Finite Group Databasebeta

← #24988734

Subgroups — #24988734

Order 11008.

1038 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 147 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
1#19134519C211yesmaximal, characteristic, complemented
2#19133099C214yesmaximal, complemented
3#19132866C211yesmaximal, characteristic, complemented
4C43×#3486C211yesmaximal, characteristic
5C2×#13063244—22no—
6#14582863C2212yes—
7C2×C43⋊#95—22no—
8C2×C43⋊#95C2212yes—
9#14582636—24no—
10#14582633—24no—
11C2×2−1+4×C43C411yescharacteristic, complemented
12C23×Q8×C43C2211yescharacteristic
13D4×Q8×C43—24no—
14C43×#548—24no—
15C43×#536—22no—
16C43×#536C412yescomplemented
17C43×#525—22no—
18C43×#525C2212yes—
19C43×#519C411yescharacteristic, complemented
20C23×Dic43—22no—
21C2×#10621877—22no—
22#13063244—44no—
23C22⋊(C43⋊C8)—24no—
24C43⋊#95—24no—
25C43⋊#95—44no—
262−1+4×C43—44no—
27C2×#41×C43—22no—
28C2×#41×C43—22no—
29C2×#41×C43C2×C412yes—
30C22×Q8×C43—22no—
31C22×Q8×C43—24no—
32C22×Q8×C43—44no—
33C22×Q8×C43C2×C411yescharacteristic
34C22×Q8×C43C2×C411yescharacteristic
35C22×Q8×C43D412yes—
36C23×C4×C43D412yes—
37C23×C4×C43C2311yescharacteristic
38C4⋊Q8×C43—24no—
39C4⋊Q8×C43—22no—
40#124×C43—24no—
41#124×C43—22no—
42#123×C43—24no—
43#123×C43—22no—
44C22⋊Q8×C43—24no—
45C22⋊Q8×C43—28no—
46C22⋊Q8×C43—24no—
47C22⋊Q8×C43D414yes—
48#121×C43—24no—
49#121×C43D414yes—
50#120×C43—22no—
51C4×Q8×C43—24no—
52C4×D4×C43—28no—
53C4×D4×C43—24no—
54#117×C43—22no—
55#117×C43C2×C412yes—
56C24×C43C2×D411yescharacteristic
57#41×C43—44no—
58#41×C43—44no—
59#41×C43—48no—
60#41×C43—44no—
61C2×Q8×C43—22no—
62C2×Q8×C43—24no—
63C2×Q8×C43—24no—
64C2×Q8×C43—22no—
65C2×Q8×C43—22no—
66C2×Q8×C43—24no—
67C2×Q8×C43—42no—
68C2×Q8×C43—42no—
69C2×Q8×C43—416no—
70C2×Q8×C43—48no—
71C2×Q8×C43C22⋊C412yes—
72C2×D4×C43—22no—
73C2×D4×C43—22no—
74C2×D4×C43—24no—
75C2×D4×C43C22⋊C412yes—
76C22×C4×C43—24no—
77C22×C4×C43—24no—
78C22×C4×C43—22no—
79C22×C4×C43—24no—
80C22×C4×C43—42no—
81C22×C4×C43—42no—
82C22×C4×C43C22⋊C412yes—
83C22×C4×C43C22⋊C412yes—
84C22×C4×C43C22×C411yescharacteristic
85C22×C4×C43C2×D411yescharacteristic
86C22×C4×C43C2×D414yes—
87C22×Dic43—24no—
88C22×Dic43—44no—
89C22×Dic43—42no—
90C22×Dic43—44no—
91C22×Dic43—42no—
92C4⋊C4×C43—24no—
93C4⋊C4×C43—28no—
94C4⋊C4×C43—24no—
95C4⋊C4×C43C22⋊C414yes—
96C22⋊C4×C43—28no—
97C22⋊C4×C43—24no—
98C22⋊C4×C43—22no—
99C22⋊C4×C43—24no—
100C42×C43—24no—
101C42×C43—22no—
102#10621877—44no—
103C2×C43⋊C8—42no—
104C23×C43—22no—
105C23×C43—22no—
106C23×C43—22no—
107C23×C43—22no—
108C23×C43C2×C22⋊C411yescharacteristic
109C23×C43#12012yes—
110Q8×C43—28no—
111Q8×C43—48no—
112Q8×C43—42no—
113Q8×C43—44no—
114Q8×C43—44no—
115D4×C43—42no—
116D4×C43—44no—
117D4×C43—44no—
118D4×C43—44no—
119D4×C43—44no—
120C2×C4×C43—22no—
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—22no—
124C2×C4×C43—24no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—22no—
129C2×C4×C43—44no—
130C2×C4×C43—48no—
131C2×C4×C43—48no—
132C2×C4×C43—44no—
133C2×C4×C43#9914yes—
134C2×C4×C43C2×C22⋊C412yes—
135C2×C4×C43#12012yes—
136C2×Dic43—48no—
137C2×Dic43—44no—
138C2×Dic43—84no—
139C2×Dic43—82no—
140C2×Dic43—82no—
141C43⋊C8—82no—
142#8455—431nomaximal, Sylow
143C22×C43—22no—
144C22×C43—24no—
145C22×C43—22no—
146C22×C43—22no—
147C22×C43—22no—
148C22×C43—22no—
149C22×C43—44no—
150C22×C43—42no—
151C22×C43#37911yescharacteristic
152C22×C43C2×#9912yes—
153C22×C43#45412yes—
154C22×C43#45612yes—
155C4×C43—22no—
156C4×C43—24no—
157C4×C43—22no—
158C4×C43—44no—
159C4×C43—42no—
160C4×C43—42no—
161Dic43—84no—
162#3486D4311yescharacteristic
163#1784—431no—
164#1504—434no—
165#1446—431no—
166C2×C43—22no—
167C2×C43—22no—
168C2×C43—22no—
169C2×C43—82no—
170C2×C43#178111yescharacteristic
171C2×C43#179312yes—
172C2×2−1+4Dic4311yescharacteristic, complemented
173C23×Q8C2×D4311yescharacteristic
174D4×Q8—24no—
175#548—24no—
176#536—22no—
177#536Dic4312yescomplemented
178#525—22no—
179#525C2×D4312yes—
180#519Dic4311yescharacteristic, complemented
181C2×#101—862no—
182#407—432no—
183C2×#95—432no—
184C2×#95—862no—
185#328—864no—
186#327—864no—
187C43#845511yescharacteristic, complemented, Sylow
1882−1+4—44no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41C2×Dic4312yes—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8C2×Dic4311yescharacteristic
196C22×Q8C2×Dic4311yescharacteristic
197C22×Q8C22⋊D4312yes—
198C23×C4—862no—
199C23×C4C22⋊D4312yes—
200C23×C4C22×D4311yescharacteristic
201C2×M16—862no—
202C4⋊Q8—24no—
203C4⋊Q8—22no—
204#124—24no—
205#124—22no—
206#123—24no—
207#123—22no—
208C22⋊Q8—24no—
209C22⋊Q8—28no—
210C22⋊Q8—24no—
211C22⋊Q8C22⋊D4314yes—
212#121—24no—
213#121C22⋊D4314yes—
214#120—22no—
215C4×Q8—24no—
216C4×D4—28no—
217C4×D4—24no—
218#117—22no—
219#117C2×Dic4312yes—
220#101—1724no—
221C22⋊C8—864no—
222#95—864no—
223#95—1724no—
224C24C2×C22⋊D4311yescharacteristic
225#41—44no—
226#41—44no—
227#41—48no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—416no—
238C2×Q8—48no—
239C2×Q8C22⋊Dic4312yes—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4C22⋊Dic4312yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—864no—
251C22×C4—1724no—
252C22×C4—1722no—
253C22×C4—1724no—
254C22×C4—1722no—
255C22×C4C22⋊Dic4312yes—
256C22×C4C22⋊Dic4312yes—
257C22×C4C22×Dic4311yescharacteristic
258C22×C4C2×C22⋊D4311yescharacteristic
259C22×C4C2×C22⋊D4314yes—
260M16—1724no—
261C2×C8—1722no—
262C4⋊C4—24no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4C22⋊Dic4314yes—
266C22⋊C4—28no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C42—24no—
271C42—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23C2×C22⋊Dic4311yescharacteristic
277C23#1306329412yes—
278Q8—28no—
279Q8—48no—
280Q8—42no—
281Q8—44no—
282Q8—44no—
283D4—42no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—44no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—3444no—
302C2×C4—3442no—
303C2×C4—3442no—
304C2×C4—1728no—
305C2×C4—1724no—
306C2×C4#1306324014yes—
307C2×C4C2×C22⋊Dic4312yes—
308C2×C4#1306329412yes—
309C8—3442no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—44no—
318C22#1458275511yescharacteristic
319C22C2×#1306324012yes—
320C22#1458307212yes—
321C22#1458308212yes—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—44no—
328C4—3444no—
329C2—22no—
330C2—22no—
331C2—22no—
332C2—82no—
333C2#1913451011yescharacteristic
334C2#1913455412yes—
subgroupG/H
centerC22#14582755
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
Fitting, prime radicalC43×#3486C2
socleC22×C43#379
cosocleC23×C4×C43C23
hypercenter#3486D43
hypocenterC43#8455
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #19134519C2semidirect
2. #19133099C2semidirect
3. #19132866C2semidirect
11. C2×2−1+4×C43C4semidirect
16. C43×#536C4semidirect
19. C43×#519C4semidirect
172. C2×2−1+4Dic43semidirect
177. #536Dic43semidirect
180. #519Dic43semidirect
187. C43#8455semidirect

Maximal subgroups

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

subgroupindex
#191345192
#191330992
#191328662
C43×#34862
#845543