Finite Group Databasebeta

← C2×#11722493 #13164724

Subgroups — C2×#11722493 #13164724

Order 1728.

3379 conjugacy classes of subgroups; only the 555 normal subgroups are shown.

normal subgroups only

This group has 3379 conjugacy classes of subgroups — too many to draw. Only its 555 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, complemented
2C2×#3828⋊C3C212yesmaximal, complemented
3C2×#88677C214yesmaximal, complemented
4#11722493C2116yesmaximal, direct product factor
5C2×#88654C214yesmaximal, complemented
6C2×#88652C214yesmaximal, complemented
7C2×#64307—32nomaximal
8C23×#272C2212yescomplemented
9C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
10C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×C4⋊#272C2214yescomplemented
12C2×C4⋊#272C2218yescomplemented
13C2×#638⋊C3C2214yescomplemented
14#3828⋊C3C22116yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
16C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
17C2×#634⋊C3C2212yescomplemented
18#88677C22132yescomplemented
19C2×#6200C2218yescomplemented
20C2×#6199C2214yescomplemented
21#88654C22132yescomplemented
22#88652C22132yescomplemented
23C2×#3828D312yescomplemented
24C2×C3×#235—32no—
25C22×C3×D12—32no—
26C22×C3×D12—61no—
27C2×D3×#67—38no—
28#64307—332no—
29C2×#3802—34no—
30C2×D3×D12—34no—
31C2×#3796—34no—
32C2×#3795—34no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×#272—44no—
35C22×#272C2314yes—
36C22×#272C2318yescomplemented
37C22×(C3⋊D3)⋊C3C2314yes—
38C2×((C3⋊D3)⋊C3)⋊C2—48no—
39C2×((C3⋊D3)⋊C3)⋊C2C23116yescomplemented
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2314yescomplemented
42C4⋊#272C23116yescomplemented
43#638⋊C3C23116yescomplemented
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C4⋊((C3⋊D3)⋊C3)C2318yescomplemented
46C4×(C3⋊D3)⋊C3C23116yescomplemented
47#634⋊C3C2318yescomplemented
48#6200C23132yescomplemented
49#6199C23116yescomplemented
50C2×#1009—92no—
51C2×#1004—91no—
52C23×C3×D3—34no—
53C23×C3×D3—62no—
54C22×D32—38no—
55C22×C32×C4—21no—
56C22×C32×C4C2×D312yescomplemented
57C2×#638C2×D314yescomplemented
58#3828C2×D3116yescomplemented
59C2×C4⋊(C3⋊D3)C2×D312yescomplemented
60C2×C4×C3⋊D3C2×D314yescomplemented
61C2×#634C2×D312yescomplemented
62C2×C3×#67—34no—
63C3×#235—316no—
64C2×C3×D12—34no—
65C2×C3×D12—34no—
66C2×C3×D12—32no—
67C2×C3×D12—316no—
68C2×C3×D12—64no—
69C2×C3×D12—68no—
70C2×C3×C4×D3—34no—
71C2×C3×Dic6—32no—
72D3×#67—364no—
73C2×#626—38no—
74C2×#625—38no—
75C2×D3×Dic3—38no—
76#3802—332no—
77D3×D12—332no—
78#3796—332no—
79#3795—332no—
80C22×H(3,𝔽3)—22no—
81C22×H(3,𝔽3)C2412yes—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×#272—28no—
84C2×#272—48no—
85C2×#272C2418yes—
86C2×(C3⋊D3)⋊C3—28no—
87C2×(C3⋊D3)⋊C3C2418yes—
88((C3⋊D3)⋊C3)⋊C2—416no—
89C4×H(3,𝔽3)C2414yes—
90(C3⋊Dic3)⋊C3C2418yes—
91C2×C3×#41—92no—
92C22×C3×D4—91no—
93C22×#67—94no—
94#1009—932no—
95C2×#239—94no—
96#1004—916no—
97C2×#237—94no—
98C2×D3×D4—94no—
99C2×D3×D4—94no—
100C2×D3×D4—98no—
101C2×#235—32no—
102C2×#235—92no—
103C2×#235—94no—
104C22×D12—32no—
105C22×D12—61no—
106C22×D12—94no—
107C23×C32—21no—
108C23×C32C22×D312yes—
109C22×C3⋊D3C22×D314yes—
110C22×C3×D3—34no—
111C22×C3×D3—38no—
112C22×C3×D3—316no—
113C22×C3×D3—64no—
114C22×C3×D3—68no—
115C22×C3×D3—128no—
116C22×C3×D3—128no—
117C2×D32—332no—
118C2×D32—1216no—
119C2×C32×C4—22no—
120C2×C32×C4—24no—
121C2×C32×C4C22×D312yes—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D318yescomplemented
124#638C22×D3116yescomplemented
125C2×C3⋊Dic3C22×D314yes—
126C4⋊(C3⋊D3)C22×D318yescomplemented
127C4×C3⋊D3C22×D3116yescomplemented
128#634C22×D318yescomplemented
129C3×#67—316no—
130C2×C3×Dic3—34no—
131C3×D12—38no—
132C3×D12—316no—
133C3×D12—316no—
134C3×D12—616no—
135C3×C4×D3—316no—
136C3×Dic6—38no—
137#626—332no—
138#625—332no—
139D3×Dic3—332no—
140C2×H(5,𝔽2)—271noSylow
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)H(5,𝔽2)12yescomplemented
143C2×H(3,𝔽3)C2511yescharacteristic
144#272—28no—
145(C3⋊D3)⋊C3—28no—
146C24×C3—92no—
147C23×D3—34no—
148C23×D3—62no—
149C23×D3—94no—
150C23×D3—98no—
151C3×#41—916no—
152C2×C3×Q8—92no—
153C2×C3×D4—92no—
154C2×C3×D4—94no—
155C2×C3×D4—94no—
156C2×C3×D4—98no—
157C22×C3×C4—32no—
158C22×C3×C4—61no—
159C22×C3×C4—94no—
160C22×C3×C4D3211yescharacteristic
161C2×#67—34no—
162C2×#67—94no—
163C2×#67—98no—
164C2×#67—932no—
165C2×#67—98no—
166C22×Dic3—94no—
167#239—932no—
168#237—932no—
169D3×D4—932no—
170D3×D4—932no—
171D3×D4—964no—
172#235—316no—
173#235—916no—
174#235—932no—
175C2×D12—34no—
176C2×D12—34no—
177C2×D12—316no—
178C2×D12—32no—
179C2×D12—64no—
180C2×D12—68no—
181C2×D12—92no—
182C2×D12—932no—
183C2×D12—98no—
184C2×C4×D3—34no—
185C2×C4×D3—94no—
186C2×C4×D3—98no—
187C2×Dic6—32no—
188C2×Dic6—92no—
189C22×C32—24no—
190C22×C32—24no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32C23×D314yes—
194C22×C32C23×D312yes—
195C2×C3⋊D3—28no—
196C2×C3⋊D3C23×D318yes—
197C2×C3×D3—38no—
198C2×C3×D3—316no—
199C2×C3×D3—68no—
200C2×C3×D3—68no—
201C2×C3×D3—616no—
202C2×C3×D3—128no—
203C2×C3×D3—1216no—
204C2×C3×D3—1216no—
205D32—1232no—
206C32×C4—24no—
207C32×C4C23×D314yes—
208C32×C4C23×D314yes—
209C3⋊Dic3C23×D318yes—
210C3×Dic3—38no—
211H(5,𝔽2)—2716no—
212C2×#41—92no—
213C2×#41—274no—
214C22×D4—91no—
215C22×D4—274no—
216C22×D4—274no—
217H(3,𝔽3)C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
218C23×C3—32no—
219C23×C3—61no—
220C23×C3—94no—
221C23×C3—94no—
222C23×C3—98no—
223C23×C3—364no—
224C23×C3C2×D3211yescharacteristic
225C22×D3—38no—
226C22×D3—34no—
227C22×D3—316no—
228C22×D3—64no—
229C22×D3—68no—
230C22×D3—94no—
231C22×D3—916no—
232C22×D3—932no—
233C22×D3—128no—
234C22×D3—128no—
235C22×D3—368no—
236C22×D3—3616no—
237C3×Q8—98no—
238C3×D4—98no—
239C3×D4—916no—
240C3×D4—916no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—38no—
244C2×C3×C4—62no—
245C2×C3×C4—64no—
246C2×C3×C4—916no—
247C2×C3×C4—94no—
248C2×C3×C4C2×D3212yes—
249C2×C3×C4C2×D3214yes—
250#67—316no—
251#67—916no—
252#67—932no—
253#67—932no—
254C2×Dic3—34no—
255C2×Dic3—94no—
256C2×Dic3—916no—
257D12—316no—
258D12—316no—
259D12—38no—
260D12—616no—
261D12—98no—
262D12—932no—
263C4×D3—316no—
264C4×D3—916no—
265C4×D3—932no—
266Dic6—38no—
267Dic6—98no—
268C2×C32—24no—
269C2×C32—24no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32#100914yescomplemented
273C2×C32C24×D312yes—
274C3⋊D3—28no—
275C3×D3—68no—
276C3×D3—616no—
277C3×D3—128no—
278C24—92no—
279C24—274no—
280#41—916no—
281#41—2732no—
282C2×Q8—92no—
283C2×D4—94no—
284C2×D4—94no—
285C2×D4—92no—
286C2×D4—98no—
287C2×D4—2732no—
288C2×D4—278no—
289C2×D4—2732no—
290C22×C4—94no—
291C22×C4—274no—
292C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
293C22×C3—22no—
294C22×C3—34no—
295C22×C3—32no—
296C22×C3—64no—
297C22×C3—64no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—98no—
301C22×C3—98no—
302C22×C3—188no—
303C22×C3—188no—
304C22×C3—368no—
305C22×C3C22×D3212yes—
306C22×C3C22×D3211yescharacteristic
307C2×D3—316no—
308C2×D3—38no—
309C2×D3—68no—
310C2×D3—616no—
311C2×D3—68no—
312C2×D3—98no—
313C2×D3—128no—
314C2×D3—1216no—
315C2×D3—1216no—
316C2×D3—188no—
317C2×D3—3616no—
318C2×D3—3632no—
319C3×C4—34no—
320C3×C4—34no—
321C3×C4—64no—
322C3×C4—98no—
323C3×C4C22×D3214yes—
324Dic3—38no—
325Dic3—98no—
326C32—21no—
327C32C2×#100912yescomplemented
328C23—94no—
329C23—94no—
330C23—98no—
331C23—2716no—
332C23—364no—
333C23—1088no—
334C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335Q8—98no—
336D4—916no—
337D4—916no—
338D4—98no—
339D4—2732no—
340C2×C4—94no—
341C2×C4—916no—
342C2×C4—2716no—
343C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
344C2×C4C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
345C2×C3—22no—
346C2×C3—32no—
347C2×C3—34no—
348C2×C3—64no—
349C2×C3—64no—
350C2×C3—61no—
351C2×C3—62no—
352C2×C3—188no—
353C2×C3—188no—
354C2×C3#6430812yes—
355C2×C3C23×D3211yescharacteristic
356D3—616no—
357D3—68no—
358D3—128no—
359D3—188no—
360C22—22no—
361C22—98no—
362C22—98no—
363C22—188no—
364C22—188no—
365C22—368no—
366C22—10816no—
367C22C22×((C3⋊D3)⋊C3)⋊C212yes—
368C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
369C4—98no—
370C4C22×((C3⋊D3)⋊C3)⋊C214yes—
371C3—32no—
372C3—61no—
373C3C2×#6430811yescharacteristic
374C2—22no—
375C2—188no—
376C2—188no—
377C2#1172249312yesdirect product factor
378C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C2×#88677C2semidirect
4. #11722493C2direct (N × H)
5. C2×#88654C2semidirect
6. C2×#88652C2semidirect
8. C23×#272C22semidirect
9. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
10. C22×C4×H(3,𝔽3)C22semidirect
11. C2×C4⋊#272C22semidirect
12. C2×C4⋊#272C22semidirect
13. C2×#638⋊C3C22semidirect
14. #3828⋊C3C22semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C4×(C3⋊D3)⋊C3C22semidirect
17. C2×#634⋊C3C22semidirect
18. #88677C22semidirect
19. C2×#6200C22semidirect
20. C2×#6199C22semidirect
21. #88654C22semidirect
22. #88652C22semidirect
23. C2×#3828D3semidirect
36. C22×#272C23semidirect
39. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
41. C2×C4×H(3,𝔽3)C23semidirect
42. C4⋊#272C23semidirect
43. #638⋊C3C23semidirect
45. C4⋊((C3⋊D3)⋊C3)C23semidirect
46. C4×(C3⋊D3)⋊C3C23semidirect
47. #634⋊C3C23semidirect
48. #6200C23semidirect
49. #6199C23semidirect
56. C22×C32×C4C2×D3semidirect
57. C2×#638C2×D3semidirect
58. #3828C2×D3semidirect
59. C2×C4⋊(C3⋊D3)C2×D3semidirect
60. C2×C4×C3⋊D3C2×D3semidirect
61. C2×#634C2×D3semidirect
123. C2×C32×C4C22×D3semidirect
124. #638C22×D3semidirect
126. C4⋊(C3⋊D3)C22×D3semidirect
127. C4×C3⋊D3C22×D3semidirect
128. #634C22×D3semidirect
142. C2×H(3,𝔽3)H(5,𝔽2)semidirect
217. H(3,𝔽3)C2×H(5,𝔽2)semidirect
272. C2×C32#1009semidirect
292. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
327. C32C2×#1009semidirect
344. C2×C4C2×((C3⋊D3)⋊C3)⋊C2semidirect
377. C2#11722493direct (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
C2×#3828⋊C32
C2×#886772
#117224932
C2×#886542
C2×#886522
C2×#643073