Finite Group Databasebeta

← C2×#11721110 #13150638

Subgroups — C2×#11721110 #13150638

Order 1728.

1311 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1311 conjugacy classes of subgroups — too many to draw. Only its 219 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
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#88450C214yesmaximal, complemented
5#11721110C218yesmaximal, direct product factor
6C2×#63864—31nomaximal
7C2×#63863—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C2×#6229—24no—
12C2×#6229C2214yescomplemented
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)—24no—
15C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
16C4×(C3⋊Dic3)⋊C3C2214yescomplemented
17#88450C22116yescomplemented
18C22×C4⋊(C3⋊D3)D311yescharacteristic
19C22×C3×D12—31no—
20C22×C3×Dic6—31no—
21C22×C3×Dic6—31no—
22C22×C3×Dic6—61no—
23C2×C4×C3⋊Dic3D311yescharacteristic, complemented
24C2×C3×C4×Dic3—31no—
25C2×#3722—34no—
26C2×(C2×D3)⋊Dic3—34no—
27#63864—38no—
28#63863—38no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—24no—
31C22×(C3⋊D3)⋊C3—28no—
32C22×(C3⋊D3)⋊C3C2314yes—
33C2×C4×H(3,𝔽3)D414yes—
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084—24no—
36C2×#1084C2314yes—
37#6229—28no—
38C2×(C3⋊Dic3)⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C4⋊((C3⋊D3)⋊C3)—28no—
41C2×#947—91no—
42C2×#936—91no—
43C2×#875—91no—
44C23×C3⋊D3C2×D312yes—
45C23×C3×D3—32no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D312yes—
50C2×C4⋊(C3⋊D3)—24no—
51C2×C4⋊(C3⋊D3)C2×D314yes—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—62no—
56C2×C3×D12—34no—
57C2×C3×D12—64no—
58C2×C3×Dic6—34no—
59C2×C3×Dic6—34no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—68no—
62C2×C3×Dic6—64no—
63C2×C3×Dic6—64no—
64C4×C3⋊Dic3C2×D314yescomplemented
65C3×C4×Dic3—34no—
66#3722—316no—
67(C2×D3)⋊Dic3—316no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)#4114yes—
70C22×H(3,𝔽3)C2411yescharacteristic
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—28no—
73C2×(C3⋊D3)⋊C3—48no—
74C4×H(3,𝔽3)C2×D414yes—
75#1084—24no—
76(C3⋊Dic3)⋊C3—24no—
77C22×C3×Q8—91no—
78C22×C3×D4—91no—
79C22×D12—31no—
80C22×D12—91no—
81C22×Dic6—31no—
82C22×Dic6—31no—
83C22×Dic6—61no—
84C2×C3×C42—91no—
85C2×#217—94no—
86#947—98no—
87#936—98no—
88C2×(C2×D3)⋊C4—94no—
89C2×(C2×D3)⋊C4—94no—
90C2×C4×Dic3—31no—
91C2×C4×Dic3—91no—
92#875—98no—
93C23×C32—21no—
94C23×C32C22×D311yescharacteristic
95C23×C32C22×D311yescharacteristic
96C22×C3⋊D3—28no—
97C22×C3⋊D3—24no—
98C22×C3⋊D3C22×D314yes—
99C22×C3×D3—34no—
100C22×C3×D3—64no—
101C22×C3×D3—68no—
102C2×C32×C4—24no—
103C2×C32×C4—22no—
104C2×C32×C4D1214yes—
105C2×C32×C4#6714yes—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D312yes—
108C2×C3⋊Dic3—24no—
109C2×C3⋊Dic3C22×D314yes—
110C4⋊(C3⋊D3)—28no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—68no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C3×D12—68no—
120C3×Dic6—616no—
121C3×Dic6—68no—
122C3×Dic6—68no—
123C2×#124—271noSylow
124C2×H(3,𝔽3)#12414yescomplemented
125C2×H(3,𝔽3)C22×D411yescharacteristic
126C2×H(3,𝔽3)C2×#4112yes—
127(C3⋊D3)⋊C3—44no—
128C24×C3—92no—
129C23×D3—32no—
130C23×D3—92no—
131C2×C3×Q8—94no—
132C2×C3×Q8—184no—
133C2×C3×D4—94no—
134C2×C3×D4—184no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—61no—
138C22×C3×C4—92no—
139C22×C3×C4—92no—
140C22×C3×C4D3211yescharacteristic
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—32no—
144C22×Dic3—62no—
145C22×Dic3—92no—
146C2×D12—34no—
147C2×D12—64no—
148C2×D12—94no—
149C2×D12—184no—
150C2×Dic6—34no—
151C2×Dic6—34no—
152C2×Dic6—64no—
153C2×Dic6—64no—
154C2×Dic6—64no—
155C2×Dic6—68no—
156C3×C42—94no—
157#217—916no—
158(C2×D3)⋊C4—916no—
159(C2×D3)⋊C4—916no—
160C4×Dic3—34no—
161C4×Dic3—94no—
162C22×C32—21no—
163C22×C32—24no—
164C22×C32—22no—
165C22×C32C2×D1212yes—
166C22×C32#23514yes—
167C22×C32#23714yes—
168C22×C32C2×#6712yes—
169C22×C32C23×D311yescharacteristic
170C22×C32C23×D311yescharacteristic
171C2×C3⋊D3—28no—
172C2×C3⋊D3—24no—
173C2×C3⋊D3—48no—
174C2×C3×D3—64no—
175C2×C3×D3—68no—
176C2×C3×D3—128no—
177C32×C4—24no—
178C32×C4C2×D1214yes—
179C32×C4C2×#6714yes—
180C3⋊Dic3—24no—
181C3×Dic3—68no—
182C3×Dic3—64no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C22×Q8—91no—
186C22×D4—91no—
187#124—278no—
188C2×C22⋊C4—274no—
189C2×C42—91no—
190H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—61no—
194C23×C3—94no—
195C23×C3—184no—
196C23×C3—188no—
197C23×C3C2×D3211yescharacteristic
198C22×D3—34no—
199C22×D3—68no—
200C22×D3—64no—
201C22×D3—94no—
202C22×D3—184no—
203C22×D3—188no—
204C3×Q8—188no—
205C3×D4—188no—
206C2×C3×C4—34no—
207C2×C3×C4—34no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—64no—
211C2×C3×C4—62no—
212C2×C3×C4—94no—
213C2×C3×C4—94no—
214C2×C3×C4—184no—
215C2×C3×C4—184no—
216C2×C3×C4#62614yes—
217C2×C3×C4C2×D3212yes—
218C2×Dic3—34no—
219C2×Dic3—34no—
220C2×Dic3—34no—
221C2×Dic3—64no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224C2×Dic3—68no—
225C2×Dic3—64no—
226C2×Dic3—94no—
227C2×Dic3—184no—
228D12—68no—
229D12—188no—
230Dic6—68no—
231Dic6—68no—
232Dic6—616no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32#87514yescomplemented
237C2×C32#93614yescomplemented
238C2×C32C22×D1211yescharacteristic
239C2×C32C2×#23512yes—
240C2×C32C2×#23712yes—
241C2×C32C22×#6711yescharacteristic
242C3⋊D3—44no—
243C3×D3—124no—
244C24—92no—
245C2×Q8—94no—
246C2×Q8—184no—
247C2×D4—94no—
248C2×D4—184no—
249C22×C4—92no—
250C22×C4—92no—
251C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C22⋊C4—2716no—
253C42—94no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—34no—
257C22×C3—34no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—61no—
261C22×C3—64no—
262C22×C3—62no—
263C22×C3—184no—
264C22×C3—188no—
265C22×C3—368no—
266C22×C3#379514yes—
267C22×C3C2×#62612yes—
268C22×C3C22×D3211yescharacteristic
269C2×D3—68no—
270C2×D3—64no—
271C2×D3—128no—
272C2×D3—184no—
273C2×D3—188no—
274C2×D3—368no—
275C3×C4—34no—
276C3×C4—34no—
277C3×C4—64no—
278C3×C4—184no—
279C3×C4—184no—
280C3×C4C2×#62614yes—
281Dic3—64no—
282Dic3—64no—
283Dic3—68no—
284Dic3—64no—
285Dic3—184no—
286C32—21no—
287C32C2×#87511yescharacteristic, complemented
288C32C2×#93611yescharacteristic, complemented
289C23—94no—
290C23—188no—
291C23—184no—
292C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
293Q8—188no—
294D4—188no—
295C2×C4—94no—
296C2×C4—94no—
297C2×C4—184no—
298C2×C4—184no—
299C2×C4#620014yes—
300C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—34no—
304C2×C3—34no—
305C2×C3—32no—
306C2×C3—32no—
307C2×C3—61no—
308C2×C3—64no—
309C2×C3—62no—
310C2×C3—364no—
311C2×C3#6386014yes—
312C2×C3C2×#379512yes—
313C2×C3C22×#62611yescharacteristic
314D3—124no—
315D3—364no—
316C22—184no—
317C22—188no—
318C22—368no—
319C22#8864814yes—
320C22C2×#620012yes—
321C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C4—184no—
323C4—184no—
324C4C2×#620014yes—
325C3—31no—
326C3—31no—
327C3—61no—
328C3C2×#6386011yescharacteristic
329C2—364no—
330C2#1172111014yesdirect product factor
331C2C2×#8864812yes—
332C2C22×#620011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#124
LCSG ▷ C22×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
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#88450C2semidirect
5. #11721110C2direct (N × H)
12. C2×#6229C22semidirect
16. C4×(C3⋊Dic3)⋊C3C22semidirect
17. #88450C22semidirect
23. C2×C4×C3⋊Dic3D3semidirect
64. C4×C3⋊Dic3C2×D3semidirect
124. C2×H(3,𝔽3)#124semidirect
190. H(3,𝔽3)C2×#124semidirect
236. C2×C32#875semidirect
237. C2×C32#936semidirect
287. C32C2×#875semidirect
288. C32C2×#936semidirect
330. C2#11721110direct (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
C2×C4×(C3⋊Dic3)⋊C32
C2×#884502
#117211102
C2×#638643
C2×#638633