Finite Group Databasebeta

← C2×#11723109 #13173428

Subgroups — C2×#11723109 #13173428

Order 1728.

712 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 109 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×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
3C2×#88988C211yesmaximal, characteristic
4#11723109C214yesmaximal, direct product factor
5C2×#63784—31nomaximal
6C22×C3⋊#157—21no—
7C2×C33×Q8C411yescharacteristic, complemented
8C3×Q8×C3⋊D3—21no—
9C3×Q8×C3⋊D3—21no—
10C3×Q8×C3⋊D3C412yescomplemented
11C3×Q8×C3⋊D3C2212yes—
12C2×C3×C4×C3⋊D3—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634—21no—
15C2×C3×#634C411yescharacteristic, complemented
16C2×C3⋊#622—21no—
17C4⋊(C3⋊#157)—21no—
18C4⋊(C3⋊#157)C2212yes—
19#88988—21no—
20#88988C2212yes—
21C2×Q8×C3⋊D3D311yescharacteristic
22C2×C3×D3×Q8—62no—
23C2×C4⋊#157—31no—
24C2×#3787—31no—
25#63784—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3⋊#157—22no—
28C2×C3⋊#157—42no—
29C33×Q8—21no—
30C33×Q8C2×C412yescomplemented
31C2×C33×C4—21no—
32C2×C33×C4C2×C411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41C3×#634—21no—
42C3×#634—22no—
43C3×#634—22no—
44C3×#634C2×C412yescomplemented
45C3⋊#622—22no—
46C2×#835—91nomaximal
47C22×#157—61no—
48C2×C32×Q8—22no—
49C2×C32×Q8—42no—
50C2×C32×Q8Dic311yescharacteristic, complemented
51Q8×C3⋊D3—21no—
52Q8×C3⋊D3—21no—
53Q8×C3⋊D3Dic312yescomplemented
54Q8×C3⋊D3C2×D312yes—
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634—21no—
58C2×#634Dic311yescharacteristic, complemented
59C3×D3×Q8—64no—
60C3×D3×Q8—64no—
61C3×D3×Q8—64no—
62C3×D3×Q8—64no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—64no—
65C2×C3×Dic6—62no—
66C2×C3×Dic6—64no—
67C2×#622—61no—
68C4⋊#157—32no—
69C4⋊#157—61no—
70#3787—32no—
71#3787—61no—
72C22×C33C22⋊C411yescharacteristic
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3SD1612yes—
75C2×C3×C3⋊D3Q1612yes—
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3⋊#157—42no—
78C33×C4—22no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C3×C3⋊Dic3C2×D411yescharacteristic
84C22×C3×Q8—91no—
85C2×D3×Q8—62no—
86C2×C4⋊Dic3—91no—
87C22×C3⋊C8—91no—
88#835—94no—
89C22×C3⋊D3#6711yescharacteristic
90C22×C3×D3—62no—
91C2×#157—62no—
92C2×#157—122no—
93C32×Q8—21no—
94C32×Q8—24no—
95C32×Q8—24no—
96C32×Q8—44no—
97C32×Q8—44no—
98C32×Q8C2×Dic312yescomplemented
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—22no—
102C2×C32×C4—44no—
103C2×C32×C4—42no—
104C2×C32×C4C2×Dic311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3#6711yescharacteristic
107C4×C3⋊D3—22no—
108C4×C3⋊D3—22no—
109C4×C3⋊D3C2×Dic311yescharacteristic
110C4×C3⋊D3#6711yescharacteristic
111C4×C3⋊D3#6711yescharacteristic
112C4×C3⋊D3C22×D311yescharacteristic
113#634—21no—
114#634—22no—
115#634—22no—
116#634C2×Dic312yescomplemented
117C2×C3×Dic3—62no—
118C2×C3×Dic3—64no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—68no—
124C3×C4×D3—68no—
125C3×Dic6—64no—
126C3×Dic6—68no—
127C3×Dic6—68no—
128C3×Dic6—64no—
129#622—62no—
130C2×#103—271noSylow
131C2×C33#10312yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C3×C3⋊D3#10312yescomplemented
134C3×C3⋊D3C2×SD1611yescharacteristic
135C3×C3⋊D3C2×Q1611yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—42no—
138C2×C3×Q8—92no—
139C2×C3×Q8—92no—
140C2×C3×Q8—91no—
141C2×C3×Q8—181no—
142C2×C3×Q8—181no—
143C2×C3×Q8—181no—
144C2×C3×Q8#15711yescharacteristic, complemented
145C22×C3×C4—91no—
146C22×C3×C4—181no—
147C22×Dic3—181no—
148D3×Q8—64no—
149D3×Q8—64no—
150D3×Q8—64no—
151D3×Q8—64no—
152C2×C4×D3—62no—
153C2×C4×D3—64no—
154C2×Dic6—62no—
155C2×Dic6—64no—
156C4⋊Dic3—92no—
157C4⋊Dic3—181no—
158C2×C3⋊C8—92no—
159C2×C3⋊C8—181no—
160C2×C3⋊C8—181no—
161C22×C32—22no—
162C22×C32—42no—
163C22×C32#21711yescharacteristic
164C2×C3⋊D3#21512yes—
165C2×C3⋊D3#21612yes—
166C2×C3⋊D3#21711yescharacteristic
167C2×C3⋊D3C2×#6711yescharacteristic
168C2×C3×D3—62no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—62no—
172#157—122no—
173C32×C4—22no—
174C32×C4—28no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—48no—
178C32×C4—42no—
179C32×C4—42no—
180C32×C4#21711yescharacteristic
181C32×C4C22×Dic311yescharacteristic
182C3⋊Dic3—22no—
183C3⋊Dic3#21711yescharacteristic
184C3⋊Dic3C2×#6711yescharacteristic
185C3×Dic3—62no—
186C3×Dic3—62no—
187C3×Dic3—68no—
188C22×Q8—91no—
189C22×C8—271no—
190C2×C4⋊C4—271no—
191#103—274no—
192C33C2×#10311yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C22×D3—62no—
195C3×Q8—24no—
196C3×Q8—21no—
197C3×Q8—24no—
198C3×Q8—44no—
199C3×Q8—44no—
200C3×Q8—92no—
201C3×Q8—182no—
202C3×Q8—182no—
203C3×Q8—181no—
204C3×Q8C2×#15712yescomplemented
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—181no—
218C2×C3×C4C2×#15711yescharacteristic
219C2×Dic3—62no—
220C2×Dic3—64no—
221C2×Dic3—182no—
222C2×Dic3—362no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—68no—
228C4×D3—68no—
229Dic6—64no—
230Dic6—68no—
231Dic6—68no—
232Dic6—64no—
233C3⋊C8—182no—
234C2×C32—22no—
235C2×C32—24no—
236C2×C32—44no—
237C2×C32—42no—
238C2×C32#83512yescomplemented
239C2×C32C2×#21711yescharacteristic
240C3⋊D3#83512yescomplemented
241C3⋊D3C2×#21511yescharacteristic
242C3⋊D3C2×#21611yescharacteristic
243C3×D3—64no—
244C3×D3—62no—
245C3×D3—62no—
246C2×Q8—92no—
247C2×Q8—92no—
248C2×Q8—91no—
249C2×Q8—181no—
250C2×Q8—181no—
251C2×Q8—181no—
252C2×Q8C3⋊#15711yescharacteristic, complemented
253C22×C4—91no—
254C22×C4—181no—
255C22×C4—541no—
256C2×C8—272no—
257C2×C8—541no—
258C2×C8—541no—
259C4⋊C4—272no—
260C4⋊C4—541no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—91no—
264C22×C3—91no—
265C22×C3—92no—
266C22×C3—92no—
267C22×C3C22⋊#15711yescharacteristic
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—64no—
271C2×D3—64no—
272C3×C4—28no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—48no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—91no—
280C3×C4—91no—
281C3×C4—182no—
282C3×C4C22⋊#15711yescharacteristic
283C3×C4C22×#15711yescharacteristic
284Dic3—62no—
285Dic3—62no—
286Dic3—68no—
287Dic3—362no—
288C32—22no—
289C32—42no—
290C32C2×#83511yescharacteristic, complemented
291C23—91no—
292Q8—21no—
293Q8—92no—
294Q8—182no—
295Q8—182no—
296Q8—181no—
297Q8C2×C3⋊#15712yescomplemented
298C2×C4—21no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—181no—
307C2×C4—542no—
308C2×C4—1082no—
309C2×C4C2×C3⋊#15711yescharacteristic
310C8—542no—
311C2×C3—24no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—91no—
316C2×C3—91no—
317C2×C3—92no—
318C2×C3#6378412yescomplemented
319C2×C3C2×C22⋊#15711yescharacteristic
320D3—64no—
321D3—62no—
322D3—62no—
323C22—92no—
324C22—92no—
325C22—91no—
326C22—91no—
327C22#8899411yescharacteristic
328C4—22no—
329C4—91no—
330C4—91no—
331C4—182no—
332C4—1082no—
333C4#8899411yescharacteristic
334C4C22×C3⋊#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3C2×#6378411yescharacteristic, complemented
338C2—91no—
339C2—92no—
340C2—91no—
341C2#1172310912yesdirect product factor
342C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×Q8C4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11723109C2direct (N × H)
7. C2×C33×Q8C4semidirect
10. C3×Q8×C3⋊D3C4semidirect
15. C2×C3×#634C4semidirect
30. C33×Q8C2×C4semidirect
44. C3×#634C2×C4semidirect
50. C2×C32×Q8Dic3semidirect
53. Q8×C3⋊D3Dic3semidirect
58. C2×#634Dic3semidirect
98. C32×Q8C2×Dic3semidirect
116. #634C2×Dic3semidirect
131. C2×C33#103semidirect
133. C3×C3⋊D3#103semidirect
144. C2×C3×Q8#157semidirect
192. C33C2×#103semidirect
204. C3×Q8C2×#157semidirect
238. C2×C32#835semidirect
240. C3⋊D3#835semidirect
252. C2×Q8C3⋊#157semidirect
290. C32C2×#835semidirect
297. Q8C2×C3⋊#157semidirect
318. C2×C3#63784semidirect
337. C3C2×#63784semidirect
341. C2#11723109direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C4⋊(C3⋊#157)2
C2×#889882
#117231092
C2×#637843
C2×#8359