Finite Group Databasebeta

← C2×#11721309 #13152057

Subgroups — C2×#11721309 #13152057

Order 1728.

1363 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 227 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
1C23×#1084C211yesmaximal, characteristic
2C22×#6196C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C212yesmaximal
4#11721309C218yesmaximal, direct product factor
5C2×#88449C211yesmaximal, characteristic, complemented
6C2×#88448C212yesmaximal, complemented
7C2×#63964—32nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—24no—
10C22×#1084C2211yescharacteristic
11C22×#1084C2214yes—
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6196—24no—
16C2×#6196C2214yescomplemented
17#3757⋊C3C2218yes—
18#88449C2214yescomplemented
19#88448C2218yescomplemented
20C23×C3×Dic3—32no—
21C23×C3×Dic3—61no—
22C22×#627—32no—
23C2×#3757D312yes—
24C2×C3×#217—32no—
25C2×#3724—32no—
26C2×Dic3⋊Dic3—32no—
27C2×Dic3⋊Dic3—32no—
28#63964—316no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)Q814yescomplemented
32C23×H(3,𝔽3)C2312yes—
33C23×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—22no—
35C2×#1084—28no—
36C2×#1084D414yes—
37C2×#1084C2312yes—
38C2×#1084C2312yes—
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×#878—92no—
46C24×C32—21no—
47C24×C32C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C22×C3⋊Dic3C2×D312yes—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—38no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—68no—
56C22×C3×Dic3—61no—
57C22×C3×Dic3—61no—
58C22×C3×Dic3—64no—
59C22×C3×Dic3—68no—
60C2×#627—38no—
61C2×#627—68no—
62#3757C2×D318yes—
63C3×#217—38no—
64#3724—38no—
65Dic3⋊Dic3—38no—
66Dic3⋊Dic3—38no—
67C22×H(3,𝔽3)—28no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)C2×D412yes—
71C22×H(3,𝔽3)C2×Q812yes—
72C22×H(3,𝔽3)C2×Q814yescomplemented
73C22×H(3,𝔽3)#4112yes—
74C22×H(3,𝔽3)C2411yescharacteristic
75#1084—22no—
76#1084C2×D414yes—
77(C3⋊Dic3)⋊C3—24no—
78(C3⋊Dic3)⋊C3—24no—
79C23×C3×C4—91no—
80C23×Dic3—32no—
81C23×Dic3—61no—
82C22×Dic6—91no—
83C22×Dic6—92no—
84C2×C3×C22⋊C4—92no—
85C2×#217—32no—
86C2×#217—92no—
87C2×C4⋊Dic3—91no—
88C2×C4⋊Dic3—92no—
89#924—98no—
90C2×Dic3⋊C4—92no—
91C2×Dic3⋊C4—92no—
92C2×Dic3⋊C4—92no—
93#878—916no—
94C23×C32—24no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—24no—
98C23×C32—24no—
99C23×C32—22no—
100C23×C32—21no—
101C23×C32Dic618yescomplemented
102C23×C32C22×D314yes—
103C23×C32C22×D312yes—
104C2×C3⋊Dic3—24no—
105C2×C3⋊Dic3—24no—
106C2×C3⋊Dic3C22×D314yes—
107C2×C3⋊Dic3C22×D314yes—
108C2×C3×Dic3—38no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—616no—
117C2×C3×Dic3—616no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122#627—616no—
123C2×C22⋊Q8—271noSylow
124C2×H(3,𝔽3)—24no—
125C2×H(3,𝔽3)C22⋊Q814yescomplemented
126C2×H(3,𝔽3)C22×D411yescharacteristic
127C2×H(3,𝔽3)C22×Q811yescharacteristic
128C2×H(3,𝔽3)C2×#4111yescharacteristic
129C24×C3—32no—
130C24×C3—61no—
131C24×C3D3211yescharacteristic
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4—92no—
135C22×C3×C4—91no—
136C22×C3×C4—94no—
137C22×C3×C4—184no—
138C22×Dic3—32no—
139C22×Dic3—38no—
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—61no—
144C22×Dic3—64no—
145C22×Dic3—61no—
146C22×Dic3—68no—
147C22×Dic3—68no—
148C22×Dic3—92no—
149C22×Dic3—92no—
150C2×Dic6—94no—
151C2×Dic6—98no—
152C2×Dic6—188no—
153C2×Dic6—184no—
154C3×C22⋊C4—98no—
155#217—38no—
156#217—98no—
157C4⋊Dic3—94no—
158C4⋊Dic3—98no—
159Dic3⋊C4—98no—
160Dic3⋊C4—98no—
161Dic3⋊C4—98no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—216no—
165C22×C32—216no—
166C22×C32—24no—
167C22×C32—24no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—24no—
172C22×C32—22no—
173C22×C32C2×Dic614yes—
174C22×C32C2×Dic618yescomplemented
175C22×C32D3×D414yes—
176C22×C32#23714yes—
177C22×C32C23×D312yes—
178C3⋊Dic3—24no—
179C3⋊Dic3—24no—
180C3×Dic3—38no—
181C3×Dic3—64no—
182C3×Dic3—64no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C3×Dic3—64no—
186C22×Q8—271no—
187C23×C4—91no—
188C22⋊Q8—278no—
189C2×C4⋊C4—271no—
190C2×C4⋊C4—272no—
191C2×C22⋊C4—92no—
192H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—38no—
196C23×C3—34no—
197C23×C3—32no—
198C23×C3—64no—
199C23×C3—64no—
200C23×C3—64no—
201C23×C3—64no—
202C23×C3—64no—
203C23×C3—62no—
204C23×C3—61no—
205C23×C3#62714yes—
206C23×C3C2×D3212yes—
207C23×C3C2×D3211yescharacteristic
208C2×C3×C4—92no—
209C2×C3×C4—94no—
210C2×C3×C4—94no—
211C2×C3×C4—94no—
212C2×C3×C4—92no—
213C2×C3×C4—188no—
214C2×C3×C4—184no—
215C2×C3×C4—184no—
216C2×C3×C4—182no—
217C2×Dic3—34no—
218C2×Dic3—38no—
219C2×Dic3—34no—
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—62no—
223C2×Dic3—64no—
224C2×Dic3—62no—
225C2×Dic3—616no—
226C2×Dic3—64no—
227C2×Dic3—616no—
228C2×Dic3—64no—
229C2×Dic3—64no—
230C2×Dic3—64no—
231C2×Dic3—94no—
232C2×Dic3—94no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235Dic6—1816no—
236Dic6—188no—
237C2×C32—28no—
238C2×C32—28no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—24no—
243C2×C32#87818yescomplemented
244C2×C32C22×Dic612yes—
245C2×C32C2×D3×D412yes—
246C2×C32C2×#23712yes—
247C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C2×Q8—274no—
249C2×Q8—544no—
250C22×C4—91no—
251C22×C4—94no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4—91no—
255C22×C4—184no—
256C4⋊C4—274no—
257C4⋊C4—278no—
258C22⋊C4—98no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—28no—
262C22×C3—32no—
263C22×C3—34no—
264C22×C3—38no—
265C22×C3—34no—
266C22×C3—34no—
267C22×C3—64no—
268C22×C3—64no—
269C22×C3—616no—
270C22×C3—64no—
271C22×C3—64no—
272C22×C3—616no—
273C22×C3—61no—
274C22×C3—62no—
275C22×C3—64no—
276C22×C3—62no—
277C22×C3—62no—
278C22×C3#380512yes—
279C22×C3C2×#62712yes—
280C22×C3C2×#62714yes—
281C22×C3#381112yes—
282C22×C3C22×D3211yescharacteristic
283C3×C4—94no—
284C3×C4—184no—
285C3×C4—184no—
286C3×C4—182no—
287Dic3—38no—
288Dic3—64no—
289Dic3—64no—
290Dic3—64no—
291Dic3—64no—
292Dic3—64no—
293Dic3—184no—
294Dic3—184no—
295C32—21no—
296C32C2×#87812yescomplemented
297C23—22no—
298C23—22no—
299C23#619614yescomplemented
300C23C2×((C3⋊D3)⋊C3)⋊C212yes—
301C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
302Q8—548no—
303C2×C4—92no—
304C2×C4—94no—
305C2×C4—94no—
306C2×C4—94no—
307C2×C4—92no—
308C2×C4—188no—
309C2×C4—184no—
310C2×C4—182no—
311C2×C4—184no—
312C2×C3—24no—
313C2×C3—32no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—38no—
317C2×C3—68no—
318C2×C3—68no—
319C2×C3—61no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3—64no—
323C2×C3#6398214yes—
324C2×C3C2×#380511yescharacteristic
325C2×C3C22×#62711yescharacteristic
326C2×C3C2×#381111yescharacteristic
327C22—22no—
328C22—22no—
329C22—28no—
330C22C2×#619612yes—
331C22C2×#619614yescomplemented
332C22#8867212yes—
333C22#8867712yes—
334C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C4—94no—
336C4—184no—
337C4—182no—
338C4—184no—
339C3—32no—
340C3—61no—
341C3C2×#6398211yescharacteristic
342C2—24no—
343C2#1172130914yesdirect product factor
344C2C22×#619611yescharacteristic
345C2C2×#8867211yescharacteristic
346C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6196C2semidirect
4. #11721309C2direct (N × H)
5. C2×#88449C2semidirect
6. C2×#88448C2semidirect
16. C2×#6196C22semidirect
18. #88449C22semidirect
19. #88448C22semidirect
31. C23×H(3,𝔽3)Q8semidirect
72. C22×H(3,𝔽3)C2×Q8semidirect
101. C23×C32Dic6semidirect
125. C2×H(3,𝔽3)C22⋊Q8semidirect
174. C22×C32C2×Dic6semidirect
192. H(3,𝔽3)C2×C22⋊Q8semidirect
243. C2×C32#878semidirect
296. C32C2×#878semidirect
299. C23#6196semidirect
331. C22C2×#6196semidirect
343. C2#11721309direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10842
C22×#61962
C2×#3757⋊C32
#117213092
C2×#884492
C2×#884482
C2×#639643