Finite Group Databasebeta

← #11713106

Subgroups — #11713106

Order 768.

438 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 67 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×#5391C211yesmaximal, characteristic
2#2351⋊C3C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73063C211yesmaximal, characteristic
5#73062C211yesmaximal, characteristic
6#73059C211yesmaximal, characteristic
7C4×#859C211yesmaximal, characteristic
8#22977—31nomaximal, Sylow
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#5391C414yescomplemented
17#4653—41nomaximal
18C2×#477—31no—
19#2365—31no—
20#2351D311yescharacteristic
21#1993—31no—
22#1949—31no—
23#1943—31no—
24#1942—31no—
25#1921—31no—
26#1918—31no—
27#1865—31no—
28#1863—31no—
29#1771—31no—
30#1763—31no—
31#1763—31no—
32C4×#103—31no—
33C22×SL(2,𝔽3)C2311yescharacteristic
34C2×C4⋊Dic3—41no—
35C2×Dic3⋊C4—41no—
36C2×C4×Dic3—41no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)C2×C412yes—
40#859—21no—
41#859C2×C412yes—
42#859C2×C412yes—
43#859Q812yescomplemented
44C3×#95—41no—
45C3⋊#95—41no—
46C3⋊#95—41no—
47C3⋊#95—41no—
48C2×C4⋊Q8—31no—
49C2×C4×Q8—61no—
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53#477—34no—
54#477—34no—
55C2×#107—31no—
56C2×#107—31no—
57C2×#107—31no—
58C2×#106—31no—
59C2×C4⋊C8—31no—
60C2×#103—31no—
61C2×#103—31no—
62C2×#103—31no—
63C2×#103—31no—
64C2×#97—31no—
65C2×#97—31no—
66C2×C4×C8—31no—
67#391—61no—
68#389—31no—
69#389—31no—
70#389—31no—
71#389—31no—
72#387—61no—
73#387—61no—
74#382—31no—
75C4×C4⋊C4—31no—
76C4×C4⋊C4—31no—
77C4×C4⋊C4—31no—
78C4×C4⋊C4—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#340—31no—
85#340—31no—
86#340—31no—
87#336—31no—
88#336—31no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×SL(2,𝔽3)C22×C411yescharacteristic
97C2×SL(2,𝔽3)C2×Q811yescharacteristic
98C2×SL(2,𝔽3)#4111yescharacteristic
99C4⋊Dic3—44no—
100Dic3⋊C4—82no—
101C4×Dic3—82no—
102C22×Q8—61no—
103C22×Q8C22×D311yescharacteristic
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—31no—
107C22×C8—31no—
108C4⋊Q8—34no—
109C4⋊Q8—62no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—32no—
113C4×Q8—61no—
114C4×Q8—61no—
115C4×Q8—124no—
116C4×Q8C4×D312yes—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—61no—
133C2×C42—61no—
134C2×C42—61no—
135#107—34no—
136#107—34no—
137#107—121no—
138#106—121no—
139C4⋊C8—34no—
140#103—32no—
141#103—32no—
142#103—32no—
143#103—32no—
144#103—32no—
145#103—32no—
146#103—61no—
147#103—61no—
148#97—62no—
149#97—121no—
150C4×C8—121no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95S411yescharacteristic
159C23×C3—41no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—81no—
163C2×C3×C4—81no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4—81no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—81no—
173C2×Dic3—81no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177C2×Dic3—81no—
178C2×Dic3—81no—
179C2×Dic3—81no—
180C2×Dic3—81no—
181SL(2,𝔽3)C4×Q811yescharacteristic, complemented
182C2×Q8—31no—
183C2×Q8—31no—
184C2×Q8—31no—
185C2×Q8—64no—
186C2×Q8—122no—
187C2×Q8—122no—
188C2×Q8C2×C4×D311yescharacteristic
189C2×Q8#23711yescharacteristic
190C2×Q8D3×Q811yescharacteristic
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4C2×S411yescharacteristic
200C22×C4C2×S411yescharacteristic
201C22×C4C2×S411yescharacteristic
202C2×C8—32no—
203C2×C8—32no—
204C2×C8—32no—
205C2×C8—61no—
206C2×C8—61no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8—61no—
210C2×C8—61no—
211C2×C8—61no—
212C2×C8—61no—
213C2×C8—61no—
214C4⋊C4—32no—
215C4⋊C4—32no—
216C4⋊C4—32no—
217C4⋊C4—32no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—64no—
229C4⋊C4—62no—
230C4⋊C4—121no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C42—32no—
234C42—32no—
235C42—61no—
236C42—61no—
237C42—61no—
238C42—61no—
239C42—122no—
240C42—122no—
241C42—122no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C3×C4—82no—
250C3×C4—82no—
251C3×C4—161no—
252Dic3—82no—
253Dic3—82no—
254Dic3—82no—
255Dic3—161no—
256C23C22×S411yescharacteristic
257Q8—31no—
258Q8—31no—
259Q8—31no—
260Q8—61no—
261Q8—128no—
262Q8#88711yescharacteristic
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—62no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4C4×S412yes—
298C2×C4#98612yes—
299C8—62no—
300C8—62no—
301C8—62no—
302C8—121no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C22C2×C4×S411yescharacteristic
311C22#591411yescharacteristic
312C22Q8×S411yescharacteristic
313C22C2×#98611yescharacteristic
314C22#592311yescharacteristic
315C22#592711yescharacteristic
316C22#592711yescharacteristic
317C4—22no—
318C4—22no—
319C4—31no—
320C4—31no—
321C4—31no—
322C4—31no—
323C4—41no—
324C4—62no—
325C4—122no—
326C4—122no—
327C4—121no—
328C4—121no—
329C4—122no—
330C4—241no—
331C3—41noSylow
332C2#8530211yescharacteristic
333C2C4×#98611yescharacteristic
334C2#8535011yescharacteristic
335C2#8537911yescharacteristic
336C2#8538911yescharacteristic
337C2#8541611yescharacteristic
338C2#8542411yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2351D3
prime radical, cosocleC22×SL(2,𝔽3)C23
hypercenter#95S4
hypocenterSL(2,𝔽3)C4×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
16. #5391C4semidirect
43. #859Q8semidirect
181. SL(2,𝔽3)C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#53912
#2351⋊C32
#730632
#730622
#730592
C4×#8592
#229773
#46534