Finite Group Databasebeta

← C2×#11721096 #13150657

Subgroups — C2×#11721096 #13150657

Order 1728.

1363 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 195 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
1C22×C4×#272C211yesmaximal, characteristic
2C22×#6196C211yesmaximal, characteristic, complemented
3C2×#3750⋊C3C212yesmaximal, complemented
4C2×#88451C212yesmaximal
5C2×#88449C211yesmaximal, characteristic, complemented
6#11721096C218yesmaximal, direct product factor
7C2×#63851—32nomaximal
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C2×C4×#272—24no—
12C2×C4×#272C2214yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6196—24no—
16C2×#6196C2214yescomplemented
17#3750⋊C3C2218yescomplemented
18#88451C2218yes—
19#88449C2214yescomplemented
20C22×C3×C4×D3—32no—
21C22×C3×C4×D3—61no—
22C22×#627—32no—
23C2×#3750D312yescomplemented
24C2×C3×Dic3⋊C4—32no—
25C2×Dic3⋊Dic3—32no—
26C2×(C2×D3)⋊Dic3—32no—
27C2×(C2×D3)⋊Dic3—32no—
28#63851—316no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—22no—
31C22×#272—22no—
32C22×#272Q814yescomplemented
33C22×#272C2312yes—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)C2312yes—
36C2×#1084D414yes—
37C2×#1084C2312yes—
38C4×#272—28no—
39C2×(C3⋊Dic3)⋊C3—24no—
40C2×(C3⋊Dic3)⋊C3—24no—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42C2×(C3⋊Dic3)⋊C3C2314yes—
43#6196—28no—
44C2×#924—91no—
45C2×#896—92no—
46C23×C3×D3—32no—
47C23×C3×D3—61no—
48C22×C32×C4—21no—
49C22×C32×C4C2×D312yes—
50C22×C3⋊Dic3C2×D312yes—
51C22×C3⋊Dic3C2×D312yes—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—61no—
56C2×C3×C4×D3—38no—
57C2×C3×C4×D3—68no—
58C2×C3×C4×D3—64no—
59C2×C3×C4×D3—68no—
60C2×#627—38no—
61C2×#627—68no—
62#3750C2×D318yescomplemented
63C3×Dic3⋊C4—38no—
64Dic3⋊Dic3—38no—
65(C2×D3)⋊Dic3—38no—
66(C2×D3)⋊Dic3—38no—
67C22×H(3,𝔽3)C2×D412yes—
68C22×H(3,𝔽3)C2×Q812yes—
69C22×H(3,𝔽3)#4112yes—
70C22×H(3,𝔽3)C2411yescharacteristic
71C2×#272—28no—
72C2×#272—22no—
73C2×#272—22no—
74C2×#272C2×Q814yescomplemented
75C4×H(3,𝔽3)—22no—
76#1084C2×D414yes—
77(C3⋊Dic3)⋊C3—24no—
78(C3⋊Dic3)⋊C3—24no—
79C23×C3×C4—91no—
80C22×C4×D3—32no—
81C22×C4×D3—61no—
82C22×Dic6—91no—
83C22×Dic6—92no—
84C2×C3×C4⋊C4—92no—
85C2×#217—92no—
86C2×(C2×D3)⋊C4—92no—
87C2×(C2×D3)⋊C4—92no—
88C2×C4⋊Dic3—91no—
89#924—98no—
90C2×Dic3⋊C4—32no—
91C2×Dic3⋊C4—92no—
92C2×Dic3⋊C4—92no—
93#896—916no—
94C23×C32—21no—
95C23×C32C22×D312yes—
96C22×C3×D3—38no—
97C22×C3×D3—34no—
98C22×C3×D3—64no—
99C22×C3×D3—62no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C22×C3×D3—64no—
103C22×C3×D3—64no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—22no—
107C2×C32×C4C22×D314yes—
108C2×C3⋊Dic3—24no—
109C2×C3⋊Dic3—24no—
110C2×C3⋊Dic3C22×D314yes—
111C2×C3⋊Dic3C22×D314yes—
112C2×C3×Dic3—38no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—64no—
120C3×C4×D3—616no—
121C3×C4×D3—616no—
122#627—616no—
123C2×C22⋊Q8—271noSylow
124C2×H(3,𝔽3)C22⋊Q814yescomplemented
125C2×H(3,𝔽3)C22×D411yescharacteristic
126C2×H(3,𝔽3)C22×Q811yescharacteristic
127C2×H(3,𝔽3)C2×#4111yescharacteristic
128#272—24no—
129C24×C3—91no—
130C23×D3—32no—
131C23×D3—61no—
132C22×C3×C4—32no—
133C22×C3×C4—61no—
134C22×C3×C4—91no—
135C22×C3×C4—92no—
136C22×C3×C4—92no—
137C22×C3×C4—94no—
138C22×C3×C4—184no—
139C22×C3×C4D3211yescharacteristic
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—61no—
144C22×Dic3—92no—
145C22×Dic3—92no—
146C2×C4×D3—38no—
147C2×C4×D3—64no—
148C2×C4×D3—68no—
149C2×C4×D3—68no—
150C2×Dic6—94no—
151C2×Dic6—98no—
152C2×Dic6—188no—
153C2×Dic6—184no—
154C3×C4⋊C4—98no—
155#217—98no—
156(C2×D3)⋊C4—98no—
157(C2×D3)⋊C4—98no—
158C4⋊Dic3—94no—
159Dic3⋊C4—38no—
160Dic3⋊C4—98no—
161Dic3⋊C4—98no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32#23514yes—
167C22×C32D3×D414yes—
168C22×C32D3×Q814yes—
169C22×C32C23×D312yes—
170C2×C3×D3—38no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—616no—
175C2×C3×D3—616no—
176C2×C3×D3—64no—
177C2×C3×D3—64no—
178C32×C4—24no—
179C32×C4—24no—
180C3⋊Dic3—24no—
181C3⋊Dic3—24no—
182C3×Dic3—38no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C3×Dic3—64no—
186C22×Q8—271no—
187C23×C4—91no—
188C22⋊Q8—278no—
189C2×C4⋊C4—92no—
190C2×C4⋊C4—271no—
191C2×C22⋊C4—272no—
192H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
193C23×C3—32no—
194C23×C3—61no—
195C23×C3—92no—
196C23×C3—94no—
197C23×C3—182no—
198C23×C3—182no—
199C23×C3C2×D3211yescharacteristic
200C22×D3—34no—
201C22×D3—38no—
202C22×D3—62no—
203C22×D3—64no—
204C22×D3—64no—
205C22×D3—64no—
206C22×D3—64no—
207C22×D3—64no—
208C2×C3×C4—22no—
209C2×C3×C4—34no—
210C2×C3×C4—64no—
211C2×C3×C4—64no—
212C2×C3×C4—62no—
213C2×C3×C4—92no—
214C2×C3×C4—94no—
215C2×C3×C4—94no—
216C2×C3×C4—94no—
217C2×C3×C4—188no—
218C2×C3×C4—184no—
219C2×C3×C4—184no—
220C2×C3×C4C2×D3212yes—
221C2×Dic3—34no—
222C2×Dic3—38no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—62no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—94no—
231C2×Dic3—184no—
232C2×Dic3—184no—
233C4×D3—616no—
234C4×D3—616no—
235Dic6—1816no—
236Dic6—188no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—24no—
241C2×C32#89618yescomplemented
242C2×C32C2×#23512yes—
243C2×C32C2×D3×D412yes—
244C2×C32C2×D3×Q812yes—
245C3×D3—68no—
246C3×D3—68no—
247C24—91no—
248C2×Q8—274no—
249C2×Q8—544no—
250C22×C4—91no—
251C22×C4—92no—
252C22×C4—92no—
253C22×C4—94no—
254C22×C4—184no—
255C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C4⋊C4—98no—
257C4⋊C4—274no—
258C22⋊C4—278no—
259C22×C3—32no—
260C22×C3—34no—
261C22×C3—34no—
262C22×C3—34no—
263C22×C3—61no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3—94no—
268C22×C3—182no—
269C22×C3—182no—
270C22×C3—188no—
271C22×C3#379712yes—
272C22×C3#379812yes—
273C22×C3#381112yes—
274C22×C3C22×D3211yescharacteristic
275C2×D3—38no—
276C2×D3—64no—
277C2×D3—64no—
278C2×D3—616no—
279C2×D3—64no—
280C2×D3—64no—
281C2×D3—616no—
282C2×D3—64no—
283C3×C4—22no—
284C3×C4—64no—
285C3×C4—64no—
286C3×C4—94no—
287C3×C4—184no—
288C3×C4—184no—
289Dic3—38no—
290Dic3—64no—
291Dic3—64no—
292Dic3—64no—
293Dic3—184no—
294Dic3—184no—
295C32—21no—
296C32C2×#89612yescomplemented
297C23—92no—
298C23—94no—
299C23—182no—
300C23—182no—
301C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
302Q8—548no—
303C2×C4—22no—
304C2×C4—92no—
305C2×C4—94no—
306C2×C4—94no—
307C2×C4—94no—
308C2×C4—188no—
309C2×C4—184no—
310C2×C4—184no—
311C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
312C2×C3—32no—
313C2×C3—38no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—61no—
317C2×C3—64no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3—184no—
321C2×C3#6386514yes—
322C2×C3C2×#379711yescharacteristic
323C2×C3C2×#379811yescharacteristic
324C2×C3C2×#381111yescharacteristic
325D3—68no—
326D3—68no—
327C22—94no—
328C22—182no—
329C22—188no—
330C22—182no—
331C22#8865012yes—
332C22#8865112yes—
333C22#8867712yes—
334C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C4—22no—
336C4—94no—
337C4—184no—
338C4—184no—
339C3—32no—
340C3—61no—
341C3C2×#6386511yescharacteristic
342C2—184no—
343C2#1172109614yesdirect product factor
344C2C2×#8865011yescharacteristic
345C2C2×#8865111yescharacteristic
346C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6196C2semidirect
3. C2×#3750⋊C3C2semidirect
5. C2×#88449C2semidirect
6. #11721096C2direct (N × H)
16. C2×#6196C22semidirect
17. #3750⋊C3C22semidirect
19. #88449C22semidirect
23. C2×#3750D3semidirect
32. C22×#272Q8semidirect
62. #3750C2×D3semidirect
74. C2×#272C2×Q8semidirect
124. C2×H(3,𝔽3)C22⋊Q8semidirect
192. H(3,𝔽3)C2×C22⋊Q8semidirect
241. C2×C32#896semidirect
296. C32C2×#896semidirect
343. C2#11721096direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#61962
C2×#3750⋊C32
C2×#884512
C2×#884492
#117210962
C2×#638513