Finite Group Databasebeta

← #11782634

Subgroups — #11782634

Order 1152.

455 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 102 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#10592769C211yesmaximal, characteristic
2C3×#5076C211yesmaximal, characteristic
3C3×#5076C211yesmaximal, characteristic
4#10590311C211yesmaximal, characteristic, complemented
5#10590309C211yesmaximal, characteristic, complemented
6#10590288C211yesmaximal, characteristic, complemented
7#10589078C211yesmaximal, characteristic, complemented
8#71258—31nomaximal
9#71257—31nomaximal
10C32×#127C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#214—22no—
13C2×C3×#214—22no—
14C3×#869C2211yescharacteristic
15C3×#869C2211yescharacteristic
16#63659—22no—
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63575—22no—
21#63574—22no—
22#63567—22no—
23C3×#492—31no—
24C3×#492—31no—
25#5093—31no—
26#5080—31no—
27#5080—31no—
28#5078—31no—
29#5076D311yescharacteristic
30#5076D311yescharacteristic
31#5070—31no—
32#5070—31no—
33#5070—61no—
34#4682—31no—
35#4681—31no—
36#4455—31no—
37#4454—31no—
38C2×C32×D4—21no—
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×#634—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C32×C42C2311yescharacteristic
45C4⋊(C3⋊Dic3)—21no—
46C4⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊C8)D412yes—
48C3×#214—44no—
49C3×#214—44no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C2×C3×C3⋊C8D412yes—
55C2×C3×C3⋊C8D412yes—
56#1714—91noSylow
57C2×C3×SD16—62no—
58C2×C3×SD16—62no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8—31no—
61C3×#127—21no—
62C3×#127C2×D311yescharacteristic
63C3×#127C2×D311yescharacteristic
64C2×#214—22no—
65C2×#214—22no—
66#869—31no—
67#869—31no—
68#869—61no—
69#869C2×D311yescharacteristic
70#869C2×D311yescharacteristic
71C3×C4×C8—31no—
72C3×C4×C8—31no—
73#832—62no—
74#832—62no—
75#832—64no—
76#818—62no—
77#816—62no—
78#810—62no—
79#808—62no—
80#808—62no—
81#807—62no—
82C4×C3⋊C8—31no—
83C4×C3⋊C8—31no—
84C4×C3⋊C8—61no—
85C4×C3⋊C8C2×D311yescharacteristic
86C4×C3⋊C8C2×D311yescharacteristic
87C23×C32—41no—
88C32×D4—42no—
89C32×D4—42no—
90C32×D4—42no—
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—41no—
95#634—42no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C3×Dic6—42no—
99C3×Dic6—42no—
100C3⋊(C3⋊C8)—22no—
101C3×C3⋊C8—22no—
102C3×C3⋊C8SD1614yescomplemented
103#500—91no—
104#498—91no—
105#492—31no—
106#492—31no—
107#487—91no—
108#486—91no—
109#322—91no—
110C2×C3×Q8—61no—
111C2×C3×Q8—61no—
112C2×C3×D4—22no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—61no—
124C2×Dic6—61no—
125C2×Dic6—121no—
126C3×SD16—124no—
127C3×SD16—124no—
128C2×C3×C8—32no—
129C2×C3×C8—32no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C42—21no—
135C3×C42C22×D311yescharacteristic
136C3×C42C22×D311yescharacteristic
137#214—44no—
138#214—44no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—62no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—121no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—62no—
152C2×C3⋊C8#6712yes—
153C2×C3⋊C8#6712yes—
154C22×C32—81no—
155C22×C32—81no—
156C22×C32—81no—
157C22×C32#12711yescharacteristic
158C32×C4C2×SD1612yes—
159C32×C4AGL(1,ℤ/8)12yes—
160C32×C4#13712yes—
161C3⋊Dic3—81no—
162C3×Dic3—81no—
163C3×Dic3—81no—
164C2×SD16—62no—
165C2×SD16—62no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—91no—
169#127D3211yescharacteristic
170#107—182no—
171#106—182no—
172#103—182no—
173#102—182no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8—91no—
177C23×C3—42no—
178C23×C3—41no—
179C23×C3—41no—
180C3×Q8—122no—
181C3×Q8—122no—
182C3×D4—44no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—42no—
186C3×D4—42no—
187C3×D4—42no—
188C3×D4—42no—
189C3×D4—82no—
190C3×D4—82no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—121no—
195C2×C3×C4—121no—
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×C3×C4D3×D411yescharacteristic
200C2×C3×C4C2×#6711yescharacteristic
201C2×C3×C4C2×#6711yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—122no—
205C2×Dic3—121no—
206C2×Dic3—121no—
207Dic6—42no—
208Dic6—42no—
209Dic6—122no—
210Dic6—122no—
211Dic6—242no—
212C3×C8—34no—
213C3×C8—62no—
214C3⋊C8—22no—
215C3⋊C8—62no—
216C3⋊C8—62no—
217C3⋊C8—122no—
218C3⋊C8#21414yescomplemented
219C2×C32—161no—
220C2×C32#49211yescharacteristic
221C2×C32#49611yescharacteristic
222C2×C32#49711yescharacteristic
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—181no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229SD16—124no—
230SD16—124no—
231C2×C8—32no—
232C2×C8—32no—
233C2×C8—92no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C42C2×D3211yescharacteristic
241C22×C3—21no—
242C22×C3—82no—
243C22×C3—82no—
244C22×C3—82no—
245C22×C3—81no—
246C22×C3—81no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3#94011yescharacteristic
252C22×C3#94011yescharacteristic
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—241no—
257C3×C4—241no—
258C3×C4D3×SD1612yes—
259C3×C4#91412yes—
260C3×C4#91512yes—
261C3×C4#91512yes—
262C3×C4#93212yes—
263C3×C4C2×#21412yes—
264Dic3—81no—
265Dic3—81no—
266Dic3—242no—
267Dic3—241no—
268Dic3—241no—
269C32#171411yescharacteristic, complemented, Sylow
270C23—41no—
271Q8—122no—
272Q8—122no—
273Q8—362no—
274D4—42no—
275D4—42no—
276D4—42no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—361no—
280C2×C4D3×#6711yescharacteristic
281C2×C4D3×#6711yescharacteristic
282C2×C4#381111yescharacteristic
283C8—34no—
284C8—62no—
285C8—182no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—162no—
290C2×C3—161no—
291C2×C3—161no—
292C2×C3#507411yescharacteristic
293C2×C3#507611yescharacteristic
294C2×C3#516711yescharacteristic
295C2×C3#516711yescharacteristic
296C2×C3#517511yescharacteristic
297C2×C3#517611yescharacteristic
298C22—81no—
299C22—81no—
300C22—81no—
301C22#6397911yescharacteristic
302C4—241no—
303C4—241no—
304C4—721no—
305C4D3×#21412yes—
306C4#6393112yes—
307C4#6393412yes—
308C3—21no—
309C3#7125711yescharacteristic, complemented
310C3#7125811yescharacteristic, complemented
311C2—161no—
312C2#1059147211yescharacteristic
313C2#1059149011yescharacteristic
314C2#1059149311yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
Fitting, prime radicalC32×#127C22
socleC22×C32#127
hypercenter#127D32
hypocenterC32#1714
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
4. #10590311C2semidirect
5. #10590309C2semidirect
6. #10590288C2semidirect
7. #10589078C2semidirect
102. C3×C3⋊C8SD16semidirect
218. C3⋊C8#214semidirect
269. C32#1714semidirect
309. C3#71257semidirect
310. C3#71258semidirect

Maximal subgroups

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

subgroupindex
#105927692
C3×#50762
#105903112
#105903092
#105902882
#105890782
#712583
#712573