Finite Group Databasebeta

← C2×#11723108 #13173410

Subgroups — C2×#11723108 #13173410

Order 1728.

776 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 93 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×C3×#3830C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×#88989C211yesmaximal, characteristic, complemented
4#11723108C214yesmaximal, direct product factor
5C2×#63783—31nomaximal
6C22×C3⋊#157—21no—
7C2×C33×D4C411yescharacteristic, complemented
8C2×C3×#638—21no—
9C22×C3×C3⋊Dic3—21no—
10C3×#3830—21no—
11C3×#3830—21no—
12C3×#3830C412yescomplemented
13C3×#3830C2212yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C411yescharacteristic, complemented
16C2×C3⋊#622—21no—
17C4×C3⋊#157—21no—
18C4×C3⋊#157C2212yescomplemented
19#88989—21no—
20#88989C2212yescomplemented
21C2×#3830D311yescharacteristic
22C2×C3×#237—62no—
23C2×C4×#157—31no—
24C2×#3788—31no—
25#63783—34no—
26C23×C33—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3⋊#157—24no—
29C2×C3⋊#157—22no—
30C33×D4—21no—
31C33×D4C2×C412yescomplemented
32C2×C33×C4C2×C411yescharacteristic
33C3×#638—22no—
34C3×#638—22no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3C2×C411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C3×#634—21no—
44C3×#634C2×C412yescomplemented
45C3⋊#622—22no—
46C2×#837—91nomaximal
47C22×#157—61no—
48C2×C32×D4—22no—
49C2×C32×D4—42no—
50C2×C32×D4Dic311yescharacteristic, complemented
51C2×#638—21no—
52C22×C3⋊Dic3—21no—
53#3830—21no—
54#3830—21no—
55#3830Dic312yescomplemented
56#3830C2×D312yes—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×#634Dic311yescharacteristic, complemented
59C2×C3×#67—64no—
60C22×C3×Dic3—64no—
61C3×#237—64no—
62C3×#237—64no—
63C3×#237—64no—
64C3×#237—64no—
65C2×C3×C4×D3—62no—
66C2×C3×Dic6—62no—
67C2×#622—61no—
68C4×#157—32no—
69C4×#157—61no—
70#3788—32no—
71#3788—61no—
72C22×C33—22no—
73C22×C33—42no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C3⋊#157—24no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C3×C3⋊Dic3C2×D411yescharacteristic
84C2×C3×#41—91no—
85C2×#237—62no—
86C2×C4×Dic3—91no—
87C2×#208—91no—
88#837—94no—
89C23×C32—21no—
90C23×C32—24no—
91C23×C32—44no—
92C22×C3⋊D3#6711yescharacteristic
93C22×C3×D3—62no—
94C2×#157—64no—
95C2×#157—62no—
96C32×D4—21no—
97C32×D4—24no—
98C32×D4—24no—
99C32×D4—44no—
100C32×D4—44no—
101C32×D4C2×Dic312yescomplemented
102C2×C32×C4—22no—
103C2×C32×C4—42no—
104C2×C32×C4C2×Dic311yescharacteristic
105#638—22no—
106#638—22no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3#6711yescharacteristic
111C4×C3⋊D3C2×Dic311yescharacteristic
112C4×C3⋊D3#6711yescharacteristic
113C4×C3⋊D3#6711yescharacteristic
114C4×C3⋊D3C22×D311yescharacteristic
115#634—21no—
116#634C2×Dic312yescomplemented
117C3×#67—68no—
118C3×#67—68no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—68no—
122C2×C3×Dic3—68no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129#622—62no—
130C2×#104—271noSylow
131C2×C33—42no—
132C2×C33#10412yescomplemented
133C2×C33C2×C22⋊C411yescharacteristic
134C3×C3⋊D3—21no—
135C3×C3⋊D3—21no—
136C3×#41—92no—
137C3×#41—92no—
138C3×#41—181no—
139C3×#41—181no—
140C2×C3×Q8—91no—
141C2×C3×D4—22no—
142C2×C3×D4—42no—
143C2×C3×D4—181no—
144C2×C3×D4#15711yescharacteristic, complemented
145C22×C3×C4—91no—
146C22×C3×C4—181no—
147C2×#67—64no—
148C22×Dic3—64no—
149C22×Dic3—181no—
150#237—64no—
151#237—64no—
152#237—64no—
153#237—64no—
154C2×C4×D3—62no—
155C2×Dic6—62no—
156C4×Dic3—92no—
157C4×Dic3—181no—
158#208—92no—
159#208—181no—
160C2×C3⋊C8—181no—
161C22×C32—22no—
162C22×C32—28no—
163C22×C32—22no—
164C22×C32—48no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32—48no—
168C22×C32—88no—
169C22×C32#21711yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3#21711yescharacteristic
172C2×C3⋊D3C2×#6711yescharacteristic
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—124no—
176#157—64no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4#21711yescharacteristic
182C32×C4C22×Dic311yescharacteristic
183C3⋊Dic3—22no—
184C3⋊Dic3#21711yescharacteristic
185C3⋊Dic3C2×#6711yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—62no—
188C3×Dic3—68no—
189C2×#41—91no—
190C2×M16—271no—
191C2×C42—271no—
192#104—274no—
193C33C2×#10411yescharacteristic, complemented, Sylow
194C23×C3—24no—
195C23×C3—21no—
196C23×C3—44no—
197C23×C3—91no—
198C22×D3—62no—
199C3×Q8—92no—
200C3×Q8—181no—
201C3×D4—24no—
202C3×D4—21no—
203C3×D4—24no—
204C3×D4—44no—
205C3×D4—44no—
206C3×D4—182no—
207C3×D4—182no—
208C3×D4C2×#15712yescomplemented
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—181no—
219C2×C3×C4C2×#15711yescharacteristic
220#67—68no—
221#67—68no—
222C2×Dic3—62no—
223C2×Dic3—64no—
224C2×Dic3—68no—
225C2×Dic3—68no—
226C2×Dic3—182no—
227C2×Dic3—184no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—62no—
232Dic6—64no—
233Dic6—64no—
234C3⋊C8—182no—
235C2×C32—22no—
236C2×C32—24no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32—42no—
240C2×C32—48no—
241C2×C32—88no—
242C2×C32#83712yescomplemented
243C2×C32C2×#21711yescharacteristic
244C3⋊D3—21no—
245C3⋊D3—21no—
246C3×D3—122no—
247C3×D3—122no—
248#41—92no—
249#41—92no—
250#41—181no—
251#41—181no—
252C2×Q8—91no—
253C2×D4—181no—
254C2×D4C3⋊#15711yescharacteristic, complemented
255C22×C4—91no—
256C22×C4—181no—
257C22×C4—541no—
258M16—272no—
259M16—541no—
260C2×C8—541no—
261C42—272no—
262C42—541no—
263C22×C3—28no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—48no—
267C22×C3—48no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—88no—
271C22×C3—91no—
272C22×C3—91no—
273C22×C3—182no—
274C22×C3C22⋊#15711yescharacteristic
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—124no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—42no—
281C3×C4—42no—
282C3×C4—91no—
283C3×C4—91no—
284C3×C4—182no—
285C3×C4C22⋊#15711yescharacteristic
286C3×C4C22×#15711yescharacteristic
287Dic3—62no—
288Dic3—62no—
289Dic3—68no—
290Dic3—184no—
291C32—22no—
292C32—42no—
293C32C2×#83711yescharacteristic, complemented
294C23—21no—
295C23—91no—
296Q8—92no—
297Q8—181no—
298D4—21no—
299D4—182no—
300D4—182no—
301D4C2×C3⋊#15712yescomplemented
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—181no—
310C2×C4—542no—
311C2×C4—544no—
312C2×C4C2×C3⋊#15711yescharacteristic
313C8—542no—
314C2×C3—24no—
315C2×C3—22no—
316C2×C3—48no—
317C2×C3—42no—
318C2×C3—44no—
319C2×C3—42no—
320C2×C3—88no—
321C2×C3—181no—
322C2×C3—181no—
323C2×C3#6378312yescomplemented
324C2×C3C2×C22⋊#15711yescharacteristic
325D3—122no—
326D3—122no—
327C22—22no—
328C22—42no—
329C22—91no—
330C22—91no—
331C22—182no—
332C22#8899411yescharacteristic
333C4—91no—
334C4—91no—
335C4—182no—
336C4—544no—
337C4#8899411yescharacteristic
338C4C22×C3⋊#15711yescharacteristic
339C3—22no—
340C3—42no—
341C3C2×#6378311yescharacteristic, complemented
342C2—42no—
343C2—181no—
344C2—181no—
345C2#1172310812yesdirect product factor
346C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×D4C4
socleC22×C33C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. C2×#88989C2semidirect
4. #11723108C2direct (N × H)
7. C2×C33×D4C4semidirect
12. C3×#3830C4semidirect
15. C2×C3×#634C4semidirect
18. C4×C3⋊#157C22semidirect
20. #88989C22semidirect
31. C33×D4C2×C4semidirect
44. C3×#634C2×C4semidirect
50. C2×C32×D4Dic3semidirect
55. #3830Dic3semidirect
58. C2×#634Dic3semidirect
101. C32×D4C2×Dic3semidirect
116. #634C2×Dic3semidirect
132. C2×C33#104semidirect
144. C2×C3×D4#157semidirect
193. C33C2×#104semidirect
208. C3×D4C2×#157semidirect
242. C2×C32#837semidirect
254. C2×D4C3⋊#157semidirect
293. C32C2×#837semidirect
301. D4C2×C3⋊#157semidirect
323. C2×C3#63783semidirect
341. C3C2×#63783semidirect
345. C2#11723108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C4×C3⋊#1572
C2×#889892
#117231082
C2×#637833
C2×#8379