Finite Group Databasebeta

← C22×#88648 #13164714

Subgroups — C22×#88648 #13164714

Order 1728.

3271 conjugacy classes of subgroups; only the 595 normal subgroups are shown.

normal subgroups only

This group has 3271 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×#6229C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C22×#6198C212yesmaximal, complemented
5C22×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
6C2×#88648C2124yesmaximal, direct product factor
7C22×#3799—31nomaximal
8C22×#3796—31nomaximal
9C23×(C3⋊D3)⋊C3C2212yes—
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2212yes—
13C2×#6229C22112yescomplemented
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C22112yescomplemented
16C2×C4×(C3⋊D3)⋊C3C22112yescomplemented
17C2×#6198C22124yescomplemented
18C2×((C3⋊D3)⋊C3)⋊C4C22124yescomplemented
19#88648C22164yesdirect product factor
20C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C4×C3⋊D3D311yescharacteristic, complemented
22C22×C3×D12—31no—
23C22×C3×C4×D3—31no—
24C22×C3×Dic6—31no—
25C22×C3×Dic6—31no—
26C22×C3×Dic6—61no—
27C22×#626—32no—
28C22×#626—32no—
29C22×#624—32no—
30C22×D3×Dic3—32no—
31C2×#3799—324no—
32C2×#3796—324no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—28no—
35C22×(C3⋊D3)⋊C3—24no—
36C22×(C3⋊D3)⋊C3C23112yes—
37C22×(C3⋊D3)⋊C3C2316yes—
38C2×C4×H(3,𝔽3)C2316yes—
39C2×#1084C23112yes—
40#6229C23116yescomplemented
41C2×(C3⋊Dic3)⋊C3C2316yes—
42C4⋊((C3⋊D3)⋊C3)C23116yescomplemented
43C4×(C3⋊D3)⋊C3C23116yescomplemented
44#6198C23132yescomplemented
45((C3⋊D3)⋊C3)⋊C4C23132yescomplemented
46C22×#239—91no—
47C22×#237—91no—
48C22×#235—91no—
49C23×C3⋊D3C2×D312yes—
50C23×C3⋊D3C2×D311yescharacteristic
51C23×C3×D3—32no—
52C23×C3×D3—31no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C3⋊Dic3C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)C2×D3112yescomplemented
58C2×C4×C3⋊D3C2×D3112yescomplemented
59C22×C3×Dic3—32no—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—62no—
63C2×C3×D12—312no—
64C2×C3×C4×D3—312no—
65C2×C3×Dic6—312no—
66C2×C3×Dic6—312no—
67C2×C3×Dic6—612no—
68C2×#626—324no—
69C2×#626—324no—
70C2×#624—324no—
71C2×D3×Dic3—324no—
72#3799—364no—
73#3796—364no—
74C22×H(3,𝔽3)#4114yescomplemented
75C22×H(3,𝔽3)C2413yes—
76C2×(C3⋊D3)⋊C3—224no—
77C2×(C3⋊D3)⋊C3—212no—
78C2×(C3⋊D3)⋊C3C2418yes—
79C2×(C3⋊D3)⋊C3C2414yes—
80C4×H(3,𝔽3)C2414yes—
81#1084C2418yes—
82(C3⋊Dic3)⋊C3C2414yes—
83C22×C3×Q8—91no—
84C22×C3×D4—91no—
85C23×C3×C4—91no—
86C22×#67—92no—
87C22×#67—92no—
88C23×Dic3—92no—
89C2×#239—924no—
90C2×#237—924no—
91C2×#235—924no—
92C22×D12—31no—
93C22×D12—91no—
94C22×D12—92no—
95C22×C4×D3—31no—
96C22×C4×D3—91no—
97C22×C4×D3—92no—
98C22×C4×D3—92no—
99C22×Dic6—31no—
100C22×Dic6—31no—
101C22×Dic6—61no—
102C23×C32—21no—
103C23×C32C22×D311yescharacteristic
104C23×C32C22×D311yescharacteristic
105C22×C3⋊D3—24no—
106C22×C3⋊D3—28no—
107C22×C3⋊D3C22×D3112yes—
108C22×C3⋊D3C22×D316yes—
109C22×C3×D3—312no—
110C22×C3×D3—36no—
111C22×C3×D3—68no—
112C22×C3×D3—64no—
113C2×C32×C4—26no—
114C2×C32×C4C22×D316yes—
115C2×C32×C4C22×D316yes—
116C2×C3⋊Dic3C22×D316yes—
117C4⋊(C3⋊D3)C22×D3116yescomplemented
118C4×C3⋊D3C22×D3116yescomplemented
119C2×C3×Dic3—36no—
120C2×C3×Dic3—312no—
121C2×C3×Dic3—312no—
122C2×C3×Dic3—612no—
123C3×D12—316no—
124C3×C4×D3—316no—
125C3×Dic6—316no—
126C3×Dic6—316no—
127C3×Dic6—616no—
128#626—332no—
129#626—332no—
130#624—332no—
131D3×Dic3—332no—
132C22×#41—271noSylow
133C2×H(3,𝔽3)C2×#4116yescomplemented
134C2×H(3,𝔽3)C2511yescharacteristic
135(C3⋊D3)⋊C3—28no—
136(C3⋊D3)⋊C3—24no—
137C24×C3—92no—
138C24×C3—91no—
139C23×D3—31no—
140C23×D3—32no—
141C23×D3—92no—
142C23×D3—91no—
143C2×C3×Q8—912no—
144C2×C3×D4—912no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—61no—
148C22×C3×C4—912no—
149C22×C3×C4—91no—
150C22×C3×C4—92no—
151C22×C3×C4D3211yescharacteristic
152C2×#67—924no—
153C2×#67—924no—
154C22×Dic3—32no—
155C22×Dic3—32no—
156C22×Dic3—31no—
157C22×Dic3—62no—
158C22×Dic3—91no—
159C22×Dic3—924no—
160#239—964no—
161#237—964no—
162#235—964no—
163C2×D12—312no—
164C2×D12—912no—
165C2×D12—924no—
166C2×C4×D3—312no—
167C2×C4×D3—912no—
168C2×C4×D3—924no—
169C2×C4×D3—924no—
170C2×Dic6—312no—
171C2×Dic6—312no—
172C2×Dic6—612no—
173C22×C32—23no—
174C22×C32—24no—
175C22×C32#23514yescomplemented
176C22×C32#23714yescomplemented
177C22×C32C23×D313yes—
178C22×C32C23×D313yes—
179C2×C3⋊D3—212no—
180C2×C3⋊D3—224no—
181C2×C3⋊D3C23×D318yes—
182C2×C3⋊D3C23×D314yes—
183C2×C3×D3—38no—
184C2×C3×D3—34no—
185C2×C3×D3—624no—
186C2×C3×D3—612no—
187C32×C4—24no—
188C32×C4C23×D314yes—
189C32×C4C23×D314yes—
190C3⋊Dic3C23×D314yes—
191C3×Dic3—34no—
192C3×Dic3—38no—
193C3×Dic3—38no—
194C3×Dic3—68no—
195C2×#41—2724no—
196C22×Q8—91no—
197C22×D4—91no—
198C22×D4—272no—
199C23×C4—91no—
200C23×C4—272no—
201H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
202C23×C3—31no—
203C23×C3—31no—
204C23×C3—61no—
205C23×C3—912no—
206C23×C3—96no—
207C23×C3—184no—
208C23×C3—188no—
209C23×C3C2×D3211yescharacteristic
210C22×D3—36no—
211C22×D3—312no—
212C22×D3—68no—
213C22×D3—64no—
214C22×D3—912no—
215C22×D3—96no—
216C22×D3—188no—
217C22×D3—184no—
218C3×Q8—916no—
219C3×D4—916no—
220C2×C3×C4—36no—
221C2×C3×C4—36no—
222C2×C3×C4—66no—
223C2×C3×C4—916no—
224C2×C3×C4—96no—
225C2×C3×C4—912no—
226C2×C3×C4C2×D3216yes—
227#67—932no—
228#67—932no—
229C2×Dic3—312no—
230C2×Dic3—312no—
231C2×Dic3—36no—
232C2×Dic3—612no—
233C2×Dic3—96no—
234C2×Dic3—932no—
235D12—316no—
236D12—916no—
237D12—932no—
238C4×D3—316no—
239C4×D3—916no—
240C4×D3—932no—
241C4×D3—932no—
242Dic6—316no—
243Dic6—316no—
244Dic6—616no—
245C2×C32—26no—
246C2×C32—21no—
247C2×C32C2×#23516yescomplemented
248C2×C32C2×#23716yescomplemented
249C2×C32C24×D311yescharacteristic
250C2×C32C24×D311yescharacteristic
251C3⋊D3—24no—
252C3⋊D3—28no—
253C3×D3—68no—
254C3×D3—64no—
255C24—91no—
256C24—92no—
257#41—2764no—
258C2×Q8—912no—
259C2×D4—912no—
260C2×D4—2724no—
261C22×C4—91no—
262C22×C4—92no—
263C22×C4—912no—
264C22×C4—2724no—
265C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
266C22×C3—33no—
267C22×C3—33no—
268C22×C3—34no—
269C22×C3—34no—
270C22×C3—63no—
271C22×C3—64no—
272C22×C3—98no—
273C22×C3—94no—
274C22×C3—1812no—
275C22×C3—1824no—
276C22×C3#379514yes—
277C22×C3C22×D3213yes—
278C2×D3—34no—
279C2×D3—38no—
280C2×D3—612no—
281C2×D3—624no—
282C2×D3—98no—
283C2×D3—94no—
284C2×D3—1824no—
285C2×D3—1812no—
286C3×C4—34no—
287C3×C4—34no—
288C3×C4—64no—
289C3×C4—94no—
290C3×C4—98no—
291C3×C4C22×D3214yes—
292Dic3—38no—
293Dic3—38no—
294Dic3—34no—
295Dic3—68no—
296Dic3—94no—
297C32—21no—
298C32C22×#23511yescharacteristic, complemented
299C32C22×#23711yescharacteristic, complemented
300C23—96no—
301C23—912no—
302C23—184no—
303C23—188no—
304C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
305Q8—916no—
306D4—916no—
307D4—2732no—
308C2×C4—96no—
309C2×C4—912no—
310C2×C4—916no—
311C2×C4—2732no—
312C2×C4C2×((C3⋊D3)⋊C3)⋊C216yes—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—36no—
316C2×C3—36no—
317C2×C3—61no—
318C2×C3—66no—
319C2×C3—184no—
320C2×C3—188no—
321C2×C3C2×#379516yes—
322C2×C3C23×D3211yescharacteristic
323D3—64no—
324D3—68no—
325D3—188no—
326D3—184no—
327C22—94no—
328C22—98no—
329C22—1812no—
330C22—1824no—
331C22#8864814yesdirect product factor
332C22C22×((C3⋊D3)⋊C3)⋊C213yes—
333C4—94no—
334C4—98no—
335C4C22×((C3⋊D3)⋊C3)⋊C214yes—
336C3—31no—
337C3—31no—
338C3—61no—
339C3C22×#379511yescharacteristic
340C2—184no—
341C2—188no—
342C2C2×#8864816yesdirect product factor
343C2C23×((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×#6229C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C22×C4×(C3⋊D3)⋊C3C2semidirect
4. C22×#6198C2semidirect
5. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
6. C2×#88648C2direct (N × H)
13. C2×#6229C22semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C4×(C3⋊D3)⋊C3C22semidirect
17. C2×#6198C22semidirect
18. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
19. #88648C22direct (N × H)
20. C22×C4⋊(C3⋊D3)D3semidirect
21. C22×C4×C3⋊D3D3semidirect
40. #6229C23semidirect
42. C4⋊((C3⋊D3)⋊C3)C23semidirect
43. C4×(C3⋊D3)⋊C3C23semidirect
44. #6198C23semidirect
45. ((C3⋊D3)⋊C3)⋊C4C23semidirect
57. C2×C4⋊(C3⋊D3)C2×D3semidirect
58. C2×C4×C3⋊D3C2×D3semidirect
74. C22×H(3,𝔽3)#41semidirect
117. C4⋊(C3⋊D3)C22×D3semidirect
118. C4×C3⋊D3C22×D3semidirect
133. C2×H(3,𝔽3)C2×#41semidirect
175. C22×C32#235semidirect
176. C22×C32#237semidirect
201. H(3,𝔽3)C22×#41semidirect
247. C2×C32C2×#235semidirect
248. C2×C32C2×#237semidirect
298. C32C22×#235semidirect
299. C32C22×#237semidirect
331. C22#88648direct (N × H)
342. C2C2×#88648direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×C4⋊((C3⋊D3)⋊C3)2
C22×C4×(C3⋊D3)⋊C32
C22×#61982
C22×((C3⋊D3)⋊C3)⋊C42
C2×#886482
C22×#37993
C22×#37963