Finite Group Databasebeta

← C2×#11721318 #13152076

Subgroups — C2×#11721318 #13152076

Order 1728.

1727 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1727 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, complemented
2C22×#638⋊C3C212yesmaximal, complemented
3C22×#6198C211yesmaximal, characteristic, complemented
4#11721318C218yesmaximal, direct product factor
5C2×#88450C212yesmaximal, complemented
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#63978—32nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×#1084—24no—
11C22×#1084C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2214yescomplemented
14C2×#638⋊C3—28no—
15C2×#638⋊C3C2218yescomplemented
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#6198—24no—
18C2×#6198C2214yescomplemented
19#88450C2218yescomplemented
20#88449C2214yescomplemented
21C22×#638D312yescomplemented
22C22×C3×#67—32no—
23C23×C3×Dic3—32no—
24C23×C3×Dic3—61no—
25C22×#626—32no—
26#63978—316no—
27C2×Dic3⋊Dic3—32no—
28C2×#3722—32no—
29C2×(C2×D3)⋊Dic3—32no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)D414yescomplemented
33C23×H(3,𝔽3)C2312yes—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3—24no—
36C22×(C3⋊D3)⋊C3—24no—
37C22×(C3⋊D3)⋊C3—24no—
38C22×(C3⋊D3)⋊C3C2314yes—
39C2×#1084—22no—
40C2×#1084—28no—
41C2×#1084D414yescomplemented
42C2×#1084C2312yes—
43C2×#1084C2312yes—
44#638⋊C3—216no—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47#6198—28no—
48C2×#939—92no—
49C2×#930—91no—
50C24×C32—21no—
51C24×C32C2×D312yes—
52C23×C3⋊D3C2×D312yes—
53C23×C3×D3—32no—
54C2×#638—28no—
55C2×#638C2×D318yescomplemented
56C22×C3⋊Dic3C2×D312yes—
57C2×C3×#67—38no—
58C2×C3×#67—68no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—38no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—68no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—64no—
67C22×C3×Dic3—68no—
68C2×#626—38no—
69C2×#626—68no—
70Dic3⋊Dic3—38no—
71#3722—38no—
72(C2×D3)⋊Dic3—38no—
73C22×H(3,𝔽3)—28no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)C2×D412yes—
77C22×H(3,𝔽3)C2×D412yes—
78C22×H(3,𝔽3)C2×D414yescomplemented
79C22×H(3,𝔽3)#4112yes—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×(C3⋊D3)⋊C3—24no—
82C2×(C3⋊D3)⋊C3—24no—
83C2×(C3⋊D3)⋊C3—24no—
84C2×(C3⋊D3)⋊C3—48no—
85#1084—22no—
86#1084C2×D414yescomplemented
87(C3⋊Dic3)⋊C3—24no—
88C22×C3×D4—92no—
89C23×C3×C4—91no—
90C22×#67—32no—
91C22×#67—92no—
92C22×#67—92no—
93C23×Dic3—32no—
94C23×Dic3—61no—
95C22×D12—91no—
96C2×#217—92no—
97#939—916no—
98#930—98no—
99C2×(C2×D3)⋊C4—92no—
100C2×(C2×D3)⋊C4—92no—
101C2×C4⋊Dic3—91no—
102C2×Dic3⋊C4—92no—
103C23×C32—24no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—24no—
107C23×C32—24no—
108C23×C32—22no—
109C23×C32—21no—
110C23×C32#6718yescomplemented
111C23×C32C22×D314yes—
112C23×C32C22×D312yes—
113C22×C3⋊D3—24no—
114C22×C3⋊D3—24no—
115C22×C3⋊D3—24no—
116C22×C3⋊D3C22×D314yes—
117C22×C3×D3—34no—
118C22×C3×D3—64no—
119C22×C3×D3—64no—
120C22×C3×D3—64no—
121#638—216no—
122C2×C3⋊Dic3—24no—
123C2×C3⋊Dic3C22×D314yes—
124C3×#67—616no—
125C2×C3×Dic3—38no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—616no—
133C2×C3×Dic3—616no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—64no—
137#626—616no—
138C2×#121—271noSylow
139C2×H(3,𝔽3)—24no—
140C2×H(3,𝔽3)#12114yescomplemented
141C2×H(3,𝔽3)C22×D411yescharacteristic
142C2×H(3,𝔽3)C22×D411yescharacteristic
143C2×H(3,𝔽3)C2×#4111yescharacteristic
144(C3⋊D3)⋊C3—44no—
145C24×C3—32no—
146C24×C3—61no—
147C24×C3—92no—
148C24×C3D3211yescharacteristic
149C23×D3—32no—
150C23×D3—92no—
151C2×C3×D4—98no—
152C2×C3×D4—188no—
153C22×C3×C4—91no—
154C22×C3×C4—92no—
155C22×C3×C4—91no—
156C22×C3×C4—94no—
157C22×C3×C4—184no—
158C2×#67—38no—
159C2×#67—68no—
160C2×#67—98no—
161C2×#67—98no—
162C2×#67—188no—
163C2×#67—188no—
164C22×Dic3—32no—
165C22×Dic3—32no—
166C22×Dic3—38no—
167C22×Dic3—32no—
168C22×Dic3—61no—
169C22×Dic3—64no—
170C22×Dic3—61no—
171C22×Dic3—68no—
172C22×Dic3—68no—
173C22×Dic3—92no—
174C2×D12—94no—
175C2×D12—184no—
176#217—98no—
177(C2×D3)⋊C4—98no—
178(C2×D3)⋊C4—98no—
179C4⋊Dic3—94no—
180Dic3⋊C4—98no—
181C22×C32—24no—
182C22×C32—24no—
183C22×C32—216no—
184C22×C32—216no—
185C22×C32—24no—
186C22×C32—24no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—24no—
191C22×C32—22no—
192C22×C32D3×D414yes—
193C22×C32#23714yes—
194C22×C32C2×#6714yes—
195C22×C32C2×#6718yescomplemented
196C22×C32C23×D312yes—
197C2×C3⋊D3—24no—
198C2×C3⋊D3—24no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—48no—
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—128no—
205C3⋊Dic3—24no—
206C3×Dic3—38no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C22×D4—92no—
212C22×D4—271no—
213C23×C4—91no—
214#121—278no—
215C2×C4⋊C4—271no—
216C2×C22⋊C4—272no—
217H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—38no—
221C23×C3—34no—
222C23×C3—32no—
223C23×C3—64no—
224C23×C3—64no—
225C23×C3—64no—
226C23×C3—64no—
227C23×C3—64no—
228C23×C3—62no—
229C23×C3—61no—
230C23×C3—94no—
231C23×C3—184no—
232C23×C3—184no—
233C23×C3—184no—
234C23×C3#62514yes—
235C23×C3C2×D3212yes—
236C23×C3C2×D3211yescharacteristic
237C22×D3—34no—
238C22×D3—64no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—94no—
242C22×D3—184no—
243C22×D3—184no—
244C22×D3—184no—
245C3×D4—1816no—
246C2×C3×C4—92no—
247C2×C3×C4—94no—
248C2×C3×C4—94no—
249C2×C3×C4—92no—
250C2×C3×C4—188no—
251C2×C3×C4—184no—
252C2×C3×C4—182no—
253#67—616no—
254#67—1816no—
255#67—1816no—
256C2×Dic3—34no—
257C2×Dic3—38no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—62no—
261C2×Dic3—64no—
262C2×Dic3—62no—
263C2×Dic3—616no—
264C2×Dic3—64no—
265C2×Dic3—616no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C2×Dic3—94no—
269C2×Dic3—184no—
270D12—188no—
271C2×C32—28no—
272C2×C32—28no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—24no—
277C2×C32#93918yescomplemented
278C2×C32C2×D3×D412yes—
279C2×C32C2×#23712yes—
280C2×C32C22×#6712yes—
281C3⋊D3—44no—
282C3×D3—124no—
283C24—92no—
284C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C2×D4—98no—
286C2×D4—188no—
287C2×D4—274no—
288C2×D4—544no—
289C22×C4—91no—
290C22×C4—94no—
291C22×C4—91no—
292C22×C4—92no—
293C22×C4—184no—
294C4⋊C4—274no—
295C22⋊C4—278no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—28no—
299C22×C3—32no—
300C22×C3—34no—
301C22×C3—34no—
302C22×C3—38no—
303C22×C3—34no—
304C22×C3—64no—
305C22×C3—64no—
306C22×C3—616no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—616no—
310C22×C3—61no—
311C22×C3—62no—
312C22×C3—62no—
313C22×C3—64no—
314C22×C3—62no—
315C22×C3—184no—
316C22×C3—184no—
317C22×C3—184no—
318C22×C3—368no—
319C22×C3#380512yes—
320C22×C3C2×#62512yes—
321C22×C3C2×#62514yes—
322C22×C3#381112yes—
323C22×C3C22×D3211yescharacteristic
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—128no—
328C2×D3—184no—
329C2×D3—184no—
330C2×D3—184no—
331C2×D3—368no—
332C3×C4—94no—
333C3×C4—184no—
334C3×C4—182no—
335Dic3—38no—
336Dic3—64no—
337Dic3—64no—
338Dic3—64no—
339Dic3—64no—
340Dic3—184no—
341C32—21no—
342C32C2×#93912yescomplemented
343C23—22no—
344C23—22no—
345C23—94no—
346C23—184no—
347C23—184no—
348C23—184no—
349C23#619814yescomplemented
350C23C2×((C3⋊D3)⋊C3)⋊C212yes—
351C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352D4—1816no—
353D4—548no—
354C2×C4—92no—
355C2×C4—94no—
356C2×C4—92no—
357C2×C4—94no—
358C2×C4—188no—
359C2×C4—184no—
360C2×C4—182no—
361C2×C3—24no—
362C2×C3—32no—
363C2×C3—32no—
364C2×C3—32no—
365C2×C3—38no—
366C2×C3—68no—
367C2×C3—68no—
368C2×C3—61no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—64no—
372C2×C3—364no—
373C2×C3#6398014yes—
374C2×C3C2×#380511yescharacteristic
375C2×C3C22×#62511yescharacteristic
376C2×C3C2×#381111yescharacteristic
377D3—124no—
378D3—364no—
379C22—22no—
380C22—22no—
381C22—28no—
382C22—184no—
383C22—184no—
384C22—184no—
385C22—368no—
386C22#8867212yes—
387C22C2×#619812yes—
388C22C2×#619814yescomplemented
389C22#8867712yes—
390C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C4—94no—
392C4—184no—
393C4—182no—
394C3—32no—
395C3—61no—
396C3C2×#6398011yescharacteristic
397C2—24no—
398C2—364no—
399C2#1172131814yesdirect product factor
400C2C2×#8867211yescharacteristic
401C2C22×#619811yescharacteristic
402C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
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
1. C23×#1084C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×#6198C2semidirect
4. #11721318C2direct (N × H)
5. C2×#88450C2semidirect
6. C2×#88449C2semidirect
13. C22×#1084C22semidirect
15. C2×#638⋊C3C22semidirect
18. C2×#6198C22semidirect
19. #88450C22semidirect
20. #88449C22semidirect
21. C22×#638D3semidirect
32. C23×H(3,𝔽3)D4semidirect
41. C2×#1084D4semidirect
55. C2×#638C2×D3semidirect
78. C22×H(3,𝔽3)C2×D4semidirect
86. #1084C2×D4semidirect
110. C23×C32#67semidirect
140. C2×H(3,𝔽3)#121semidirect
195. C22×C32C2×#67semidirect
217. H(3,𝔽3)C2×#121semidirect
277. C2×C32#939semidirect
342. C32C2×#939semidirect
349. C23#6198semidirect
388. C22C2×#6198semidirect
399. C2#11721318direct (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×#638⋊C32
C22×#61982
#117213182
C2×#884502
C2×#884492
C2×#639783