Finite Group Databasebeta

← #11920008

Subgroups — #11920008

Order 1152.

384 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 50 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⋊C9)⋊C2)C211yesmaximal, characteristic
2C4⋊C4×C22⋊C9C211yesmaximal, characteristic, complemented
3C2×(C22⋊C9)⋊C8C211yesmaximal, characteristic, complemented
4#71560—31nomaximal
5C22×(C22⋊C9)⋊C2—21no—
6C2×C4×C22⋊C9—21no—
7C2×C4×C22⋊C9C2211yescharacteristic
8C4⋊((C22⋊C9)⋊C2)—21no—
9C4⋊((C22⋊C9)⋊C2)C412yescomplemented
10(C22⋊C9)⋊C8—21no—
11#63369—41nomaximal
12C22×C3×C4⋊C4D311yescharacteristic
13C2×#930—31no—
14C2×#830—31no—
15C2×#809—31no—
16C2×#809—31no—
17#4529—31no—
18#4529—31no—
19C22×C22⋊C9D411yescharacteristic
20C2×(C22⋊C9)⋊C2—22no—
21C2×(C22⋊C9)⋊C2—41no—
22C2×(C22⋊C9)⋊C2—41no—
23C2×D36—41no—
24C4×C22⋊C9—41no—
25C4×C22⋊C9C2×C411yescharacteristic
26C4×C22⋊C9D411yescharacteristic
27C9×C4⋊C4—41no—
28C2×C9⋊C8—41no—
29#1874—91noSylow
30C23×C3×C4—21no—
31C23×C3×C4C2×D311yescharacteristic
32C22×#67—61no—
33C22×D12—31no—
34C2×C3×C4⋊C4—31no—
35C2×C3×C4⋊C4—31no—
36C2×C3×C4⋊C4—61no—
37#930—32no—
38#930—32no—
39#930—61no—
40#930—61no—
41C2×(C2×D3)⋊C4—61no—
42C2×C4⋊Dic3—31no—
43C22×C3⋊C8—31no—
44C22×C3⋊C8—31no—
45#830—61no—
46#830—61no—
47#809—62no—
48#809—61no—
49#809—61no—
50C22×D9—81no—
51C2×C22⋊C9C22⋊C411yescharacteristic
52C2×C22⋊C9D811yescharacteristic
53C2×C22⋊C9SD1611yescharacteristic
54(C22⋊C9)⋊C2—42no—
55C2×C4×C9—41no—
56C2×C4×C9—81no—
57D36—42no—
58D36—81no—
59C9⋊C8—81no—
60C2×#121—91no—
61C22×C4⋊C4D911yescharacteristic, complemented
62C2×#102—91no—
63C2×#102—91no—
64C2×C22⋊C8—91no—
65#340—91no—
66#340—91no—
67C24×C3#6711yescharacteristic
68C23×D3—61no—
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4—31no—
73C22×C3×C4—41no—
74C22×C3×C4—61no—
75C22×C3×C4—61no—
76C22×C3×C4—121no—
77C22×C3×C4C4×D311yescharacteristic
78C22×C3×C4D1211yescharacteristic
79C2×#67—62no—
80C2×#67—62no—
81C2×#67—121no—
82C2×#67—121no—
83C2×#67—121no—
84C2×#67—121no—
85C22×Dic3—61no—
86C2×D12—32no—
87C2×D12—61no—
88C2×D12—61no—
89C2×D12—61no—
90C2×D12—61no—
91C2×D12—61no—
92C3×C4⋊C4—32no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4—61no—
95C3×C4⋊C4S411yescharacteristic
96(C2×D3)⋊C4—62no—
97(C2×D3)⋊C4—62no—
98C4⋊Dic3—32no—
99C4⋊Dic3—61no—
100C2×C3⋊C8—61no—
101C2×C3⋊C8—61no—
102C2×C3⋊C8—61no—
103C2×C3⋊C8—61no—
104C2×C3⋊C8—61no—
105C2×C3⋊C8—61no—
106C22×C9—41no—
107C2×D9—82no—
108C2×D9—161no—
109C2×D9—161no—
110C22⋊C9#10211yescharacteristic, complemented
111C4×C9—41no—
112C4×C9—41no—
113C4×C9—161no—
114C22×D4—91no—
115C22×D4—181no—
116C23×C4—21no—
117C23×C4C2×D911yescharacteristic
118C22×C8—91no—
119C22×C8—91no—
120#121—92no—
121#121—92no—
122#121—181no—
123#121—181no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—91no—
128C2×C22⋊C4—181no—
129#102—182no—
130#102—181no—
131#102—181no—
132C22⋊C8—181no—
133C22⋊C8—181no—
134C23×C3—31no—
135C23×C3—31no—
136C23×C3—31no—
137C23×C3—31no—
138C23×C3(C2×D3)⋊C411yescharacteristic
139C23×C3#21311yescharacteristic
140C23×C3#21511yescharacteristic
141C22×D3—62no—
142C22×D3—121no—
143C22×D3—121no—
144C22×D3—121no—
145C22×D3—121no—
146C22×D3—121no—
147C22×D3—121no—
148C2×C3×C4—21no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×C3×C4—121no—
159C2×C3×C4—121no—
160C2×C3×C4—121no—
161C2×C3×C4—121no—
162C2×C3×C4C2×S411yescharacteristic
163#67—122no—
164#67—122no—
165#67—122no—
166#67—122no—
167C2×Dic3—62no—
168C2×Dic3—121no—
169C2×Dic3—121no—
170D12—62no—
171D12—62no—
172D12—121no—
173D12—121no—
174C3⋊C8—121no—
175C3⋊C8—121no—
176C2×C9—41no—
177C2×C9—41no—
178C2×C9—41no—
179D9—162no—
180C24—181no—
181C24#61111yescharacteristic
182C2×D4—92no—
183C2×D4—181no—
184C2×D4—181no—
185C2×D4—181no—
186C2×D4—181no—
187C2×D4—182no—
188C2×D4—182no—
189C2×D4—181no—
190C2×D4—361no—
191C2×D4—361no—
192C2×D4—361no—
193C2×D4—361no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—41no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—121no—
202C22×C4—181no—
203C22×C4C4×D911yescharacteristic
204C22×C4D3611yescharacteristic
205C2×C8—181no—
206C2×C8—181no—
207C2×C8—181no—
208C2×C8—181no—
209C2×C8—181no—
210C2×C8—181no—
211C4⋊C4—32no—
212C4⋊C4—31no—
213C4⋊C4—61no—
214C4⋊C4—92no—
215C4⋊C4—181no—
216C4⋊C4(C22⋊C9)⋊C211yescharacteristic, complemented
217C22⋊C4—182no—
218C22⋊C4—182no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—61no—
229C22×C3#80911yescharacteristic
230C22×C3GL(2,ℤ/4)11yescharacteristic
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—122no—
234C2×D3—241no—
235C2×D3—241no—
236C2×D3—241no—
237C2×D3—241no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—41no—
241C3×C4—121no—
242C3×C4C4×S411yescharacteristic
243C3×C4C4⋊S411yescharacteristic
244Dic3—122no—
245C9—41noSylow
246C23—31no—
247C23—31no—
248C23—31no—
249C23—31no—
250C23—182no—
251C23—361no—
252C23—361no—
253C23—361no—
254C23—361no—
255C23—361no—
256C23—361no—
257C23(C2×D9)⋊C411yescharacteristic
258C23#367311yescharacteristic
259C23#367511yescharacteristic
260D4—182no—
261D4—182no—
262D4—361no—
263D4—362no—
264D4—362no—
265D4—362no—
266D4—362no—
267D4—361no—
268C2×C4—21no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—182no—
283C2×C4—361no—
284C2×C4—361no—
285C2×C4C2×(C22⋊C9)⋊C211yescharacteristic
286C8—361no—
287C8—361no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3(C2×S4)⋊C411yescharacteristic
293C2×C3#541511yescharacteristic
294C2×C3#541711yescharacteristic
295D3—242no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—61no—
306C22—362no—
307C22—362no—
308C22—362no—
309C22—721no—
310C22—721no—
311C22—721no—
312C22—721no—
313C22#6336911yescharacteristic, complemented
314C22#6369411yescharacteristic
315C4—31no—
316C4—31no—
317C4—41no—
318C4—121no—
319C4—362no—
320C4C4×(C22⋊C9)⋊C211yescharacteristic
321C4C4⋊((C22⋊C9)⋊C2)11yescharacteristic
322C3#7305311yescharacteristic
323C2—31no—
324C2—31no—
325C2—31no—
326C2—31no—
327C2—722no—
328C2(C2×(C22⋊C9)⋊C2)⋊C411yescharacteristic
329C2#1058951711yescharacteristic
330C2#1058951911yescharacteristic
subgroupG/H
centerC22#63694
commutatorC4×C22⋊C9C2×C4
FrattiniC2×C3×C4C2×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3#67
prime radicalC2×C4⋊((C22⋊C9)⋊C2)C2
cosocleC2×C4×C22⋊C9C22
hypercenterC4⋊C4(C22⋊C9)⋊C2
hypocenterC22⋊C9#102
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C22⋊C9C2semidirect
3. C2×(C22⋊C9)⋊C8C2semidirect
9. C4⋊((C22⋊C9)⋊C2)C4semidirect
61. C22×C4⋊C4D9semidirect
110. C22⋊C9#102semidirect
216. C4⋊C4(C22⋊C9)⋊C2semidirect
313. C22#63369semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C22⋊C9)⋊C2)2
C4⋊C4×C22⋊C92
C2×(C22⋊C9)⋊C82
#715603
#633694