Finite Group Databasebeta

← #11878875

Subgroups — #11878875

Order 1152.

1152 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 241 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#10596199C211yesmaximal, characteristic
2C3⋊D3×C4⋊Q8C211yesmaximal, characteristic
3C32×#494C211yesmaximal, characteristic, complemented
4#10592885C214yesmaximal
5#10592881C214yesmaximal, complemented
6#10592791C211yesmaximal, characteristic, complemented
7#10592778C211yesmaximal, characteristic, complemented
8#10592384C211yesmaximal, characteristic
9#10592380C211yesmaximal, characteristic, complemented
10#81802—34nomaximal
11C2×Q8×C3⋊D3C2212yes—
12C2×C32×Q16C2214yes—
13C32×C4⋊Q8C2211yescharacteristic
14C32×C4⋊Q8C2211yescharacteristic
15#64155C2212yes—
16Q8×C3⋊Dic3C2212yes—
17#64153C2212yes—
18C2×#3756C2214yes—
19#64106—21no—
20#64101—21no—
21C4⋊C4×C3⋊D3—21no—
22C4⋊C4×C3⋊D3—21no—
23#64099—21no—
24#64097—21no—
25#64097—21no—
26#64096—21no—
27C42×C3⋊D3C2211yescharacteristic
28#64078C2211yescharacteristic
29#64078C2211yescharacteristic
30C32×C4×C8C2211yescharacteristic
31#63662C2214yes—
32#63647C2212yes—
33#63643C2214yes—
34#63641C2212yes—
35C4×C3⋊(C3⋊C8)C2211yescharacteristic
36#5728—34no—
37D3×C4⋊Q8—34no—
38C3×#494D314yescomplemented
39#5186—316no—
40#5182—316no—
41#5092—34no—
42#5079—34no—
43#4685—34no—
44#4681—34no—
45C2×C32×Q8C2312yes—
46C2×C32×Q8C2312yes—
47Q8×C3⋊D3—44no—
48C2×C4×C3⋊D3—22no—
49C2×C4×C3⋊D3D411yescharacteristic
50C2×C4×C3⋊D3D411yescharacteristic
51C2×C4×C3⋊D3C2311yescharacteristic
52C2×#634—21no—
53C2×#634—22no—
54C2×#634C2312yes—
55C32×Q16—44no—
56C2×C32×C8C2312yes—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C42C2311yescharacteristic
62#3756—44no—
63#3752—22no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)—21no—
67C4⋊(C3⋊Dic3)—22no—
68C4⋊(C3⋊Dic3)C2312yes—
69#3750—22no—
70#3750—22no—
71#3750—22no—
72C4×C3⋊Dic3—22no—
73C4×C3⋊Dic3D411yescharacteristic
74C4×C3⋊Dic3D411yescharacteristic
75C4×C3⋊Dic3C2311yescharacteristic
76C2×C3⋊(C3⋊C8)C2312yes—
77#3209—91noSylow
78C2×D3×Q8—38no—
79C2×C3×Q16C2×D3116yes—
80C3×C4⋊Q8C2×D314yes—
81C3×C4⋊Q8C2×D314yes—
82#946—38no—
83Q8×Dic3—38no—
84#944—38no—
85C2×#216—316no—
86#897—64no—
87#892—64no—
88D3×C4⋊C4—64no—
89D3×C4⋊C4—64no—
90#890—64no—
91#888—64no—
92#888—64no—
93#887—64no—
94C42×D3—34no—
95#869—34no—
96#869—34no—
97C3×C4×C8C2×D314yes—
98#835—316no—
99(C2×D3)⋊C8—38no—
100#816—316no—
101Dic3⋊C8—38no—
102C4×C3⋊C8—34no—
103C22×C3⋊D3C2×D411yescharacteristic
104C32×Q8—24no—
105C32×Q8—24no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2411yescharacteristic
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3C2×D411yescharacteristic
114C2×C3⋊Dic3C2×D411yescharacteristic
115C2×C3⋊Dic3C2×D411yescharacteristic
116C4×C3⋊D3—22no—
117C4×C3⋊D3—22no—
118C4×C3⋊D3—44no—
119C4×C3⋊D3Q1614yescomplemented
120#634—24no—
121#634—44no—
122#634—42no—
123C32×C8—22no—
124C3⋊(C3⋊C8)—22no—
125#533—91no—
126C2×C4⋊Q8—91no—
127#494—91no—
128#494C3⋊D311yescharacteristic, complemented
129#487—91no—
130#487—91no—
131#462—94no—
132#451—94no—
133#431—91no—
134C2×C3×Q8C22×D318yes—
135C2×C3×Q8C22×D318yes—
136D3×Q8—1216no—
137C2×C4×D3—34no—
138C2×C4×D3—34no—
139C2×C4×D3—34no—
140C2×C4×D3—68no—
141C2×Dic6—38no—
142C2×Dic6—68no—
143C2×Dic6—64no—
144C3×Q16—416no—
145C2×C3×C8C22×D318yes—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C3×C42C22×D314yes—
151#216—1216no—
152(C2×D3)⋊C4—68no—
153C4⋊Dic3—38no—
154C4⋊Dic3—68no—
155C4⋊Dic3—64no—
156C4⋊Dic3—64no—
157C4⋊Dic3—64no—
158Dic3⋊C4—68no—
159Dic3⋊C4—68no—
160Dic3⋊C4—68no—
161C4×Dic3—34no—
162C4×Dic3—34no—
163C4×Dic3—34no—
164C4×Dic3—68no—
165C2×C3⋊C8—38no—
166C22×C32C22×D411yescharacteristic
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3C2×Q1612yes—
170C32×C4—42no—
171C32×C4—42no—
172C32×C4C2×Q1612yes—
173C32×C4#13712yes—
174C32×C4H(5,𝔽2)12yes—
175C3⋊Dic3—21no—
176C3⋊Dic3—21no—
177C3⋊Dic3—42no—
178C3⋊Dic3—42no—
179C3⋊Dic3C2×Q1612yes—
180C22×Q8—92no—
181C2×Q16—94no—
182C2×Q16C2×C3⋊D314yes—
183C4⋊Q8—92no—
184C4⋊Q8—91no—
185C4⋊Q8—91no—
186C4⋊Q8—181no—
187C4⋊Q8—181no—
188C4⋊Q8C2×C3⋊D311yescharacteristic
189C4⋊Q8C2×C3⋊D311yescharacteristic
190#125—181no—
191C22⋊Q8—92no—
192C22⋊Q8—181no—
193C4×Q8—92no—
194C4×Q8—181no—
195#117—181no—
196C2×C4⋊C4—181no—
197C2×C4⋊C4—181no—
198C2×C42—91no—
199C4⋊C8—92no—
200#103—94no—
201#103—94no—
202C22⋊C8—92no—
203C4×C8—91no—
204C4×C8C2×C3⋊D311yescharacteristic
205C22×D3—34no—
206C3×Q8—216no—
207C3×Q8—216no—
208C2×C3×C4—28no—
209C2×C3×C4—28no—
210C2×C3×C4D3×D414yes—
211C2×C3×C4D3×D414yes—
212C2×C3×C4C23×D314yes—
213C2×Dic3—34no—
214C2×Dic3—34no—
215C2×Dic3—34no—
216C2×Dic3—68no—
217C2×Dic3—68no—
218C4×D3—316no—
219C4×D3—68no—
220C4×D3—68no—
221C4×D3—1216no—
222Dic6—616no—
223Dic6—1216no—
224Dic6—128no—
225C3×C8—28no—
226C3⋊C8—68no—
227C2×C32#53511yescharacteristic
228C2×C32C22×Q1611yescharacteristic
229C2×C32C2×#13711yescharacteristic
230C3⋊D3—22no—
231C2×Q8—92no—
232C2×Q8—182no—
233C2×Q8—181no—
234C2×Q8—364no—
235C2×Q8C22×C3⋊D312yes—
236C2×Q8C22×C3⋊D312yes—
237C22×C4—91no—
238C22×C4—91no—
239C22×C4—91no—
240C22×C4—182no—
241Q16—44no—
242Q16—364no—
243C2×C8—92no—
244C2×C8C22×C3⋊D312yes—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—92no—
250C4⋊C4—182no—
251C4⋊C4—182no—
252C4⋊C4—182no—
253C4⋊C4—182no—
254C4⋊C4—181no—
255C4⋊C4—181no—
256C4⋊C4—181no—
257C22⋊C4—182no—
258C42—91no—
259C42—91no—
260C42—91no—
261C42—182no—
262C42C22×C3⋊D311yescharacteristic
263C22×C3C2×D3×D414yes—
264C2×D3—38no—
265C2×D3—64no—
266C2×D3—64no—
267C3×C4—48no—
268C3×C4—48no—
269C3×C4D3×Q1618yes—
270C3×C4#91818yes—
271C3×C4#100418yes—
272Dic3—38no—
273Dic3—64no—
274Dic3—64no—
275Dic3—128no—
276Dic3—128no—
277C32#320911yescharacteristic, complemented, Sylow
278C23—91no—
279Q8—24no—
280Q8—24no—
281Q8—184no—
282Q8—364no—
283Q8—362no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—91no—
287C2×C4—94no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—364no—
295C2×C4D4×C3⋊D311yescharacteristic
296C2×C4D4×C3⋊D311yescharacteristic
297C2×C4C23×C3⋊D311yescharacteristic
298C8—22no—
299C8—182no—
300C2×C3#571514yes—
301C2×C3C2×D3×Q1614yes—
302C2×C3C2×#91814yes—
303D3—68no—
304C22—92no—
305C22—181no—
306C22—181no—
307C22C2×D4×C3⋊D311yescharacteristic
308C4—42no—
309C4—42no—
310C4—92no—
311C4—181no—
312C4—181no—
313C4—362no—
314C4—362no—
315C4Q16×C3⋊D312yes—
316C4#6412712yes—
317C4#6436112yes—
318C3#8180214yescomplemented
319C2—182no—
320C2#1059618611yescharacteristic
321C2C2×Q16×C3⋊D311yescharacteristic
322C2C2×#6412711yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#494C2
socleC22×C32C22×D4
prime radicalC22×C3⋊D3C2×D4
hypercenter#494C3⋊D3
hypocenterC32#3209
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. C32×#494C2semidirect
5. #10592881C2semidirect
6. #10592791C2semidirect
7. #10592778C2semidirect
9. #10592380C2semidirect
38. C3×#494D3semidirect
119. C4×C3⋊D3Q16semidirect
128. #494C3⋊D3semidirect
277. C32#3209semidirect
318. C3#81802semidirect

Maximal subgroups

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

subgroupindex
#105961992
C3⋊D3×C4⋊Q82
C32×#4942
#105928852
#105928812
#105927912
#105927782
#105923842
#105923802
#818023