Finite Group Databasebeta

← C22×#88934 #13181112

Subgroups — C22×#88934 #13181112

Order 1728.

1416 conjugacy classes of subgroups; only the 279 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 279 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
1C23×C3⋊#157C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic
3C22×C3×#624C211yesmaximal, characteristic
4C2×#88934C2112yesmaximal, direct product factor
5C22×#3773—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157C418yescomplemented
8C22×C3⋊#157C2216yes—
9C22×C3×C3⋊Dic3—21no—
10C22×C32×Dic3—21no—
11C2×#6294—23no—
12C2×#6294C2216yes—
13C2×C3×#624—23no—
14C2×C3×#624C2216yes—
15#88934C22116yesdirect product factor
16C23×#157—31no—
17C22×C3×C4×D3—61no—
18C22×#624—31no—
19C22×#624D311yescharacteristic
20C22×D3×Dic3—61no—
21C2×#3773—312no—
22C23×C33D411yescharacteristic
23C22×C3×C3⋊D3C2×C414yes—
24C22×C3×C3⋊D3D413yes—
25C22×C3×C3⋊D3Q814yes—
26C22×C3×C3⋊D3C2313yes—
27C2×C3⋊#157C2×C4124yescomplemented
28C2×C3⋊#157C2314yes—
29C2×C3×C3⋊Dic3—26no—
30C2×C32×Dic3—26no—
31#6294—26no—
32#6294C2314yes—
33C3×#624—26no—
34C3×#624C2314yes—
35C22×Dic3⋊C4—91nomaximal
36C23×C3⋊D3C2×D311yescharacteristic
37C23×C3×D3—61no—
38C23×C3×D3—61no—
39C22×#157—38no—
40C22×#157—36no—
41C22×C32×C4—61no—
42C22×C3⋊Dic3—21no—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—21no—
45C22×C3×Dic3—41no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3—121no—
49C2×C3×C4×D3—66no—
50C2×C3×C4×D3—66no—
51C2×#624—23no—
52C2×#624—36no—
53C2×#624—63no—
54C2×#624C2×D316yes—
55C2×D3×Dic3—66no—
56C2×D3×Dic3—66no—
57#3773—316no—
58C22×C33C4⋊C414yescomplemented
59C22×C33C2×D413yes—
60C2×C3×C3⋊D3C4⋊C4112yescomplemented
61C2×C3×C3⋊D3C22×C416yes—
62C2×C3×C3⋊D3C2×D413yes—
63C2×C3×C3⋊D3C2×Q816yes—
64C2×C3×C3⋊D3C2411yescharacteristic
65C3⋊#157C22×C418yescomplemented
66C3×C3⋊Dic3—24no—
67C32×Dic3—24no—
68C23×C3×C4—91no—
69C23×Dic3—91no—
70C23×Dic3—91no—
71C22×C4×D3—61no—
72C22×C4×D3—181no—
73C2×Dic3⋊C4—912no—
74C23×C32—21no—
75C23×C32—21no—
76C23×C32—41no—
77C23×C32—41no—
78C23×C32#6711yescharacteristic
79C22×C3⋊D3Dic614yes—
80C22×C3⋊D3C4×D314yes—
81C22×C3⋊D3#6713yes—
82C22×C3⋊D3C22×D313yes—
83C22×C3×D3—64no—
84C22×C3×D3—64no—
85C22×C3×D3—63no—
86C22×C3×D3—63no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C22×C3×D3—63no—
90C22×C3×D3—63no—
91C2×#157—324no—
92C2×#157—34no—
93C2×C32×C4—66no—
94C2×C3⋊Dic3—26no—
95C2×C3×Dic3—26no—
96C2×C3×Dic3—26no—
97C2×C3×Dic3—46no—
98C2×C3×Dic3—66no—
99C2×C3×Dic3—66no—
100C2×C3×Dic3—126no—
101C3×C4×D3—64no—
102C3×C4×D3—612no—
103#624—26no—
104#624—34no—
105#624—66no—
106#624C22×D314yes—
107D3×Dic3—64no—
108D3×Dic3—612no—
109C22×C4⋊C4—271noSylow
110C2×C33C2×C4⋊C416yescomplemented
111C2×C33C22×D411yescharacteristic
112C3×C3⋊D3C2×C4⋊C416yescomplemented
113C3×C3⋊D3C23×C411yescharacteristic
114C3×C3⋊D3C22×Q811yescharacteristic
115C24×C3—91no—
116C23×D3—61no—
117C23×D3—61no—
118C22×C3×C4—61no—
119C22×C3×C4—61no—
120C22×C3×C4—96no—
121C22×C3×C4—121no—
122C22×C3×C4—183no—
123C22×C3×C4—181no—
124C22×Dic3—21no—
125C22×Dic3—61no—
126C22×Dic3—61no—
127C22×Dic3—96no—
128C22×Dic3—98no—
129C22×Dic3—96no—
130C22×Dic3—121no—
131C22×Dic3—183no—
132C2×C4×D3—66no—
133C2×C4×D3—66no—
134C2×C4×D3—186no—
135C2×C4×D3—186no—
136Dic3⋊C4—916no—
137C22×C32—23no—
138C22×C32—23no—
139C22×C32—24no—
140C22×C32—24no—
141C22×C32—44no—
142C22×C32—44no—
143C22×C32—43no—
144C22×C32—43no—
145C22×C32Dic3⋊C414yescomplemented
146C22×C32C2×#6713yes—
147C2×C3⋊D3Dic3⋊C4112yescomplemented
148C2×C3⋊D3C2×Dic616yes—
149C2×C3⋊D3C2×C4×D316yes—
150C2×C3⋊D3C2×#6713yes—
151C2×C3⋊D3C23×D311yescharacteristic
152C2×C3×D3—612no—
153C2×C3×D3—612no—
154C2×C3×D3—66no—
155C2×C3×D3—66no—
156C2×C3×D3—63no—
157C2×C3×D3—63no—
158C2×C3×D3—66no—
159C2×C3×D3—66no—
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162#157—38no—
163C32×C4—64no—
164C3⋊Dic3—24no—
165C3×Dic3—24no—
166C3×Dic3—24no—
167C3×Dic3—44no—
168C3×Dic3—64no—
169C3×Dic3—64no—
170C3×Dic3—124no—
171C23×C4—91no—
172C23×C4—271no—
173C23×C4—271no—
174C2×C4⋊C4—2712no—
175C33C22×C4⋊C411yescharacteristic, complemented, Sylow
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—93no—
181C23×C3—93no—
182C23×C3—94no—
183C23×C3—94no—
184C23×C3GO+(4,𝔽2)11yescharacteristic
185C22×D3—64no—
186C22×D3—64no—
187C22×D3—64no—
188C22×D3—64no—
189C22×D3—63no—
190C22×D3—63no—
191C22×D3—63no—
192C22×D3—63no—
193C2×C3×C4—66no—
194C2×C3×C4—66no—
195C2×C3×C4—94no—
196C2×C3×C4—126no—
197C2×C3×C4—186no—
198C2×C3×C4—186no—
199C2×Dic3—26no—
200C2×Dic3—66no—
201C2×Dic3—66no—
202C2×Dic3—94no—
203C2×Dic3—924no—
204C2×Dic3—94no—
205C2×Dic3—126no—
206C2×Dic3—186no—
207C4×D3—64no—
208C4×D3—612no—
209C4×D3—184no—
210C4×D3—1812no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—26no—
214C2×C32—26no—
215C2×C32—46no—
216C2×C32—46no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32C2×Dic3⋊C416yescomplemented
220C2×C32C22×#6711yescharacteristic
221C3⋊D3C2×Dic3⋊C416yescomplemented
222C3⋊D3C22×Dic611yescharacteristic
223C3⋊D3C22×C4×D311yescharacteristic
224C3×D3—66no—
225C3×D3—66no—
226C3×D3—61no—
227C3×D3—61no—
228C3×D3—61no—
229C3×D3—61no—
230C24—91no—
231C22×C4—61no—
232C22×C4—96no—
233C22×C4—183no—
234C22×C4—181no—
235C22×C4—276no—
236C22×C4—276no—
237C22×C4—278no—
238C22×C4—543no—
239C4⋊C4—2716no—
240C22×C3—23no—
241C22×C3—23no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—43no—
245C22×C3—43no—
246C22×C3—44no—
247C22×C3—44no—
248C22×C3—91no—
249C22×C3—93no—
250C22×C3—96no—
251C22×C3—96no—
252C22×C3—912no—
253C22×C3#377314yescomplemented
254C22×C3C2×GO+(4,𝔽2)13yes—
255C2×D3—63no—
256C2×D3—61no—
257C2×D3—63no—
258C2×D3—61no—
259C2×D3—612no—
260C2×D3—612no—
261C2×D3—66no—
262C2×D3—66no—
263C2×D3—66no—
264C2×D3—66no—
265C3×C4—64no—
266C3×C4—64no—
267C3×C4—124no—
268C3×C4—184no—
269Dic3—24no—
270Dic3—64no—
271Dic3—64no—
272Dic3—98no—
273Dic3—124no—
274C32—21no—
275C32—21no—
276C32—41no—
277C32—41no—
278C32C22×Dic3⋊C411yescharacteristic, complemented
279C23—93no—
280C23—93no—
281C23—94no—
282C23—94no—
283C23D32⋊D311yescharacteristic
284C2×C4—66no—
285C2×C4—94no—
286C2×C4—186no—
287C2×C4—186no—
288C2×C4—274no—
289C2×C4—274no—
290C2×C4—2724no—
291C2×C4—546no—
292C2×C3—26no—
293C2×C3—26no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—46no—
297C2×C3—46no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—91no—
301C2×C3—91no—
302C2×C3—96no—
303C2×C3C2×#377316yescomplemented
304C2×C3C22×GO+(4,𝔽2)11yescharacteristic
305D3—66no—
306D3—66no—
307D3—61no—
308D3—61no—
309D3—61no—
310D3—61no—
311C22—96no—
312C22—96no—
313C22—912no—
314C22—91no—
315C22—93no—
316C22#8893414yesdirect product factor
317C22C2×D32⋊D313yes—
318C4—64no—
319C4—184no—
320C4—278no—
321C3—21no—
322C3—21no—
323C3—41no—
324C3—41no—
325C3C22×#377311yescharacteristic, complemented
326C2—91no—
327C2—96no—
328C2—91no—
329C2C2×#8893416yesdirect product factor
330C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutatorC3×C3⋊D3C23×C4
FrattiniC2C22×D32⋊D3
Fitting, socleC23×C33D4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C4⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×#88934C2direct (N × H)
7. C22×C3⋊#157C4semidirect
15. #88934C22direct (N × H)
27. C2×C3⋊#157C2×C4semidirect
58. C22×C33C4⋊C4semidirect
60. C2×C3×C3⋊D3C4⋊C4semidirect
65. C3⋊#157C22×C4semidirect
110. C2×C33C2×C4⋊C4semidirect
112. C3×C3⋊D3C2×C4⋊C4semidirect
145. C22×C32Dic3⋊C4semidirect
147. C2×C3⋊D3Dic3⋊C4semidirect
175. C33C22×C4⋊C4semidirect
219. C2×C32C2×Dic3⋊C4semidirect
221. C3⋊D3C2×Dic3⋊C4semidirect
253. C22×C3#3773semidirect
278. C32C22×Dic3⋊C4semidirect
303. C2×C3C2×#3773semidirect
316. C22#88934direct (N × H)
325. C3C22×#3773semidirect
329. C2C2×#88934direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C22×#62942
C22×C3×#6242
C2×#889342
C22×#37733
C22×Dic3⋊C49