Finite Group Databasebeta

← C2×#11720836 #13149805

Subgroups — C2×#11720836 #13149805

Order 1728.

1119 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 211 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C212yesmaximal, complemented
4C2×#88434C211yesmaximal, characteristic, complemented
5C2×#88430C212yesmaximal, complemented
6#11720836C218yesmaximal, direct product factor
7C2×D3×ΓL(1,𝔽25)—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
9C2×C4×#272C2211yescharacteristic
10C2×C4×(C3⋊D3)⋊C3C2212yes—
11C2×#6199C411yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4C412yes—
13C4×((C3⋊D3)⋊C3)⋊C2C414yes—
14C4×((C3⋊D3)⋊C3)⋊C2C2214yes—
15C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
16C2×#6180C2211yescharacteristic, complemented
17C8×#272C2214yescomplemented
18C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
19#3743⋊C3C2218yescomplemented
20#88434C2214yescomplemented
21#88430C2218yescomplemented
22C2×C4×D32—32no—
23C2×#3743D312yescomplemented
24C2×C3×ΓL(1,𝔽25)—32no—
25C2×C3×D3×C8—32no—
26C2×C3×D3×C8—61no—
27C2×#3712—32no—
28C2×#3711—32no—
29C2×D3×C3⋊C8—32no—
30D3×ΓL(1,𝔽25)—316no—
31C22×#272C2×C411yescharacteristic
32C22×(C3⋊D3)⋊C3C2×C412yes—
33C2×((C3⋊D3)⋊C3)⋊C2—24no—
34C2×((C3⋊D3)⋊C3)⋊C2C2×C414yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084C2×C411yescharacteristic
37C4×#272C2×C412yes—
38C4×#272C2312yes—
39C2×(C3⋊Dic3)⋊C3C2×C412yes—
40C4×(C3⋊D3)⋊C3C2×C414yes—
41C4×(C3⋊D3)⋊C3C2314yes—
42#6199C2×C414yes—
43((C3⋊D3)⋊C3)⋊C4C2×C418yes—
44C8×H(3,𝔽3)C2312yescomplemented
45#6180C2312yescomplemented
46(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
47C2×D3×M16—92no—
48C22×ΓL(1,𝔽25)—91no—
49C22×D32—32no—
50C2×C4×C3⋊D3C2×D312yes—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—32no—
53C2×C3×C4×D3—61no—
54C2×#624—32no—
55C2×D3×Dic3—32no—
56C2×D3×Dic3—32no—
57C4×D32—38no—
58C4×D32—38no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D312yescomplemented
61C2×C3⋊(C3⋊C8)C2×D312yescomplemented
62#3743C2×D318yescomplemented
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66C3×ΓL(1,𝔽25)—38no—
67C3×D3×C8—38no—
68C3×D3×C8—64no—
69#3712—38no—
70#3711—38no—
71D3×C3⋊C8—38no—
72C22×H(3,𝔽3)C22×C411yescharacteristic
73C2×#272M1614yescomplemented
74C2×#272C22×C412yes—
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3C22×C414yes—
77((C3⋊D3)⋊C3)⋊C2—28no—
78C4×H(3,𝔽3)C22×C411yescharacteristic
79C4×H(3,𝔽3)C2411yescharacteristic
80#1084C22×C412yes—
81(C3⋊Dic3)⋊C3C22×C414yes—
82C22×C4×D3—91no—
83C22×C4×D3—92no—
84C2×C3×M16—92no—
85C22×C3×C8—91no—
86C2×#208—92no—
87C22×C3⋊C8—91no—
88D3×M16—916no—
89C2×ΓL(1,𝔽25)—32no—
90C2×ΓL(1,𝔽25)—92no—
91C2×ΓL(1,𝔽25)—98no—
92C2×ΓL(1,𝔽25)—92no—
93C2×ΓL(1,𝔽25)—92no—
94C2×D3×C8—32no—
95C2×D3×C8—61no—
96C2×D3×C8—92no—
97C22×C3⋊D3C4×D312yes—
98C22×C3×D3—32no—
99C22×C3×D3—32no—
100C22×C3×D3—61no—
101C2×D32—38no—
102C2×D32—68no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D312yes—
105C2×C3⋊Dic3C4×D312yes—
106C4×C3⋊D3C4×D314yes—
107C4×C3⋊D3C22×D314yes—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—61no—
111C3×C4×D3—34no—
112C3×C4×D3—34no—
113C3×C4×D3—34no—
114C3×C4×D3—34no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117#624—38no—
118D3×Dic3—38no—
119D3×Dic3—38no—
120C32×C8—22no—
121C32×C8C22×D314yescomplemented
122C3⋊(C3⋊C8)C22×D314yescomplemented
123C3×C3⋊C8—34no—
124C3×C3⋊C8—34no—
125C3×C3⋊C8—62no—
126C22×M16—271noSylow
127C2×H(3,𝔽3)C2×M1612yescomplemented
128C2×H(3,𝔽3)C23×C411yescharacteristic
129#272C2×M1614yescomplemented
130(C3⋊D3)⋊C3—24no—
131C23×D3—91no—
132C23×D3—92no—
133C22×C3×C4—92no—
134C22×C3×C4—91no—
135C22×Dic3—91no—
136C22×Dic3—92no—
137C2×C4×D3—32no—
138C2×C4×D3—32no—
139C2×C4×D3—61no—
140C2×C4×D3—92no—
141C2×C4×D3—94no—
142C2×C4×D3—94no—
143C2×C4×D3—98no—
144C2×C4×D3—98no—
145C2×C4×D3—92no—
146C2×C4×D3—92no—
147C2×C4×D3—92no—
148C3×M16—98no—
149C2×C3×C8—32no—
150C2×C3×C8—61no—
151C2×C3×C8—92no—
152C2×C3×C8—91no—
153C2×C3×C8—94no—
154C2×C3×C8D3211yescharacteristic
155#208—98no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—92no—
160C2×C3⋊C8—94no—
161ΓL(1,𝔽25)—38no—
162ΓL(1,𝔽25)—98no—
163ΓL(1,𝔽25)—98no—
164ΓL(1,𝔽25)—98no—
165D3×C8—38no—
166D3×C8—64no—
167D3×C8—98no—
168C22×C32—21no—
169C22×C32C2×C4×D312yes—
170C2×C3⋊D3—24no—
171C2×C3⋊D3C2×C4×D314yes—
172C2×C3×D3—38no—
173C2×C3×D3—34no—
174C2×C3×D3—34no—
175C2×C3×D3—64no—
176C2×C3×D3—62no—
177C2×C3×D3—64no—
178D32—616no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4C2×C4×D312yes—
182C32×C4C23×D312yes—
183C3⋊Dic3C2×C4×D314yes—
184C3×Dic3—34no—
185C3×Dic3—34no—
186C3×Dic3—62no—
187C23×C4—271no—
188C2×M16—92no—
189C2×M16—272no—
190C2×M16—278no—
191C22×C8—91no—
192C22×C8—271no—
193H(3,𝔽3)C22×M1611yescharacteristic, complemented, Sylow
194C23×C3—92no—
195C23×C3—91no—
196C22×D3—32no—
197C22×D3—32no—
198C22×D3—61no—
199C22×D3—92no—
200C22×D3—94no—
201C22×D3—98no—
202C22×D3—188no—
203C22×D3—184no—
204C2×C3×C4—32no—
205C2×C3×C4—61no—
206C2×C3×C4—94no—
207C2×C3×C4—94no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—92no—
212C2×C3×C4C2×D3211yescharacteristic
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—61no—
216C2×Dic3—92no—
217C2×Dic3—94no—
218C2×Dic3—98no—
219C4×D3—34no—
220C4×D3—34no—
221C4×D3—34no—
222C4×D3—34no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—94no—
226C4×D3—94no—
227C4×D3—98no—
228C4×D3—98no—
229C4×D3—98no—
230C3×C8—34no—
231C3×C8—62no—
232C3×C8—94no—
233C3×C8—92no—
234C3×C8C2×D3212yes—
235C3⋊C8—34no—
236C3⋊C8—34no—
237C3⋊C8—62no—
238C3⋊C8—94no—
239C2×C32—21no—
240C2×C32—22no—
241C2×C32D3×M1614yescomplemented
242C2×C32C22×C4×D312yes—
243C3⋊D3—24no—
244C3×D3—38no—
245C3×D3—64no—
246C3×D3—64no—
247C24—271no—
248C22×C4—91no—
249C22×C4—92no—
250C22×C4—274no—
251C22×C4—274no—
252C22×C4—272no—
253C22×C4—271no—
254M16—98no—
255M16—278no—
256C2×C8—91no—
257C2×C8—92no—
258C2×C8—94no—
259C2×C8—274no—
260C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
261C22×C3—32no—
262C22×C3—61no—
263C22×C3—94no—
264C22×C3—92no—
265C22×C3—94no—
266C22×C3—184no—
267C22×C3C4×D3211yescharacteristic
268C2×D3—34no—
269C2×D3—38no—
270C2×D3—34no—
271C2×D3—62no—
272C2×D3—64no—
273C2×D3—64no—
274C2×D3—94no—
275C2×D3—184no—
276C2×D3—1816no—
277C2×D3—188no—
278C3×C4—32no—
279C3×C4—32no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—94no—
283C3×C4—92no—
284C3×C4C4×D3211yescharacteristic
285C3×C4C22×D3211yescharacteristic
286Dic3—34no—
287Dic3—34no—
288Dic3—62no—
289Dic3—94no—
290C32—21no—
291C32C2×D3×M1612yescomplemented
292C23—91no—
293C23—92no—
294C23—274no—
295C23—544no—
296C2×C4—91no—
297C2×C4—92no—
298C2×C4—92no—
299C2×C4—94no—
300C2×C4—92no—
301C2×C4—94no—
302C2×C4—278no—
303C2×C4—274no—
304C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C8—92no—
306C8—94no—
307C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
308C2×C3—34no—
309C2×C3—32no—
310C2×C3—62no—
311C2×C3—61no—
312C2×C3—94no—
313C2×C3—184no—
314C2×C3#6379112yes—
315C2×C3C2×C4×D3211yescharacteristic
316D3—38no—
317D3—64no—
318D3—64no—
319D3—184no—
320C22—94no—
321C22—92no—
322C22—94no—
323C22—184no—
324C22—548no—
325C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C4—92no—
327C4—94no—
328C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
330C3—32no—
331C3—61no—
332C3C2×#6379111yescharacteristic
333C2—94no—
334C2—184no—
335C2#1172083612yesdirect product factor
336C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C4
cosocleC4×H(3,𝔽3)C24
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. C2×#3743⋊C3C2semidirect
4. C2×#88434C2semidirect
5. C2×#88430C2semidirect
6. #11720836C2direct (N × H)
15. C2×C8×H(3,𝔽3)C22semidirect
16. C2×#6180C22semidirect
17. C8×#272C22semidirect
18. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
19. #3743⋊C3C22semidirect
20. #88434C22semidirect
21. #88430C22semidirect
23. C2×#3743D3semidirect
44. C8×H(3,𝔽3)C23semidirect
45. #6180C23semidirect
46. (C3⋊(C3⋊C8))⋊C3C23semidirect
60. C2×C32×C8C2×D3semidirect
61. C2×C3⋊(C3⋊C8)C2×D3semidirect
62. #3743C2×D3semidirect
73. C2×#272M16semidirect
121. C32×C8C22×D3semidirect
122. C3⋊(C3⋊C8)C22×D3semidirect
127. C2×H(3,𝔽3)C2×M16semidirect
129. #272C2×M16semidirect
193. H(3,𝔽3)C22×M16semidirect
241. C2×C32D3×M16semidirect
260. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
291. C32C2×D3×M16semidirect
307. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
335. C2#11720836direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×C8×#2722
C2×#3743⋊C32
C2×#884342
C2×#884302
#117208362
C2×D3×ΓL(1,𝔽25)3