Finite Group Databasebeta

← C22×#88649 #13164711

Subgroups — C22×#88649 #13164711

Order 1728.

3351 conjugacy classes of subgroups; only the 595 normal subgroups are shown.

normal subgroups only

This group has 3351 conjugacy classes of subgroups — too many to draw. Only its 595 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
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C22×#634⋊C3C211yesmaximal, characteristic, complemented
4C22×#6200C212yesmaximal, complemented
5C22×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
6C2×#88649C2124yesmaximal, direct product factor
7C22×#3799—31nomaximal
8C22×#3795—31nomaximal
9C23×#272C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C2×C4×#272C22112yescomplemented
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×C4⋊((C3⋊D3)⋊C3)C22112yescomplemented
16C2×#634⋊C3C22112yescomplemented
17C2×#6200C22124yescomplemented
18C2×((C3⋊D3)⋊C3)⋊C4C22124yescomplemented
19#88649C22164yesdirect product factor
20C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×#634D311yescharacteristic, complemented
22C22×C3×D12—31no—
23C22×C3×C4×D3—31no—
24C22×C3×C4×D3—31no—
25C22×C3×C4×D3—61no—
26C22×C3×Dic6—31no—
27C22×#626—32no—
28C22×#625—32no—
29C22×#624—32no—
30C22×D3×Dic3—32no—
31C2×#3799—324no—
32C2×#3795—324no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×#272—24no—
35C22×#272C2316yes—
36C22×(C3⋊D3)⋊C3—28no—
37C22×(C3⋊D3)⋊C3C23112yes—
38C2×C4×H(3,𝔽3)C2316yes—
39C2×#1084C2316yes—
40C4×#272C23116yescomplemented
41C2×(C3⋊Dic3)⋊C3C23112yes—
42C4⋊((C3⋊D3)⋊C3)C23116yescomplemented
43#634⋊C3C23116yescomplemented
44#6200C23132yescomplemented
45((C3⋊D3)⋊C3)⋊C4C23132yescomplemented
46C22×#239—91no—
47C22×#237—91no—
48C22×#235—91no—
49C23×C3⋊D3C2×D312yes—
50C23×C3×D3—32no—
51C23×C3×D3—31no—
52C23×C3×D3—31no—
53C23×C3×D3—61no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3C2×D312yes—
58C2×C4⋊(C3⋊D3)C2×D3112yescomplemented
59C2×#634C2×D3112yescomplemented
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C2×C3×D12—312no—
65C2×C3×C4×D3—312no—
66C2×C3×C4×D3—312no—
67C2×C3×C4×D3—612no—
68C2×C3×Dic6—312no—
69C2×#626—324no—
70C2×#625—324no—
71C2×#624—324no—
72C2×D3×Dic3—324no—
73#3799—364no—
74#3795—364no—
75C22×H(3,𝔽3)#4114yescomplemented
76C22×H(3,𝔽3)C2413yes—
77C2×#272—212no—
78C2×#272C2414yes—
79C2×(C3⋊D3)⋊C3—224no—
80C2×(C3⋊D3)⋊C3C2418yes—
81C4×H(3,𝔽3)C2414yes—
82#1084C2414yes—
83(C3⋊Dic3)⋊C3C2418yes—
84C22×C3×Q8—91no—
85C22×C3×D4—91no—
86C23×C3×C4—91no—
87C22×#67—92no—
88C22×#67—92no—
89C23×Dic3—92no—
90C2×#239—924no—
91C2×#237—924no—
92C2×#235—924no—
93C22×D12—31no—
94C22×D12—91no—
95C22×D12—92no—
96C22×C4×D3—31no—
97C22×C4×D3—31no—
98C22×C4×D3—61no—
99C22×C4×D3—92no—
100C22×C4×D3—92no—
101C22×Dic6—31no—
102C22×Dic6—91no—
103C23×C32—21no—
104C23×C32C22×D311yescharacteristic
105C23×C32C22×D311yescharacteristic
106C22×C3⋊D3—28no—
107C22×C3⋊D3C22×D3112yes—
108C22×C3×D3—312no—
109C22×C3×D3—36no—
110C22×C3×D3—36no—
111C22×C3×D3—66no—
112C22×C3×D3—68no—
113C22×C3×D3—64no—
114C22×C3×D3—68no—
115C22×C3×D3—64no—
116C2×C32×C4—26no—
117C2×C32×C4C22×D316yes—
118C2×C32×C4C22×D316yes—
119C2×C3⋊Dic3C22×D3112yes—
120C4⋊(C3⋊D3)C22×D3116yescomplemented
121#634C22×D3116yescomplemented
122C2×C3×Dic3—312no—
123C2×C3×Dic3—36no—
124C2×C3×Dic3—36no—
125C2×C3×Dic3—66no—
126C3×D12—316no—
127C3×C4×D3—316no—
128C3×C4×D3—316no—
129C3×C4×D3—616no—
130C3×Dic6—316no—
131#626—332no—
132#625—332no—
133#624—332no—
134D3×Dic3—332no—
135C22×#41—271noSylow
136C2×H(3,𝔽3)C2×#4116yescomplemented
137C2×H(3,𝔽3)C2511yescharacteristic
138#272—24no—
139(C3⋊D3)⋊C3—28no—
140C24×C3—92no—
141C24×C3—91no—
142C23×D3—31no—
143C23×D3—31no—
144C23×D3—32no—
145C23×D3—61no—
146C23×D3—92no—
147C2×C3×Q8—912no—
148C2×C3×D4—912no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—92no—
153C22×C3×C4—91no—
154C22×C3×C4—912no—
155C22×C3×C4D3211yescharacteristic
156C2×#67—924no—
157C2×#67—924no—
158C22×Dic3—31no—
159C22×Dic3—31no—
160C22×Dic3—32no—
161C22×Dic3—61no—
162C22×Dic3—92no—
163C22×Dic3—924no—
164#239—964no—
165#237—964no—
166#235—964no—
167C2×D12—312no—
168C2×D12—912no—
169C2×D12—924no—
170C2×C4×D3—312no—
171C2×C4×D3—312no—
172C2×C4×D3—612no—
173C2×C4×D3—924no—
174C2×C4×D3—924no—
175C2×Dic6—312no—
176C2×Dic6—912no—
177C22×C32—23no—
178C22×C32—24no—
179C22×C32#23714yescomplemented
180C22×C32#23914yescomplemented
181C22×C32C23×D313yes—
182C22×C32C23×D313yes—
183C2×C3⋊D3—224no—
184C2×C3⋊D3C23×D318yes—
185C2×C3×D3—38no—
186C2×C3×D3—34no—
187C2×C3×D3—34no—
188C2×C3×D3—64no—
189C2×C3×D3—624no—
190C2×C3×D3—612no—
191C2×C3×D3—624no—
192C2×C3×D3—612no—
193C32×C4—24no—
194C32×C4C23×D314yes—
195C32×C4C23×D314yes—
196C3⋊Dic3C23×D318yes—
197C3×Dic3—38no—
198C3×Dic3—34no—
199C3×Dic3—34no—
200C3×Dic3—64no—
201C2×#41—2724no—
202C22×Q8—91no—
203C22×D4—91no—
204C22×D4—272no—
205C23×C4—91no—
206C23×C4—272no—
207H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—61no—
211C23×C3—912no—
212C23×C3—96no—
213C23×C3—188no—
214C23×C3—184no—
215C23×C3C2×D3211yescharacteristic
216C22×D3—36no—
217C22×D3—36no—
218C22×D3—312no—
219C22×D3—66no—
220C22×D3—64no—
221C22×D3—64no—
222C22×D3—68no—
223C22×D3—68no—
224C22×D3—912no—
225C22×D3—188no—
226C3×Q8—916no—
227C3×D4—916no—
228C2×C3×C4—36no—
229C2×C3×C4—36no—
230C2×C3×C4—66no—
231C2×C3×C4—912no—
232C2×C3×C4—96no—
233C2×C3×C4—916no—
234C2×C3×C4C2×D3216yes—
235#67—932no—
236#67—932no—
237C2×Dic3—36no—
238C2×Dic3—36no—
239C2×Dic3—312no—
240C2×Dic3—66no—
241C2×Dic3—912no—
242C2×Dic3—932no—
243D12—316no—
244D12—916no—
245D12—932no—
246C4×D3—316no—
247C4×D3—316no—
248C4×D3—616no—
249C4×D3—932no—
250C4×D3—932no—
251Dic6—316no—
252Dic6—916no—
253C2×C32—26no—
254C2×C32—21no—
255C2×C32C2×#23716yescomplemented
256C2×C32C2×#23916yescomplemented
257C2×C32C24×D311yescharacteristic
258C2×C32C24×D311yescharacteristic
259C3⋊D3—28no—
260C3×D3—68no—
261C3×D3—64no—
262C3×D3—68no—
263C3×D3—64no—
264C24—91no—
265C24—92no—
266#41—2764no—
267C2×Q8—912no—
268C2×D4—912no—
269C2×D4—2724no—
270C22×C4—91no—
271C22×C4—92no—
272C22×C4—912no—
273C22×C4—2724no—
274C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C22×C3—33no—
276C22×C3—33no—
277C22×C3—34no—
278C22×C3—34no—
279C22×C3—63no—
280C22×C3—64no—
281C22×C3—98no—
282C22×C3—94no—
283C22×C3—1824no—
284C22×C3—1812no—
285C22×C3#379614yes—
286C22×C3C22×D3213yes—
287C2×D3—34no—
288C2×D3—34no—
289C2×D3—38no—
290C2×D3—64no—
291C2×D3—612no—
292C2×D3—612no—
293C2×D3—624no—
294C2×D3—624no—
295C2×D3—98no—
296C2×D3—1824no—
297C3×C4—34no—
298C3×C4—34no—
299C3×C4—64no—
300C3×C4—98no—
301C3×C4—94no—
302C3×C4C22×D3214yes—
303Dic3—34no—
304Dic3—34no—
305Dic3—38no—
306Dic3—64no—
307Dic3—98no—
308C32—21no—
309C32C22×#23711yescharacteristic, complemented
310C32C22×#23911yescharacteristic, complemented
311C23—96no—
312C23—912no—
313C23—184no—
314C23—188no—
315C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
316Q8—916no—
317D4—916no—
318D4—2732no—
319C2×C4—96no—
320C2×C4—912no—
321C2×C4—916no—
322C2×C4—2732no—
323C2×C4C2×((C3⋊D3)⋊C3)⋊C216yes—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—36no—
327C2×C3—36no—
328C2×C3—61no—
329C2×C3—66no—
330C2×C3—188no—
331C2×C3—184no—
332C2×C3C2×#379616yes—
333C2×C3C23×D3211yescharacteristic
334D3—64no—
335D3—64no—
336D3—68no—
337D3—68no—
338D3—188no—
339C22—94no—
340C22—98no—
341C22—1812no—
342C22—1824no—
343C22#8864914yesdirect product factor
344C22C22×((C3⋊D3)⋊C3)⋊C213yes—
345C4—94no—
346C4—98no—
347C4C22×((C3⋊D3)⋊C3)⋊C214yes—
348C3—31no—
349C3—31no—
350C3—61no—
351C3C22×#379611yescharacteristic
352C2—184no—
353C2—188no—
354C2C2×#8864916yesdirect product factor
355C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×#272C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C22×#634⋊C3C2semidirect
4. C22×#6200C2semidirect
5. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
6. C2×#88649C2direct (N × H)
13. C2×C4×#272C22semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×#634⋊C3C22semidirect
17. C2×#6200C22semidirect
18. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
19. #88649C22direct (N × H)
20. C22×C4⋊(C3⋊D3)D3semidirect
21. C22×#634D3semidirect
40. C4×#272C23semidirect
42. C4⋊((C3⋊D3)⋊C3)C23semidirect
43. #634⋊C3C23semidirect
44. #6200C23semidirect
45. ((C3⋊D3)⋊C3)⋊C4C23semidirect
58. C2×C4⋊(C3⋊D3)C2×D3semidirect
59. C2×#634C2×D3semidirect
75. C22×H(3,𝔽3)#41semidirect
120. C4⋊(C3⋊D3)C22×D3semidirect
121. #634C22×D3semidirect
136. C2×H(3,𝔽3)C2×#41semidirect
179. C22×C32#237semidirect
180. C22×C32#239semidirect
207. H(3,𝔽3)C22×#41semidirect
255. C2×C32C2×#237semidirect
256. C2×C32C2×#239semidirect
309. C32C22×#237semidirect
310. C32C22×#239semidirect
343. C22#88649direct (N × H)
354. C2C2×#88649direct (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×C4⋊((C3⋊D3)⋊C3)2
C22×#634⋊C32
C22×#62002
C22×((C3⋊D3)⋊C3)⋊C42
C2×#886492
C22×#37993
C22×#37953