Finite Group Databasebeta

← C2×#11722380 #13164474

Subgroups — C2×#11722380 #13164474

Order 1728.

808 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88613C212yesmaximal
3#11722380C214yesmaximal, direct product factor
4C2×C3×A4⋊Q8—33nomaximal
5C2×C3×#924—31nomaximal
6C22×C22⋊H(3,𝔽3)C2211yescharacteristic
7C2×#6229—41nomaximal
8C4×C22⋊H(3,𝔽3)C2212yes—
9#88613C2214yes—
10C23×C32×C4D311yescharacteristic
11C22×C3×Dic6—31no—
12C2×C3×C4×A4D313yes—
13C2×C3×SL(2,ℤ/4)—36no—
14C3×A4⋊Q8—312no—
15C2×C3×#217—32no—
16C2×C3×C4⋊Dic3—31no—
17C3×#924—34no—
18C3×#924—34no—
19C2×C3×Dic3⋊C4—32no—
20C2×C22⋊H(3,𝔽3)Q812yescomplemented
21C2×C22⋊H(3,𝔽3)C2311yescharacteristic
22C2×C4×H(3,𝔽3)—41no—
23C2×#1084—42no—
24#6229—44no—
25C2×A4⋊Q8—33no—
26C2×C3×C22⋊Q8—91no—
27C2×#924—31no—
28C24×C32C2×D311yescharacteristic
29C22×C3×A4C2×D313yes—
30C22×C32×C4—31no—
31C22×C32×C4—31no—
32C22×C32×C4—32no—
33C22×C32×C4C2×D312yes—
34C22×C3×Dic3—32no—
35C22×C3×Dic3—32no—
36C2×C3×Dic6—34no—
37C2×C3×Dic6—61no—
38C2×C3×Dic6—62no—
39C2×C3×Dic6—61no—
40C2×C3×Dic6—123no—
41C3×C4×A4C2×D316yes—
42C3×SL(2,ℤ/4)—312no—
43C3×#217—34no—
44C3×#217—34no—
45C3×C4⋊Dic3—34no—
46C3×Dic3⋊C4—38no—
47C22×H(3,𝔽3)—41no—
48C22⋊H(3,𝔽3)C2×Q811yescharacteristic, complemented
49C4×H(3,𝔽3)—42no—
50#1084—44no—
51C22×C3×Q8—91no—
52C23×C3×C4—31no—
53C23×C3×C4C3⋊D311yescharacteristic
54C22×Dic6—31no—
55C2×C4×A4—33no—
56C2×SL(2,ℤ/4)—36no—
57A4⋊Q8—312no—
58C3×C22⋊Q8—94no—
59C3×C22⋊Q8—94no—
60C2×C3×C4⋊C4—92no—
61C2×C3×C4⋊C4—91no—
62C2×C3×C22⋊C4—92no—
63C2×#217—32no—
64C2×C4⋊Dic3—31no—
65#924—34no—
66#924—34no—
67C2×Dic3⋊C4—32no—
68C23×C32—31no—
69C23×C32—32no—
70C23×C32—31no—
71C23×C32Dic612yescomplemented
72C23×C32C22×D311yescharacteristic
73C2×C3×A4Dic616yescomplemented
74C2×C3×A4C22×D313yes—
75C2×C32×C4—31no—
76C2×C32×C4—32no—
77C2×C32×C4—32no—
78C2×C32×C4—32no—
79C2×C32×C4—43no—
80C2×C32×C4—61no—
81C2×C32×C4S411yescharacteristic
82C2×C3×Dic3—34no—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—126no—
88C3×Dic6—64no—
89C3×Dic6—64no—
90C3×Dic6—1212no—
91C2×C22⋊Q8—91noSylow
92C2×H(3,𝔽3)—42no—
93C2×H(3,𝔽3)—41no—
94C24×C3—31no—
95C24×C3C2×C3⋊D311yescharacteristic
96C22×A4—33no—
97C2×C3×Q8—94no—
98C2×C3×Q8—181no—
99C2×C3×Q8—182no—
100C2×C3×Q8—181no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—31no—
104C22×C3×C4—32no—
105C22×C3×C4—31no—
106C22×C3×C4—32no—
107C22×C3×C4—32no—
108C22×C3×C4—61no—
109C22×C3×C4—61no—
110C22×C3×C4—62no—
111C22×C3×C4—92no—
112C22×C3×C4—92no—
113C22×C3×C4C2×C3⋊D312yes—
114C22×Dic3—32no—
115C22×Dic3—32no—
116C2×Dic6—34no—
117C2×Dic6—61no—
118C2×Dic6—62no—
119C2×Dic6—61no—
120C2×Dic6—123no—
121C4×A4—36no—
122SL(2,ℤ/4)—312no—
123C3×C4⋊C4—98no—
124C3×C4⋊C4—94no—
125C3×C22⋊C4—94no—
126C3×C22⋊C4—94no—
127#217—34no—
128#217—34no—
129C4⋊Dic3—34no—
130Dic3⋊C4—38no—
131C22×C32—32no—
132C22×C32—31no—
133C22×C32—32no—
134C22×C32—32no—
135C22×C32—31no—
136C22×C32—31no—
137C22×C32—43no—
138C22×C32—61no—
139C22×C32C2×Dic611yescharacteristic, complemented
140C22×C32C2×S411yescharacteristic
141C3×A4C2×Dic613yescomplemented
142C32×C4—32no—
143C32×C4—46no—
144C32×C4C2×S412yes—
145C3×Dic3—64no—
146C3×Dic3—64no—
147C3×Dic3—1212no—
148C22×Q8—91no—
149C23×C4#27211yescharacteristic
150C22⋊Q8—94no—
151C22⋊Q8—94no—
152C2×C4⋊C4—92no—
153C2×C4⋊C4—91no—
154C2×C22⋊C4—92no—
155H(3,𝔽3)—41noSylow
156C23×C3—31no—
157C23×C3—31no—
158C23×C3—31no—
159C23×C3—32no—
160C23×C3—31no—
161C23×C3—32no—
162C23×C3—32no—
163C23×C3—31no—
164C23×C3—61no—
165C23×C3—61no—
166C23×C3—62no—
167C23×C3#63412yes—
168C23×C3C22×C3⋊D311yescharacteristic
169C2×A4—36no—
170C2×A4—33no—
171C3×Q8—184no—
172C3×Q8—184no—
173C2×C3×C4—32no—
174C2×C3×C4—31no—
175C2×C3×C4—32no—
176C2×C3×C4—31no—
177C2×C3×C4—32no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—94no—
190C2×C3×C4—94no—
191C2×C3×C4—123no—
192C2×C3×C4—184no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4C3⋊S411yescharacteristic
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—126no—
202Dic6—64no—
203Dic6—64no—
204Dic6—1212no—
205C2×C32—31no—
206C2×C32—31no—
207C2×C32—32no—
208C2×C32—46no—
209C2×C32—43no—
210C2×C32A4⋊Q812yescomplemented
211C2×C32C22×S411yescharacteristic
212C24C2×#27211yescharacteristic
213C2×Q8—94no—
214C2×Q8—181no—
215C2×Q8—182no—
216C2×Q8—181no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—32no—
220C22×C4—92no—
221C22×C4—92no—
222C22×C4C2×#27212yes—
223C4⋊C4—98no—
224C4⋊C4—94no—
225C22⋊C4—94no—
226C22⋊C4—94no—
227C22×C3—32no—
228C22×C3—31no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—32no—
234C22×C3—32no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—32no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—62no—
244C22×C3—61no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—62no—
250C22×C3—123no—
251C22×C3C2×#63411yescharacteristic
252C22×C3C2×C3⋊S411yescharacteristic
253A4—33no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—62no—
258C3×C4—126no—
259C3×C4—184no—
260C3×C4—184no—
261C3×C4C2×C3⋊S412yes—
262Dic3—64no—
263Dic3—64no—
264Dic3—1212no—
265C32—43no—
266C32C2×A4⋊Q811yescharacteristic, complemented
267C23—31no—
268C23—31no—
269C23—32no—
270C23#622912yescomplemented
271C23C22×#27211yescharacteristic
272Q8—184no—
273Q8—184no—
274C2×C4—31no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—61no—
279C2×C4—94no—
280C2×C4—94no—
281C2×C4—182no—
282C2×C4—184no—
283C2×C4—182no—
284C2×C4#625811yescharacteristic
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—32no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—32no—
291C2×C3—31no—
292C2×C3—32no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—62no—
296C2×C3—123no—
297C2×C3—126no—
298C2×C3#6425912yes—
299C2×C3C22×C3⋊S411yescharacteristic
300C22—32no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—32no—
305C22—32no—
306C22—61no—
307C22C2×#622911yescharacteristic, complemented
308C22C2×#625811yescharacteristic
309C4—32no—
310C4—184no—
311C4—184no—
312C4C2×#625812yes—
313C3—31no—
314C3—123no—
315C3C2×#6425911yescharacteristic
316C2—31no—
317C2—31no—
318C2—32no—
319C2#1172238012yesdirect product factor
320C2C22×#625811yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC24×C3C2×C3⋊D3
prime radicalC22×C22⋊H(3,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
hypocenterC22⋊H(3,𝔽3)C2×Q8
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. #11722380C2direct (N × H)
20. C2×C22⋊H(3,𝔽3)Q8semidirect
48. C22⋊H(3,𝔽3)C2×Q8semidirect
71. C23×C32Dic6semidirect
73. C2×C3×A4Dic6semidirect
139. C22×C32C2×Dic6semidirect
141. C3×A4C2×Dic6semidirect
210. C2×C32A4⋊Q8semidirect
266. C32C2×A4⋊Q8semidirect
270. C23#6229semidirect
307. C22C2×#6229semidirect
319. C2#11722380direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
C2×#886132
#117223802
C2×C3×A4⋊Q83
C2×C3×#9243
C2×#62294