Finite Group Databasebeta

← #11711985

Subgroups — #11711985

Order 768.

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

normal subgroups only

This group has 390 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#3480⋊C3C211yesmaximal, characteristic
2#85322C211yesmaximal, characteristic, complemented
3#73094C211yesmaximal, characteristic, complemented
4#73093C211yesmaximal, characteristic, complemented
5#73090C211yesmaximal, characteristic, complemented
6#73089C211yesmaximal, characteristic, complemented
7C2×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
8#18521—31nomaximal, Sylow
9#557⋊C3C2211yescharacteristic
10C4⋊(#41⋊C3)C2211yescharacteristic
11#548⋊C3—21no—
12C4×#41⋊C3—21no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5419—21no—
15#5391C2211yescharacteristic
16C4⋊#226C2211yescharacteristic
17C4×#226—21no—
18#4998—41nomaximal
19SL(2,𝔽3)⋊C8C2211yescharacteristic
20SL(2,𝔽3)⋊C8C2211yescharacteristic
21#3480D311yescharacteristic
22#3167—31no—
23#3104—31no—
24#3076—31no—
25#1653—31no—
26#1652—31no—
27#1641—31no—
28#1640—31no—
29#1638—31no—
30#1636—31no—
31#1631—31no—
32#1630—31no—
33#1540—31no—
34#1476—31no—
35C2×Q8⋊C8—31no—
36C2×#41⋊C3—21no—
37C22×SL(2,𝔽3)D411yescharacteristic
38C2×#226—21no—
39C3×#117—41no—
40#926—41no—
41C22×C3⋊C8—41no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)D411yescharacteristic
44C4×SL(2,𝔽3)C2311yescharacteristic
45#859—21no—
46#808—41no—
47#808—41no—
48#807—41no—
49#807—41no—
50#559—31no—
51#557C2×D311yescharacteristic
52#556—31no—
53#553—61no—
54#550C2×D311yescharacteristic
55#548—21no—
56#533—31no—
57#532—31no—
58#529—61no—
59C4×#41—21no—
60C4×#41—31no—
61C2×C4×Q8C2×D311yescharacteristic
62#498—31no—
63#492—31no—
64#487—31no—
65#486—31no—
66#484—61no—
67#479—61no—
68#478—61no—
69#477—31no—
70#477—31no—
71#477—31no—
72#476—31no—
73#475—31no—
74#474—31no—
75#465—62no—
76#462—31no—
77#462—31no—
78#451—61no—
79C4×Q16—61no—
80C4×Q16—61no—
81C4×SD16—62no—
82#431—31no—
83#427—61no—
84C2×C4⋊C8—31no—
85#416—61no—
86C2×C4×C8—31no—
87#332—31no—
88#332—31no—
89#331—31no—
90#331—31no—
91Q8⋊C8—31no—
92Q8⋊C8—31no—
93Q8⋊C8—31no—
94Q8⋊C8—31no—
95Q8⋊C8—31no—
96Q8⋊C8—31no—
97D4⋊C8—31no—
98D4⋊C8—31no—
99C22×C3×C4—41no—
100#41⋊C3—41no—
101C2×SL(2,𝔽3)—21no—
102C2×SL(2,𝔽3)C2×D411yescharacteristic
103#226—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C22⋊C4—81no—
107C3×C42—81no—
108#217—81no—
109C4⋊Dic3—41no—
110C4⋊Dic3—41no—
111C4×Dic3—81no—
112C2×C3⋊C8—41no—
113C2×C3⋊C8—41no—
114C2×C3⋊C8—44no—
115C2×#41—21no—
116C2×#41—31no—
117C22×Q8#6711yescharacteristic
118C2×Q16—61no—
119C2×Q16—61no—
120C2×SD16—62no—
121C22×C8—61no—
122C22×C8—61no—
123C4⋊Q8—31no—
124C4⋊Q8—31no—
125C4⋊Q8—31no—
126#127—31no—
127#126—62no—
128#125—61no—
129#125—61no—
130#124—61no—
131#124—61no—
132#123—62no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136#121—32no—
137#121—61no—
138#121—61no—
139C4×Q8—21no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×Q8—31no—
143C4×Q8—31no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8#6711yescharacteristic
147C4×Q8C22×D311yescharacteristic
148C4×D4—32no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C4×D4—61no—
153C4×D4—61no—
154C4×D4—61no—
155#117—61no—
156#117—61no—
157#117—61no—
158#117S411yescharacteristic
159C2×C4⋊C4—31no—
160C2×C42—31no—
161C2×C42—61no—
162#107—61no—
163#107—61no—
164#107—61no—
165#107—61no—
166#106—62no—
167#106—61no—
168#106—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8—31no—
173C4⋊C8—61no—
174#103—62no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179#103—61no—
180#103—61no—
181#102—61no—
182#102—61no—
183#102—62no—
184C22⋊C8—61no—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—61no—
188C4×C8—61no—
189C23×C3—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198SL(2,𝔽3)#13511yescharacteristic
199C3⋊C8—42no—
200C3⋊C8—42no—
201#41—41no—
202#41—62no—
203#41—62no—
204#41—121no—
205C2×Q8—21no—
206C2×Q8—31no—
207C2×Q8—31no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8C2×#6711yescharacteristic
212C2×D4—32no—
213C2×D4—31no—
214C2×D4—61no—
215C2×D4—61no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4C2×S411yescharacteristic
222Q16—121no—
223Q16—122no—
224Q16—121no—
225SD16—124no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—122no—
233C2×C8—122no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—32no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4C2×S411yescharacteristic
248C4⋊C4C2×S411yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—62no—
253C22⋊C4—62no—
254C22⋊C4—121no—
255C42—21no—
256C42—31no—
257C42—31no—
258C42—31no—
259C42—31no—
260C42—61no—
261C42—61no—
262C42—121no—
263C22×C3—41no—
264C22×C3—42no—
265C22×C3—81no—
266C22×C3—81no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—42no—
270C3×C4—161no—
271C3×C4—161no—
272Dic3—161no—
273Dic3—161no—
274C23—61no—
275C23GL(2,ℤ/4)11yescharacteristic
276Q8—31no—
277Q8—61no—
278Q8—61no—
279Q8—61no—
280Q8—61no—
281Q8—121no—
282Q8—121no—
283Q8—122no—
284Q8#95011yescharacteristic
285D4—61no—
286D4—61no—
287D4—121no—
288D4—121no—
289D4—122no—
290D4—122no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—241no—
307C2×C4#98412yes—
308C2×C4#98512yes—
309C2×C4GL(2,ℤ/4)11yescharacteristic
310C2×C4C22×S411yescharacteristic
311C8—62no—
312C8—121no—
313C8—121no—
314C8—121no—
315C8—121no—
316C8—121no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—82no—
321C22—21no—
322C22—21no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22#542512yes—
327C22C2×GL(2,ℤ/4)11yescharacteristic
328C4—31no—
329C4—31no—
330C4—41no—
331C4—41no—
332C4—61no—
333C4—61no—
334C4—61no—
335C4—121no—
336C4—241no—
337C4—241no—
338C4#542512yes—
339C4C2×#98411yescharacteristic
340C4C2×#98511yescharacteristic
341C3—41noSylow
342C2—22no—
343C2—241no—
344C2#8551411yescharacteristic
345C2C2×#542511yescharacteristic
346C2#8554411yescharacteristic
subgroupG/H
centerC2×C4GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C23
FrattiniC2×C4C22×S4
Fitting#3480D3
socleC22C2×GL(2,ℤ/4)
prime radical#3480⋊C3C2
hypercenter#117S4
hypocenterSL(2,𝔽3)#135
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #85322C2semidirect
3. #73094C2semidirect
4. #73093C2semidirect
5. #73090C2semidirect
6. #73089C2semidirect
7. C2×SL(2,𝔽3)⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#3480⋊C32
#853222
#730942
#730932
#730902
#730892
C2×SL(2,𝔽3)⋊C82
#185213
#49984