—
310C4—31no—
311C4—61no—
312C4—61no—
313C4—61no—
314C4—121no—
315C4—121no—
316C4—241no—
317C4—241no—
318C3—161noSylow
319C2—22no—
320C2C2×#593011yescharacteristic
321C2#8750711yescharacteristic
322C2#8751611yescharacteristic
subgroupG/H
center, Frattini, socleC22C2×#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C22
FittingC2×Q82D3
prime radicalC2×Q8⋊SL(2,𝔽3)C2
hypercenterC23#1020
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #85553C2semidirect
3. #85552C2semidirect
198. C2×Q8#226semidirect

Maximal subgroups

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

subgroupindex
C2×Q8⋊SL(2,𝔽3)2
#855532
#855522
#187693
#54204
#54194