Finite Group Databasebeta

← #11712998

Subgroups — #11712998

Order 768.

435 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 57 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×#5424C211yesmaximal, characteristic
2#2880⋊C3C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic
4#73122C212yesmaximal, complemented
5#73121C212yesmaximal, complemented
6#22587—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C2×#131⋊C3C2212yes—
9#5424C411yescharacteristic, complemented
10#5424C411yescharacteristic, complemented
11#5424C2211yescharacteristic
12#5424C2211yescharacteristic
13C2×GU(2,𝔽3)C2212yes—
14C2×#859C411yescharacteristic, complemented
15C2×#859C411yescharacteristic, complemented
16#4718—41nomaximal
17#2924—31no—
18C2×#416—31no—
19#2880D311yescharacteristic
20#1937—32no—
21#1929—32no—
22#1866—32no—
23#1826—31no—
24#1810—31no—
25#1789—31no—
26#1764—32no—
27C8×C22⋊C4—31no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3C2×C411yescharacteristic
30C2×#41⋊C3C2311yescharacteristic
31C22×SL(2,𝔽3)C2×C411yescharacteristic
32C2×C4×Dic3—41no—
33#131⋊C3—22no—
34GU(2,𝔽3)—22no—
35#859C2×C411yescharacteristic
36#859C2×C411yescharacteristic
37#859C2×C411yescharacteristic
38#859C2×C411yescharacteristic
39C3×C22⋊C8—41no—
40#830—41no—
41Dic3⋊C8—42no—
42C8×Dic3—42no—
43C22×#41C2×D311yescharacteristic
44C2×#131C2×D312yes—
45C23×C8—31no—
46C2×#117—31no—
47#433—62no—
48#432—61no—
49#431—61no—
50C2×#108—32no—
51#424—32no—
52C2×#104—32no—
53C2×#104—32no—
54#416—31no—
55#416—31no—
56#416—31no—
57#416—31no—
58#416—31no—
59#416—31no—
60#416—31no—
61#416—31no—
62C2×#103—31no—
63C2×#103—31no—
64C2×#102—31no—
65C2×#102—31no—
66#408—32no—
67#407—31no—
68#407—31no—
69C2×C22⋊C8—31no—
70#405—32no—
71C2×C4×C8—32no—
72C4×C22⋊C4—31no—
73#341—32no—
74#339—32no—
75#339—32no—
76#336—32no—
77C22×C3×C4—41no—
78C22×Dic3—41no—
79C22×Dic3—41no—
80#41⋊C3—21no—
81#41⋊C3#4112yes—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)C22×C411yescharacteristic
84C2×C3×C8—42no—
85C4×Dic3—41no—
86C4×Dic3—41no—
87C4×Dic3—41no—
88C4×Dic3—41no—
89C2×C3⋊C8—42no—
90C2×#41—22no—
91C2×#41—31no—
92C2×#41—31no—
93C2×#41—62no—
94C2×#41C4×D311yescharacteristic
95C2×#41C22×D311yescharacteristic
96C22×Q8C4×D311yescharacteristic
97C22×D4—31no—
98C23×C4—31no—
99#131—22no—
100#131—62no—
101C2×M16—32no—
102C2×M16—32no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—31no—
107C22×C8—31no—
108C22×C8—31no—
109C22×C8—32no—
110C22×C8—122no—
111#117—31no—
112#117—31no—
113#117—31no—
114#117—31no—
115#117—61no—
116#117—61no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C22⋊C4—61no—
120C2×C22⋊C4—61no—
121C2×C42—32no—
122C2×C42—61no—
123#108—122no—
124C4⋊C8—62no—
125C4⋊C8—62no—
126#104—62no—
127#104—62no—
128#104—122no—
129#103—31no—
130#103—31no—
131#103—31no—
132#103—31no—
133#103—31no—
134#103—31no—
135#103—31no—
136#103—31no—
137#102—31no—
138#102—31no—
139#102—31no—
140#102—31no—
141#102—31no—
142#102—31no—
143#102—31no—
144#102—31no—
145C22⋊C8—31no—
146C22⋊C8—32no—
147C22⋊C8—61no—
148C22⋊C8—61no—
149C22⋊C8—61no—
150C22⋊C8S411yescharacteristic
151#97—62no—
152C4×C8—62no—
153C4×C8—122no—
154#95—61no—
155C23×C3—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—82no—
159C2×Dic3—41no—
160C2×Dic3—41no—
161C2×Dic3—41no—
162C2×Dic3—41no—
163C2×Dic3—44no—
164C2×Dic3—82no—
165SL(2,𝔽3)#11711yescharacteristic
166C3×C8—82no—
167C3⋊C8—82no—
168C24—31no—
169#41—21no—
170#41—32no—
171#41—62no—
172#41—62no—
173#41—62no—
174#41—61no—
175#41—121no—
176#41#23712yes—
177C2×Q8—21no—
178C2×Q8—31no—
179C2×Q8—61no—
180C2×Q8C2×C4×D311yescharacteristic
181C2×D4—31no—
182C2×D4—31no—
183C2×D4—61no—
184C2×D4—62no—
185C2×D4—121no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4—61no—
196C22×C4C2×S411yescharacteristic
197M16—62no—
198M16—62no—
199M16—62no—
200M16—62no—
201C2×C8—31no—
202C2×C8—31no—
203C2×C8—31no—
204C2×C8—31no—
205C2×C8—32no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—62no—
210C2×C8—62no—
211C2×C8—121no—
212C2×C8—121no—
213C2×C8—121no—
214C2×C8—121no—
215C2×C8C2×S412yes—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C22⋊C4—61no—
223C22⋊C4—61no—
224C22⋊C4—61no—
225C22⋊C4—61no—
226C22⋊C4—64no—
227C42—62no—
228C42—62no—
229C42—61no—
230C42—61no—
231C42—61no—
232C42—61no—
233C22×C3—41no—
234C22×C3—42no—
235C22×C3—81no—
236C22×C3—81no—
237C3×C4—42no—
238C3×C4—81no—
239Dic3—42no—
240Dic3—42no—
241Dic3—42no—
242Dic3—81no—
243C23—31no—
244C23—31no—
245C23—62no—
246C23—62no—
247C23—121no—
248C23C4×S411yescharacteristic
249Q8—31no—
250Q8—61no—
251Q8—61no—
252Q8#87711yescharacteristic
253D4—32no—
254D4—61no—
255D4—61no—
256D4—61no—
257D4—122no—
258C2×C4—22no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—31no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—64no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—241no—
278C2×C4C4×S411yescharacteristic
279C2×C4C22×S411yescharacteristic
280C8—22no—
281C8—61no—
282C8—61no—
283C8—61no—
284C8—61no—
285C8—62no—
286C8—122no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—82no—
291C22—21no—
292C22—21no—
293C22—32no—
294C22—61no—
295C22—61no—
296C22—61no—
297C22—61no—
298C22—61no—
299C22—124no—
300C22#540412yes—
301C22C2×C4×S411yescharacteristic
302C4—21no—
303C4—31no—
304C4—31no—
305C4—61no—
306C4—124no—
307C4—121no—
308C4—121no—
309C4—121no—
310C4—121no—
311C4#591412yes—
312C3—41noSylow
313C2—22no—
314C2—62no—
315C2—121no—
316C2#8529211yescharacteristic
317C2C2×#540411yescharacteristic
318C2#8545211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2880D3
socleC22C2×C4×S4
prime radicalC22×#41⋊C3C22
cosocleC2×#41⋊C3C23
hypercenterC22⋊C8S4
hypocenterSL(2,𝔽3)#117
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. #73122C2semidirect
5. #73121C2semidirect
9. #5424C4semidirect
10. #5424C4semidirect
14. C2×#859C4semidirect
15. C2×#859C4semidirect

Maximal subgroups

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

subgroupindex
C2×#54242
#2880⋊C32
#731492
#731222
#731212
#225873
#47184