Finite Group Databasebeta

← C2×#11722763 #13167059

Subgroups — C2×#11722763 #13167059

Order 1728.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3⋊#157C211yesmaximal, characteristic
2C2×#88808C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4#11722763C214yesmaximal, direct product factor
5C2×#63732—31nomaximal
6C22×C3⋊#157C411yescharacteristic
7C22×C3⋊#157C411yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C4×C3⋊#157C412yes—
10C4×C3⋊#157C2212yes—
11C2×C3×C3⋊(C3⋊C8)—21no—
12C2×C32×C3⋊C8—21no—
13#88808—21no—
14#88808C2212yes—
15C3×#3710—21no—
16C3×#3710C2212yes—
17C2×C4×#157—31no—
18C2×C3×D3×C8—61no—
19C2×#3710—31no—
20C2×#3710D311yescharacteristic
21C2×D3×C3⋊C8—61no—
22#63732—34no—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C3⋊#157—22no—
25C2×C3⋊#157C814yescomplemented
26C2×C3⋊#157C2×C412yes—
27C2×C3⋊#157C2×C412yes—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3Q811yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3Q811yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C3×C3⋊(C3⋊C8)—22no—
35C32×C3⋊C8—22no—
36C2×Dic3⋊C8—91nomaximal
37C22×#157—31no—
38C22×#157—31no—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—61no—
41C2×C3×C4×D3—61no—
42C4×#157—32no—
43C4×#157—32no—
44C2×C32×C8—61no—
45C2×C3⋊(C3⋊C8)—21no—
46C2×C3×C3⋊C8—21no—
47C2×C3×C3⋊C8—21no—
48C2×C3×C3⋊C8—41no—
49C2×C3×C3⋊C8—61no—
50C2×C3×C3⋊C8—61no—
51C2×C3×C3⋊C8—121no—
52C3×D3×C8—62no—
53C3×D3×C8—62no—
54#3710—21no—
55#3710—32no—
56#3710—61no—
57#3710C2×D312yes—
58D3×C3⋊C8—62no—
59D3×C3⋊C8—62no—
60C22×C33C4⋊C411yescharacteristic
61C2×C3×C3⋊D3C4⋊C411yescharacteristic
62C2×C3×C3⋊D3C2×C812yes—
63C2×C3×C3⋊D3M1612yes—
64C2×C3×C3⋊D3C22×C411yescharacteristic
65C3⋊#157—22no—
66C3⋊#157C2×C814yescomplemented
67C33×C4C4⋊C411yescharacteristic
68C33×C4C2×D411yescharacteristic
69C3×C3⋊Dic3C4⋊C411yescharacteristic
70C3×C3⋊Dic3C2×Q811yescharacteristic
71C22×C3×C8—91no—
72C2×C4×Dic3—91no—
73C22×C3⋊C8—91no—
74C2×D3×C8—61no—
75C2×D3×C8—181no—
76Dic3⋊C8—94no—
77C22×C3⋊D3C4×D311yescharacteristic
78C22×C3×D3—61no—
79C22×C3×D3—61no—
80C2×#157—34no—
81C2×#157—32no—
82C2×#157—32no—
83C2×#157—62no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4#6711yescharacteristic
89C2×C3⋊Dic3Dic611yescharacteristic
90C4×C3⋊D3Dic611yescharacteristic
91C4×C3⋊D3C4×D311yescharacteristic
92C4×C3⋊D3#6711yescharacteristic
93C4×C3⋊D3C22×D311yescharacteristic
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C3×C4×D3—61no—
97C3×C4×D3—61no—
98C3×C4×D3—61no—
99C3×C4×D3—61no—
100C3×C4×D3—61no—
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×C4×D3—61no—
104C32×C8—62no—
105C3⋊(C3⋊C8)—22no—
106C3×C3⋊C8—22no—
107C3×C3⋊C8—22no—
108C3×C3⋊C8—42no—
109C3×C3⋊C8—62no—
110C3×C3⋊C8—62no—
111C3×C3⋊C8—122no—
112C2×C4⋊C8—271noSylow
113C2×C33C4⋊C812yescomplemented
114C2×C33C2×C4⋊C411yescharacteristic
115C3×C3⋊D3C4⋊C812yescomplemented
116C3×C3⋊D3C22×C811yescharacteristic
117C3×C3⋊D3C2×M1611yescharacteristic
118C22×C3×C4—91no—
119C22×Dic3—91no—
120C22×Dic3—91no—
121C2×C4×D3—61no—
122C2×C4×D3—61no—
123C2×C3×C8—61no—
124C2×C3×C8—61no—
125C2×C3×C8—92no—
126C2×C3×C8—121no—
127C2×C3×C8—181no—
128C2×C3×C8—181no—
129C4×Dic3—92no—
130C4×Dic3—92no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—92no—
135C2×C3⋊C8—121no—
136C2×C3⋊C8—181no—
137D3×C8—62no—
138D3×C8—62no—
139D3×C8—182no—
140D3×C8—182no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32Dic3⋊C411yescharacteristic
146C2×C3⋊D3D3×C812yes—
147C2×C3⋊D3ΓL(1,𝔽25)12yes—
148C2×C3⋊D3Dic3⋊C411yescharacteristic
149C2×C3⋊D3C2×C4×D311yescharacteristic
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C2×C3×D3—61no—
157C2×C3×D3—61no—
158#157—34no—
159#157—62no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4—41no—
167C32×C4—41no—
168C32×C4Dic3⋊C411yescharacteristic
169C32×C4C2×#6711yescharacteristic
170C3⋊Dic3Dic3⋊C411yescharacteristic
171C3⋊Dic3C2×Dic611yescharacteristic
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C22×C8—91no—
177C22×C8—271no—
178C2×C42—271no—
179C4⋊C8—274no—
180C33C2×C4⋊C811yescharacteristic, complemented, Sylow
181C23×C3—91no—
182C22×D3—61no—
183C22×D3—61no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—91no—
189C2×C3×C4—91no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4GO+(4,𝔽2)11yescharacteristic
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—94no—
197C2×Dic3—92no—
198C2×Dic3—92no—
199C2×Dic3—182no—
200C4×D3—61no—
201C4×D3—61no—
202C4×D3—61no—
203C4×D3—61no—
204C4×D3—61no—
205C4×D3—61no—
206C4×D3—61no—
207C4×D3—61no—
208C3×C8—62no—
209C3×C8—62no—
210C3×C8—122no—
211C3×C8—182no—
212C3⋊C8—22no—
213C3⋊C8—62no—
214C3⋊C8—62no—
215C3⋊C8—122no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32Dic3⋊C812yescomplemented
225C2×C32C2×Dic3⋊C411yescharacteristic
226C3⋊D3Dic3⋊C812yescomplemented
227C3⋊D3C2×D3×C811yescharacteristic
228C3⋊D3C2×ΓL(1,𝔽25)11yescharacteristic
229C3×D3—62no—
230C3×D3—62no—
231C3×D3—61no—
232C3×D3—61no—
233C3×D3—61no—
234C3×D3—61no—
235C22×C4—91no—
236C22×C4—271no—
237C22×C4—271no—
238C2×C8—61no—
239C2×C8—92no—
240C2×C8—181no—
241C2×C8—181no—
242C2×C8—272no—
243C2×C8—541no—
244C42—272no—
245C42—272no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—91no—
251C22×C3—91no—
252C22×C3—92no—
253C22×C3—92no—
254C22×C3#377311yescharacteristic
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—91no—
272C3×C4—91no—
273C3×C4#377311yescharacteristic
274C3×C4C2×GO+(4,𝔽2)11yescharacteristic
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—61no—
279Dic3—94no—
280Dic3—182no—
281C32—21no—
282C32—21no—
283C32—41no—
284C32—41no—
285C32C2×Dic3⋊C811yescharacteristic, complemented
286C23—91no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—272no—
293C2×C4—274no—
294C2×C4—272no—
295C2×C4—542no—
296C2×C4D32⋊D311yescharacteristic
297C8—62no—
298C8—182no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—42no—
304C2×C3—42no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—91no—
308C2×C3—91no—
309C2×C3—92no—
310C2×C3#6373212yescomplemented
311C2×C3C2×#377311yescharacteristic
312D3—62no—
313D3—62no—
314D3—61no—
315D3—61no—
316D3—61no—
317D3—61no—
318C22—92no—
319C22—92no—
320C22—91no—
321C22—91no—
322C22#8893411yescharacteristic
323C4—91no—
324C4—91no—
325C4—274no—
326C4—542no—
327C4#8893411yescharacteristic
328C4C2×D32⋊D311yescharacteristic
329C3—21no—
330C3—21no—
331C3—41no—
332C3—41no—
333C3C2×#6373211yescharacteristic, complemented
334C2—91no—
335C2—91no—
336C2—92no—
337C2#1172276312yesdirect product factor
338C2C2×#8893411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊D3C22×C8
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypocenterC33C2×C4⋊C8
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722763C2direct (N × H)
25. C2×C3⋊#157C8semidirect
66. C3⋊#157C2×C8semidirect
113. C2×C33C4⋊C8semidirect
115. C3×C3⋊D3C4⋊C8semidirect
180. C33C2×C4⋊C8semidirect
224. C2×C32Dic3⋊C8semidirect
226. C3⋊D3Dic3⋊C8semidirect
285. C32C2×Dic3⋊C8semidirect
310. C2×C3#63732semidirect
333. C3C2×#63732semidirect
337. C2#11722763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
C2×#888082
C2×C3×#37102
#117227632
C2×#637323
C2×Dic3⋊C89