Finite Group Databasebeta

← #24988356

Subgroups — #24988356

Order 11008.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14583622C211yesmaximal, characteristic
2#19133277C212yesmaximal, complemented
3#19133019C212yesmaximal
4#19132919C211yesmaximal, characteristic, complemented
5C43×#1801C211yesmaximal, characteristic, complemented
6#14583622—24no—
7#14583622—24no—
8C2×#13063325C2211yescharacteristic
9C2×#13063320—22no—
10C2×#13063313C412yes—
11C2×#13063301—21no—
12C2×C4×Dic86C412yes—
13C2×C4×D172C2211yescharacteristic
14C2×#13063283—21no—
15#14582952C2212yes—
16#14582647—24no—
17#14582626—24no—
18#14582625—22no—
19C2×C4×Q8×C43C2211yescharacteristic
20C43×#423C2212yes—
21C2×#101×C43—21no—
22C43×#343—21no—
23C22×Dic86C2×C412yes—
24C22×D172C2×C411yescharacteristic
25C22×C4×D43—21no—
26#13063325—22no—
27#13063325D414yes—
28#13063320—216no—
29#13063313—24no—
30#13063313—24no—
31#13063301—28no—
32C4×Dic86—24no—
33C4×Dic86—24no—
34C4×D172—22no—
35C4×D172D414yes—
36#13063283—28no—
37C2×C4⋊Dic43—21no—
38C2×C4⋊Dic43—21no—
39C2×C4⋊Dic43C2×C411yescharacteristic
40C2×Dic43⋊C4—22no—
41C2×(C2×D43)⋊C4—22no—
42C2×(C2×D43)⋊C4—22no—
43C2×(C2×D43)⋊C4—21no—
44C2×C4×Dic43—22no—
45C4⋊(C43⋊C8)—22no—
46C22⋊(C43⋊C8)—22no—
47C4×C43⋊C8—22no—
48C22×Q8×C43D411yescharacteristic
49C2×M16×C43—22no—
50C4×Q8×C43—22no—
51C4×Q8×C43—22no—
52C2×C4⋊C4×C43D412yes—
53C2×C4⋊C4×C43D411yescharacteristic
54C2×C42×C43C2×C412yes—
55C2×C42×C43C2311yescharacteristic
56C4⋊C8×C43—22no—
57C4⋊C8×C43—22no—
58#101×C43—42no—
59C23×D43—21no—
60C2×Q8×C43—22no—
61C2×Q8×C43—22no—
62C2×Q8×C43—21no—
63C2×Q8×C43—21no—
64C22×C4×C43C22⋊C412yes—
65C22×C4×C43C22⋊C412yes—
66C22×C4×C43C22×C411yescharacteristic
67C22×C4×C43C2×D411yescharacteristic
68C22×C4×C43C2×D411yescharacteristic
69C22×Dic43—21no—
70C22×Dic43—22no—
71C2×D172—21no—
72C2×D172—42no—
73C2×D172C22⋊C412yes—
74C2×C4×D43—24no—
75C2×C4×D43—42no—
76C2×C4×D43—41no—
77C2×C4×D43—41no—
78C2×Dic86—22no—
79C2×Dic86—44no—
80C2×Dic86C22⋊C414yes—
81M16×C43—42no—
82M16×C43—42no—
83C2×C8×C43—42no—
84C4⋊C4×C43—22no—
85C4⋊C4×C43—24no—
86C42×C43—22no—
87C42×C43—22no—
88C42×C43C2×D412yes—
89C42×C43C2×D412yes—
90(C2×D43)⋊C4—28no—
91(C2×D43)⋊C4—28no—
92(C2×D43)⋊C4—24no—
93C4⋊Dic43—21no—
94C4⋊Dic43—24no—
95C4⋊Dic43—24no—
96C4⋊Dic43C22⋊C412yes—
97Dic43⋊C4—28no—
98C4×Dic43—28no—
99C2×C43⋊C8—42no—
100C23×C43C2×C22⋊C411yescharacteristic
101C22×D43—22no—
102C22×D43—42no—
103C22×D43—42no—
104C22×D43—41no—
105C22×D43—41no—
106Q8×C43—42no—
107Q8×C43—42no—
108C2×C4×C43—22no—
109C2×C4×C43—22no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—22no—
113C2×C4×C43—41no—
114C2×C4×C43—42no—
115C2×C4×C43C2×C22⋊C411yescharacteristic
116C2×C4×C43C2×C22⋊C411yescharacteristic
117C2×C4×C43#12012yes—
118C2×C4×C43#12012yes—
119C2×C4×C43#13512yes—
120C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C43#13711yescharacteristic
122C2×Dic43—22no—
123C2×Dic43—24no—
124C2×Dic43—44no—
125C2×Dic43—41no—
126C2×Dic43—41no—
127D172—24no—
128D172—42no—
129C4×D43—44no—
130C4×D43—44no—
131Dic86—44no—
132Dic86—44no—
133C8×C43—82no—
134C43⋊C8—82no—
135#8394—431nomaximal, Sylow
136C22×C43—21no—
137C22×C43—21no—
138C22×C43#37911yescharacteristic
139C22×C43#41611yescharacteristic
140C22×C43#41711yescharacteristic
141C2×D43—42no—
142C2×D43—42no—
143C2×D43—42no—
144C2×D43—82no—
145C2×D43—82no—
146C4×C43—22no—
147C4×C43—21no—
148C4×C43—42no—
149C4×C43—42no—
150C4×C43#41312yes—
151C4×C43#44912yes—
152C4×C43#45212yes—
153Dic43—44no—
154Dic43—42no—
155C2×#556—431no—
156#1801D4311yescharacteristic, complemented
157#1548—432no—
158#1486—432no—
159#1461—431no—
160C2×C43—21no—
161C2×C43—21no—
162C2×C43#178511yescharacteristic
163C2×C43#178811yescharacteristic
164C2×C43#179611yescharacteristic
165D43—82no—
166#556—864no—
167#556—864no—
168C2×C4⋊Q8—431no—
169C2×#126—862no—
170C2×#124—432no—
171C2×C22⋊Q8—861no—
172C2×C4×Q8—432no—
173C2×C4×Q8C2×D4311yescharacteristic
174C2×C4×D4—431no—
175C2×#117—861no—
176#431—432no—
177#423C2×D4312yes—
178C2×#101—21no—
179#343—21no—
180#333—864no—
181D4⋊C8—864no—
182#324—862no—
183C43#839411yescharacteristic, complemented, Sylow
184C22×Q8—432no—
185C22×Q8C22⋊D4311yescharacteristic
186C22×D4—431no—
187C23×C4—861no—
188C2×M16—22no—
189C4⋊Q8—434no—
190C4⋊Q8—862no—
191#126—8616no—
192#124—864no—
193#124—864no—
194C22⋊Q8—868no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—864no—
198C4×Q8—864no—
199C4×D4—434no—
200C4×D4—862no—
201#117—868no—
202C2×C4⋊C4—431no—
203C2×C4⋊C4—862no—
204C2×C4⋊C4—861no—
205C2×C4⋊C4—861no—
206C2×C4⋊C4D17212yes—
207C2×C4⋊C4C22⋊D4311yescharacteristic
208C2×C22⋊C4—862no—
209C2×C22⋊C4—862no—
210C2×C22⋊C4—861no—
211C2×C42—862no—
212C2×C42C4×D4312yes—
213C2×C42C22×D4311yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—862no—
217#101—42no—
218C22⋊C8—862no—
219C4×C8—862no—
220C24—861no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—434no—
226C2×Q8—862no—
227C2×Q8—1724no—
228C2×D4—432no—
229C2×D4—861no—
230C2×D4—1722no—
231C22×C4—861no—
232C22×C4—862no—
233C22×C4—864no—
234C22×C4—1722no—
235C22×C4—1721no—
236C22×C4—1721no—
237C22×C4(C2×D43)⋊C412yes—
238C22×C4(C2×D43)⋊C412yes—
239C22×C4C2×C4×D4311yescharacteristic
240C22×C4C2×D17211yescharacteristic
241C22×C4C2×C22⋊D4311yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—1722no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—432no—
249C4⋊C4—868no—
250C4⋊C4—861no—
251C4⋊C4—864no—
252C4⋊C4—864no—
253C22⋊C4—868no—
254C22⋊C4—868no—
255C22⋊C4—864no—
256C42—22no—
257C42—22no—
258C42—868no—
259C42D4×D4312yes—
260C42D4×D4312yes—
261C23—862no—
262C23—1722no—
263C23—1722no—
264C23—1721no—
265C23—1721no—
266C23C2×(C2×D43)⋊C411yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—1724no—
270Q8—1724no—
271D4—864no—
272D4—1722no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—862no—
281C2×C4—864no—
282C2×C4—1724no—
283C2×C4—1721no—
284C2×C4—1721no—
285C2×C4—1724no—
286C2×C4—1724no—
287C2×C4C22⋊C4×D4311yescharacteristic
288C2×C4C22⋊C4×D4311yescharacteristic
289C2×C4#1306329312yes—
290C2×C4#1306329512yes—
291C2×C4#1306334512yes—
292C2×C4#1306335111yescharacteristic
293C2×C4#1306335611yescharacteristic
294C8—82no—
295C8—3442no—
296C22—21no—
297C22—21no—
298C22—3442no—
299C22—3442no—
300C22—1722no—
301C22—1722no—
302C22—1722no—
303C22#1458275411yescharacteristic
304C22#1458289311yescharacteristic
305C22#1458289811yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—1724no—
311C4—1722no—
312C4#1458288212yes—
313C4#1458304212yes—
314C4#1458305912yes—
315C2—21no—
316C2—21no—
317C2—3442no—
318C2#1913452211yescharacteristic
319C2#1913453311yescharacteristic
320C2#1913457111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1801C2
socleC22×C43#379
prime radicalC22×D172C2×C4
cosocleC2×C42×C43C23
hypercenter#1801D43
hypocenterC43#8394
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #19133277C2semidirect
4. #19132919C2semidirect
5. C43×#1801C2semidirect
156. #1801D43semidirect
183. C43#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#145836222
#191332772
#191330192
#191329192
C43×#18012
#839443