Finite Group Databasebeta

← #11920207

Subgroups — #11920207

Order 1152.

614 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 74 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)⋊C2C211yesmaximal, characteristic
2C4⋊C4×C22⋊C9C211yesmaximal, characteristic, complemented
3(C2×(C22⋊C9)⋊C2)⋊C4C212yesmaximal
4C4⋊((C22⋊C9)⋊C4)C211yesmaximal, characteristic, complemented
5C4×(C22⋊C9)⋊C4C212yesmaximal, complemented
6#82161—31nomaximal
7C22×(C22⋊C9)⋊C2C2211yescharacteristic
8C2×C4×C22⋊C9C2212yes—
9C2×C4×C22⋊C9C2211yescharacteristic
10C2×(C22⋊C9)⋊C4C2212yes—
11C2×(C22⋊C9)⋊C4C2211yescharacteristic
12C4×(C22⋊C9)⋊C2C414yescomplemented
13#63454—41nomaximal
14C22×C3×C4⋊C4D311yescharacteristic
15C2×C4×#67—31no—
16C2×#892—31no—
17#5213—32no—
18#5212—31no—
19C4×#217—32no—
20#4990—32no—
21#4987—31no—
22#4978—32no—
23#4977—31no—
24Dic3×C4⋊C4—31no—
25C22×C22⋊C9C2311yescharacteristic
26C2×(C22⋊C9)⋊C2—21no—
27C2×(C22⋊C9)⋊C2—21no—
28C2×(C22⋊C9)⋊C2C2×C412yes—
29C2×C4×D9—41no—
30C4×C22⋊C9—22no—
31C4×C22⋊C9C2×C412yes—
32(C22⋊C9)⋊C4—22no—
33(C22⋊C9)⋊C4C2×C412yes—
34C9×C4⋊C4—41no—
35(C2×D9)⋊C4—42no—
36C4⋊Dic9—41no—
37C4×Dic9—42no—
38#2352—91noSylow
39C23×C3×C4C2×D312yes—
40C23×C3×C4C2×D311yescharacteristic
41C22×#67—31no—
42C22×C4×D3—31no—
43C2×C3×C4⋊C4—31no—
44C2×C3×C4⋊C4—31no—
45C2×C3×C4⋊C4—32no—
46C2×#217—31no—
47C2×#217—32no—
48C4×#67—34no—
49C4×#67—34no—
50C2×(C2×D3)⋊C4—31no—
51C2×(C2×D3)⋊C4—32no—
52C2×(C2×D3)⋊C4—32no—
53C2×C4⋊Dic3—31no—
54C2×C4⋊Dic3—31no—
55C2×Dic3⋊C4—31no—
56C2×C4×Dic3—31no—
57C2×C4×Dic3—32no—
58C2×C4×Dic3—32no—
59#892—61no—
60#892—62no—
61#892—61no—
62C3⋊#95—32no—
63C3⋊#95—32no—
64C3⋊#95—32no—
65C3⋊#95—32no—
66C3⋊#95—32no—
67C22×D9—41no—
68C2×C22⋊C9C22×C411yescharacteristic
69C2×C22⋊C9#4111yescharacteristic
70C2×C22⋊C9#4111yescharacteristic
71(C22⋊C9)⋊C2—22no—
72C2×C4×C9—41no—
73C2×C4×C9—42no—
74C2×Dic9—41no—
75C2×Dic9—42no—
76C4×D9—44no—
77C2×C4×D4—91no—
78C2×#117—91no—
79C22×C4⋊C4D911yescharacteristic, complemented
80#388—92no—
81#387—92no—
82#386—92no—
83#382—91no—
84#380—91no—
85#380—91no—
86C4×C4⋊C4—91no—
87C4×C22⋊C4—92no—
88C24×C3C22×D311yescharacteristic
89C23×D3—31no—
90C22×C3×C4—22no—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—32no—
94C22×C3×C4—32no—
95C22×C3×C4—32no—
96C22×C3×C4—62no—
97C22×C3×C4C4×D312yes—
98C2×#67—32no—
99C2×#67—32no—
100C2×#67—61no—
101C2×#67—61no—
102C2×#67—61no—
103C2×#67—61no—
104C22×Dic3—31no—
105C22×Dic3—31no—
106C22×Dic3—31no—
107C22×Dic3—32no—
108C22×Dic3—32no—
109C2×C4×D3—34no—
110C2×C4×D3—61no—
111C2×C4×D3—62no—
112C2×C4×D3—61no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4S411yescharacteristic
117#217—32no—
118#217—32no—
119#217—62no—
120#217—62no—
121(C2×D3)⋊C4—34no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—122no—
125C4⋊Dic3—61no—
126C4⋊Dic3—62no—
127C4⋊Dic3—61no—
128Dic3⋊C4—34no—
129C4×Dic3—34no—
130C4×Dic3—62no—
131C4×Dic3—62no—
132C4×Dic3—122no—
133C22×C9—41no—
134C2×D9—42no—
135C2×D9—81no—
136C2×D9—81no—
137C22⋊C9#11711yescharacteristic, complemented
138C4×C9—42no—
139C4×C9—82no—
140Dic9—42no—
141Dic9—82no—
142C22×D4—91no—
143C23×C4—91no—
144C23×C4C2×D912yes—
145C23×C4C2×D911yescharacteristic
146C4×D4—94no—
147C4×D4—94no—
148#117—181no—
149#117—182no—
150#117—181no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—91no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—91no—
157C2×C22⋊C4—91no—
158C2×C22⋊C4—91no—
159C2×C22⋊C4—92no—
160C2×C22⋊C4—92no—
161C2×C22⋊C4—92no—
162C2×C42—91no—
163C2×C42—92no—
164C2×C42—92no—
165#95—92no—
166#95—92no—
167#95—92no—
168#95—92no—
169#95—92no—
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—31no—
174C23×C3C2×C4×D311yescharacteristic
175C23×C3#23711yescharacteristic
176C23×C3#23911yescharacteristic
177C22×D3—32no—
178C22×D3—61no—
179C22×D3—61no—
180C22×D3—61no—
181C22×D3—61no—
182C22×D3—61no—
183C22×D3—61no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—61no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4C2×S412yes—
195C2×C3×C4C2×S411yescharacteristic
196#67—62no—
197#67—62no—
198#67—62no—
199#67—62no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—32no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C4×D3—64no—
216C4×D3—64no—
217C2×C9—41no—
218C2×C9—41no—
219C2×C9—41no—
220D9—82no—
221C24—91no—
222C24C22×D911yescharacteristic
223C2×D4—92no—
224C2×D4—92no—
225C2×D4—181no—
226C2×D4—181no—
227C2×D4—181no—
228C2×D4—181no—
229C22×C4—22no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—32no—
235C22×C4—62no—
236C22×C4—94no—
237C22×C4—91no—
238C22×C4—91no—
239C22×C4—91no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4—181no—
243C22×C4—182no—
244C22×C4—181no—
245C22×C4C4×D912yes—
246C4⋊C4—32no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—94no—
250C4⋊C4—181no—
251C4⋊C4—182no—
252C4⋊C4—181no—
253C4⋊C4(C22⋊C9)⋊C211yescharacteristic, complemented
254C22⋊C4—94no—
255C22⋊C4—92no—
256C22⋊C4—92no—
257C22⋊C4—182no—
258C22⋊C4—182no—
259C22⋊C4—182no—
260C22⋊C4—182no—
261C22⋊C4—362no—
262C42—94no—
263C42—182no—
264C42—182no—
265C42—362no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—31no—
275C22×C3—61no—
276C22×C3#89211yescharacteristic
277C22×C3C22×S411yescharacteristic
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—121no—
285C3×C4—22no—
286C3×C4—32no—
287C3×C4—62no—
288C3×C4C4×S412yes—
289Dic3—62no—
290Dic3—62no—
291Dic3—62no—
292Dic3—122no—
293Dic3—122no—
294C9—41noSylow
295C23—31no—
296C23—31no—
297C23—31no—
298C23—31no—
299C23—92no—
300C23—181no—
301C23—181no—
302C23—181no—
303C23—181no—
304C23—181no—
305C23—181no—
306C23C2×C4×D911yescharacteristic
307C23#369911yescharacteristic
308C23#370111yescharacteristic
309D4—182no—
310D4—182no—
311D4—182no—
312D4—182no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—92no—
326C2×C4—184no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—184no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—182no—
339C2×C4—182no—
340C2×C4C2×(C22⋊C9)⋊C212yes—
341C2×C4C2×(C22⋊C9)⋊C211yescharacteristic
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3C2×C4×S411yescharacteristic
347C2×C3#591411yescharacteristic
348C2×C3#591911yescharacteristic
349D3—122no—
350C22—31no—
351C22—31no—
352C22—31no—
353C22—31no—
354C22—31no—
355C22—31no—
356C22—31no—
357C22—31no—
358C22—31no—
359C22—61no—
360C22—182no—
361C22—182no—
362C22—182no—
363C22—361no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22#6345411yescharacteristic, complemented
368C22C22×(C22⋊C9)⋊C211yescharacteristic
369C4—22no—
370C4—32no—
371C4—62no—
372C4—182no—
373C4—182no—
374C4—182no—
375C4—362no—
376C4—362no—
377C4C4×(C22⋊C9)⋊C212yes—
378C3#8530711yescharacteristic
379C2—31no—
380C2—31no—
381C2—31no—
382C2—31no—
383C2—362no—
384C2C2×C4×(C22⋊C9)⋊C211yescharacteristic
385C2#1059308911yescharacteristic
386C2#1059309411yescharacteristic
subgroupG/H
centerC22C22×(C22⋊C9)⋊C2
commutatorC2×C22⋊C9C22×C4
FrattiniC22×C3C22×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×(C22⋊C9)⋊C2C22
cosocleC22×C22⋊C9C23
hypercenterC4⋊C4(C22⋊C9)⋊C2
hypocenterC22⋊C9#117
LCSG ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C22⋊C9C2semidirect
4. C4⋊((C22⋊C9)⋊C4)C2semidirect
5. C4×(C22⋊C9)⋊C4C2semidirect
12. C4×(C22⋊C9)⋊C2C4semidirect
79. C22×C4⋊C4D9semidirect
137. C22⋊C9#117semidirect
253. C4⋊C4(C22⋊C9)⋊C2semidirect
367. C22#63454semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C22⋊C9)⋊C22
C4⋊C4×C22⋊C92
(C2×(C22⋊C9)⋊C2)⋊C42
C4⋊((C22⋊C9)⋊C4)2
C4×(C22⋊C9)⋊C42
#821613
#634544