Finite Group Databasebeta

← #117×C22⋊C9 #11919671

Subgroups — #117×C22⋊C9 #11919671

Order 1152.

804 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×C4×C22⋊C9C211yesmaximal, characteristic
2C4⋊C4×C22⋊C9C212yesmaximal, complemented
3C22⋊C4×C22⋊C9C212yesmaximal
4C42×C22⋊C9C212yesmaximal, complemented
5C22×C3×#117C311yesmaximal, characteristic
6C23×C22⋊C9C2211yescharacteristic
7C2×C4×C22⋊C9C414yescomplemented
8C2×C4×C22⋊C9C2214yes—
9C2×C4×C22⋊C9C2211yescharacteristic
10C2×C4×C22⋊C9C2211yescharacteristic
11C9×#117—41nomaximal
12C24×C3×C4C2×C311yescharacteristic
13C2×C3×#117—31no—
14C2×C3×#117—32no—
15C2×C3×#117—32no—
16C2×C3×#117—32no—
17C2×C3×#117—31no—
18C22×C3×C4⋊C4C2×C312yes—
19C22×C3×C22⋊C4C2×C312yes—
20C22×C3×C42C2×C312yes—
21C22×C22⋊C9—22no—
22C22×C22⋊C9C2×C412yes—
23C22×C22⋊C9C2311yescharacteristic
24C22×C4×C9—41no—
25C4×C22⋊C9—24no—
26C4×C22⋊C9C2×C412yes—
27C4×C22⋊C9C2×C412yes—
28C9×C4⋊C4—42no—
29C9×C22⋊C4—42no—
30C42×C9—42no—
31C22×#117C911yescharacteristic, complemented, Sylow
32C25×C3C22×C311yescharacteristic
33C23×C3×C4—31no—
34C23×C3×C4—34no—
35C23×C3×C4—31no—
36C23×C3×C4—31no—
37C23×C3×C4—31no—
38C23×C3×C4C3×C414yes—
39C23×C3×C4C22×C314yes—
40C23×C3×C4C22×C311yescharacteristic
41C23×C3×C4C22×C311yescharacteristic
42C3×#117—32no—
43C3×#117—32no—
44C3×#117—38no—
45C3×#117—32no—
46C3×#117—31no—
47C3×#117—32no—
48C3×#117—32no—
49C3×#117—32no—
50C3×#117A411yescharacteristic
51C2×C3×C4⋊C4—32no—
52C2×C3×C4⋊C4—32no—
53C2×C3×C4⋊C4—34no—
54C2×C3×C22⋊C4—32no—
55C2×C3×C22⋊C4—32no—
56C2×C3×C22⋊C4—34no—
57C2×C3×C42—32no—
58C2×C3×C42—34no—
59C2×C3×C42—32no—
60C23×C9—41no—
61C2×C22⋊C9—22no—
62C2×C22⋊C9C22×C411yescharacteristic
63C2×C22⋊C9#4112yes—
64C2×C4×C9—44no—
65C2×C4×C9—41no—
66C2×C4×C9—44no—
67C2×C4×C9—41no—
68C24×C4C2×C911yescharacteristic
69C2×#117—31no—
70C2×#117—32no—
71C2×#117—32no—
72C2×#117—32no—
73C2×#117—31no—
74C22×C4⋊C4C2×C912yescomplemented
75C22×C22⋊C4C2×C912yes—
76C22×C42C2×C912yescomplemented
77C24×C3—22no—
78C24×C3—31no—
79C24×C3—32no—
80C24×C3—31no—
81C24×C3—62no—
82C24×C3C2×C3×C412yes—
83C24×C3C23×C311yescharacteristic
84C22×C3×C4—24no—
85C22×C3×C4—32no—
86C22×C3×C4—34no—
87C22×C3×C4—34no—
88C22×C3×C4—34no—
89C22×C3×C4—31no—
90C22×C3×C4—34no—
91C22×C3×C4—34no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—34no—
99C22×C3×C4—34no—
100C22×C3×C4—34no—
101C22×C3×C4—32no—
102C22×C3×C4—32no—
103C22×C3×C4—34no—
104C22×C3×C4—34no—
105C22×C3×C4—64no—
106C22×C3×C4C2×C3×C412yes—
107C22×C3×C4C2×C3×C412yes—
108C22×C3×C4C2×A411yescharacteristic
109C3×C4⋊C4—34no—
110C3×C4⋊C4—34no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4C2×A412yes—
113C3×C22⋊C4—34no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4—34no—
116C3×C22⋊C4C2×A412yes—
117C3×C42—34no—
118C3×C42—34no—
119C3×C42—32no—
120C3×C42C2×A412yes—
121C22×C9—42no—
122C22×C9—41no—
123C22×C9—82no—
124C22⋊C9#11711yescharacteristic, direct product factor
125C4×C9—42no—
126C4×C9—42no—
127C4×C9—84no—
128C25C22×C911yescharacteristic
129C23×C4—31no—
130C23×C4—31no—
131C23×C4—31no—
132C23×C4—31no—
133C23×C4—34no—
134C23×C4C4×C914yescomplemented
135C23×C4C22×C914yes—
136C23×C4C22×C911yescharacteristic
137C23×C4C22×C911yescharacteristic
138#117—32no—
139#117—32no—
140#117—38no—
141#117—32no—
142#117—31no—
143#117—32no—
144#117—32no—
145#117—32no—
146#117C22⋊C911yescharacteristic, direct product factor
147C2×C4⋊C4—32no—
148C2×C4⋊C4—32no—
149C2×C4⋊C4—34no—
150C2×C22⋊C4—32no—
151C2×C22⋊C4—32no—
152C2×C22⋊C4—34no—
153C2×C42—32no—
154C2×C42—32no—
155C2×C42—34no—
156C23×C3—22no—
157C23×C3—31no—
158C23×C3—32no—
159C23×C3—31no—
160C23×C3—32no—
161C23×C3—32no—
162C23×C3—32no—
163C23×C3—32no—
164C23×C3—31no—
165C23×C3—62no—
166C23×C3—62no—
167C23×C3—62no—
168C23×C3—64no—
169C23×C3—62no—
170C23×C3—62no—
171C23×C3—64no—
172C23×C3C22×C3×C411yescharacteristic
173C23×C3C3×#4112yes—
174C23×C3C22×A411yescharacteristic
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—38no—
178C2×C3×C4—34no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—32no—
182C2×C3×C4—34no—
183C2×C3×C4—32no—
184C2×C3×C4—34no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—34no—
190C2×C3×C4—64no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—64no—
194C2×C3×C4C4×A414yes—
195C2×C3×C4C22×A414yes—
196C2×C3×C4C22×A411yescharacteristic
197C2×C3×C4C22×A411yescharacteristic
198C2×C9—41no—
199C2×C9—42no—
200C2×C9—82no—
201C24—22no—
202C24—31no—
203C24—32no—
204C24—31no—
205C24—62no—
206C24C2×C4×C912yes—
207C24C23×C911yescharacteristic
208C22×C4—24no—
209C22×C4—32no—
210C22×C4—34no—
211C22×C4—34no—
212C22×C4—34no—
213C22×C4—31no—
214C22×C4—34no—
215C22×C4—34no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—34no—
223C22×C4—34no—
224C22×C4—34no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—34no—
228C22×C4—34no—
229C22×C4—64no—
230C22×C4C2×C4×C912yes—
231C22×C4C2×C4×C912yes—
232C22×C4C2×C22⋊C911yescharacteristic
233C4⋊C4—34no—
234C4⋊C4—34no—
235C4⋊C4—32no—
236C4⋊C4C2×C22⋊C912yescomplemented
237C22⋊C4—34no—
238C22⋊C4—32no—
239C22⋊C4—34no—
240C22⋊C4C2×C22⋊C912yes—
241C42—34no—
242C42—34no—
243C42—32no—
244C42C2×C22⋊C912yescomplemented
245C22×C3—22no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—62no—
255C22×C3—64no—
256C22×C3—62no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—64no—
260C22×C3—62no—
261C22×C3C3×#11711yescharacteristic
262C22×C3C2×C4×A412yes—
263C22×C3C23×A411yescharacteristic
264C3×C4—24no—
265C3×C4—32no—
266C3×C4—32no—
267C3×C4—64no—
268C3×C4C2×C4×A412yes—
269C3×C4C2×C4×A412yes—
270C9—41noSylow
271C23—22no—
272C23—31no—
273C23—32no—
274C23—31no—
275C23—32no—
276C23—32no—
277C23—31no—
278C23—32no—
279C23—32no—
280C23—62no—
281C23—62no—
282C23—62no—
283C23—64no—
284C23—62no—
285C23—62no—
286C23—64no—
287C23C22×C4×C911yescharacteristic
288C23C9×#4112yes—
289C23C22×C22⋊C911yescharacteristic
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—38no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—34no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—34no—
301C2×C4—34no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—34no—
305C2×C4—64no—
306C2×C4—64no—
307C2×C4—64no—
308C2×C4—64no—
309C2×C4C4×C22⋊C914yescomplemented
310C2×C4C22×C22⋊C914yes—
311C2×C4C22×C22⋊C911yescharacteristic
312C2×C4C22×C22⋊C911yescharacteristic
313C2×C3—22no—
314C2×C3—31no—
315C2×C3—32no—
316C2×C3—31no—
317C2×C3—62no—
318C2×C3C22×C4×A411yescharacteristic
319C2×C3A4×#4112yes—
320C22—22no—
321C22—31no—
322C22—32no—
323C22—32no—
324C22—31no—
325C22—31no—
326C22—32no—
327C22—32no—
328C22—32no—
329C22—62no—
330C22—62no—
331C22—64no—
332C22—62no—
333C22—62no—
334C22—64no—
335C22—62no—
336C22C9×#11711yescharacteristic, complemented
337C22C2×C4×C22⋊C912yes—
338C22C23×C22⋊C911yescharacteristic
339C4—24no—
340C4—32no—
341C4—32no—
342C4—64no—
343C4C2×C4×C22⋊C912yes—
344C4C2×C4×C22⋊C912yes—
345C3A4×#11711yescharacteristic
346C2—22no—
347C2—31no—
348C2—31no—
349C2—32no—
350C2—62no—
351C2C22×C4×C22⋊C911yescharacteristic
352C2#41×C22⋊C912yes—
subgroupG/H
centerC2×C3×C4C22×A4
commutatorC23C22×C4×C9
FrattiniC22×C3C23×A4
FittingC22×C3×#117C3
socle, cosocleC24×C3C23×C3
prime radicalC25×C3C22×C3
hypercenterC3×#117A4
hypocenterC22C9×#117
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C22⋊C9C2semidirect
4. C42×C22⋊C9C2semidirect
7. C2×C4×C22⋊C9C4semidirect
31. C22×#117C9semidirect
74. C22×C4⋊C4C2×C9semidirect
76. C22×C42C2×C9semidirect
124. C22⋊C9#117direct (N × H)
134. C23×C4C4×C9semidirect
146. #117C22⋊C9direct (N × H)
236. C4⋊C4C2×C22⋊C9semidirect
244. C42C2×C22⋊C9semidirect
309. C2×C4C4×C22⋊C9semidirect
336. C22C9×#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C92
C4⋊C4×C22⋊C92
C22⋊C4×C22⋊C92
C42×C22⋊C92
C22×C3×#1173
C9×#1174