Finite Group Databasebeta

← #24988376

Subgroups — #24988376

Order 11008.

782 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14583609C211yesmaximal, characteristic
2#19133281C212yesmaximal, complemented
3#19133019C212yesmaximal
4#19132919C211yesmaximal, characteristic, complemented
5C43×#1801C211yesmaximal, characteristic, complemented
6#14583609—24no—
7#14583609—24no—
8C2×#13063325C2211yescharacteristic
9C2×#13063317C412yes—
10C2×#13063303—22no—
11C2×#13063301—21no—
12C2×C4×D172C412yes—
13C2×C4×D172C2211yescharacteristic
14C2×C4⋊C4×D43—21no—
15#14582952C2212yes—
16#14582644—24no—
17#14582626—24no—
18#14582625—22no—
19C2×C4×Q8×C43C2211yescharacteristic
20C43×#423C2212yes—
21C2×#101×C43—21no—
22C43×#343—21no—
23C22×Dic86—21no—
24C22×D172C2×C411yescharacteristic
25C22×C4×D43—21no—
26C22×C4×D43—22no—
27#13063325—22no—
28#13063325D414yes—
29#13063317—24no—
30#13063317—24no—
31#13063303—216no—
32#13063301—28no—
33C4×D172—24no—
34C4×D172—22no—
35C4×D172—24no—
36C4×D172D414yes—
37C2×C4⋊Dic43—21no—
38C2×C4⋊Dic43C2×C412yes—
39C2×C4⋊Dic43C2×C411yescharacteristic
40C2×Dic43⋊C4—22no—
41C2×Dic43⋊C4—22no—
42C4⋊C4×D43—28no—
43C2×(C2×D43)⋊C4—22no—
44C2×(C2×D43)⋊C4—21no—
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—24no—
73C2×D172C22⋊C412yes—
74C2×C4×D43—24no—
75C2×C4×D43—28no—
76C2×C4×D43—42no—
77C2×C4×D43—41no—
78C2×C4×D43—41no—
79C2×C4×D43—84no—
80C2×Dic86—24no—
81C2×Dic86—82no—
82M16×C43—42no—
83M16×C43—42no—
84C2×C8×C43—42no—
85C4⋊C4×C43—22no—
86C4⋊C4×C43—24no—
87C42×C43—22no—
88C42×C43—22no—
89C42×C43C2×D412yes—
90C42×C43C2×D412yes—
91(C2×D43)⋊C4—28no—
92(C2×D43)⋊C4—24no—
93C4⋊Dic43—21no—
94C4⋊Dic43—22no—
95C4⋊Dic43—24no—
96C4⋊Dic43C22⋊C414yes—
97C4⋊Dic43C22⋊C412yes—
98Dic43⋊C4—28no—
99Dic43⋊C4—28no—
100C2×C43⋊C8—42no—
101C23×C43C2×C22⋊C411yescharacteristic
102C22×D43—24no—
103C22×D43—22no—
104C22×D43—42no—
105C22×D43—41no—
106C22×D43—41no—
107Q8×C43—42no—
108Q8×C43—42no—
109C2×C4×C43—22no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—41no—
115C2×C4×C43—42no—
116C2×C4×C43C2×C22⋊C411yescharacteristic
117C2×C4×C43C2×C22⋊C411yescharacteristic
118C2×C4×C43#12012yes—
119C2×C4×C43#12012yes—
120C2×C4×C43#13512yes—
121C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C43#13711yescharacteristic
123C2×Dic43—22no—
124C2×Dic43—24no—
125C2×Dic43—44no—
126C2×Dic43—41no—
127C2×Dic43—41no—
128D172—24no—
129D172—22no—
130D172#10114yescomplemented
131C4×D43—44no—
132C4×D43—44no—
133C4×D43—88no—
134Dic86—84no—
135C8×C43—82no—
136C43⋊C8—82no—
137#8398—431nomaximal, Sylow
138C22×C43—21no—
139C22×C43—21no—
140C22×C43#37911yescharacteristic
141C22×C43#41611yescharacteristic
142C22×C43#41711yescharacteristic
143C2×D43—24no—
144C2×D43—44no—
145C2×D43—44no—
146C2×D43—42no—
147C2×D43—42no—
148C4×C43—22no—
149C4×C43—21no—
150C4×C43—42no—
151C4×C43—42no—
152C4×C43C2×#10112yes—
153C4×C43#44912yes—
154C4×C43#45212yes—
155Dic43—44no—
156Dic43—42no—
157C2×#554—431no—
158#1801D4311yescharacteristic, complemented
159#1549—432no—
160#1486—432no—
161#1461—431no—
162C2×C43—21no—
163C2×C43—21no—
164C2×C43#178411yescharacteristic
165C2×C43#178811yescharacteristic
166C2×C43#179511yescharacteristic
167D43—44no—
168#554—864no—
169#554—864no—
170C2×C4⋊Q8—431no—
171C2×#125—432no—
172C2×C22⋊Q8—861no—
173C2×C22⋊Q8—862no—
174C2×C4×Q8C2×D4311yescharacteristic
175C2×C4×D4—431no—
176C2×C4×D4—432no—
177C22×C4⋊C4—861no—
178#431—432no—
179#423C2×D4312yes—
180C2×#101—21no—
181#343—21no—
182#332—864no—
183D4⋊C8—864no—
184#324—862no—
185C43#839811yescharacteristic, complemented, Sylow
186C22×Q8—861no—
187C22×Q8C22⋊D4311yescharacteristic
188C22×D4—431no—
189C23×C4—862no—
190C23×C4—861no—
191C2×M16—22no—
192C4⋊Q8—434no—
193C4⋊Q8—862no—
194#125—864no—
195#125—864no—
196C22⋊Q8—8616no—
197C22⋊Q8—868no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—434no—
201C4×D4—864no—
202C4×D4—864no—
203C4×D4—862no—
204C2×C4⋊C4—431no—
205C2×C4⋊C4—432no—
206C2×C4⋊C4—868no—
207C2×C4⋊C4—862no—
208C2×C4⋊C4—862no—
209C2×C4⋊C4—861no—
210C2×C4⋊C4D17212yes—
211C2×C4⋊C4C22⋊D4311yescharacteristic
212C2×C22⋊C4—862no—
213C2×C22⋊C4—861no—
214C2×C42C4×D4312yes—
215C2×C42C22×D4311yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—862no—
219#101—42no—
220C22⋊C8—862no—
221C4×C8—862no—
222C24—861no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—864no—
228C2×Q8—3442no—
229C2×D4—432no—
230C2×D4—864no—
231C2×D4—861no—
232C22×C4—868no—
233C22×C4—861no—
234C22×C4—862no—
235C22×C4—864no—
236C22×C4—3444no—
237C22×C4—1722no—
238C22×C4—1721no—
239C22×C4—1721no—
240C22×C4(C2×D43)⋊C412yes—
241C22×C4(C2×D43)⋊C412yes—
242C22×C4C2×C4×D4311yescharacteristic
243C22×C4C2×D17211yescharacteristic
244C22×C4C2×C22⋊D4311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—1722no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—432no—
252C4⋊C4—434no—
253C4⋊C4—868no—
254C4⋊C4—868no—
255C4⋊C4—861no—
256C4⋊C4—862no—
257C4⋊C4—864no—
258C22⋊C4—868no—
259C22⋊C4—864no—
260C42—22no—
261C42—22no—
262C42D4×D4312yes—
263C42D4×D4312yes—
264C23—864no—
265C23—862no—
266C23—1722no—
267C23—1721no—
268C23—1721no—
269C23C2×(C2×D43)⋊C411yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—3444no—
273D4—434no—
274D4—864no—
275D4—862no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—862no—
284C2×C4—864no—
285C2×C4—3448no—
286C2×C4—1724no—
287C2×C4—1721no—
288C2×C4—1721no—
289C2×C4—1724no—
290C2×C4—1724no—
291C2×C4C22⋊C4×D4311yescharacteristic
292C2×C4C22⋊C4×D4311yescharacteristic
293C2×C4#1306329312yes—
294C2×C4#1306329512yes—
295C2×C4#1306334512yes—
296C2×C4#1306335111yescharacteristic
297C2×C4#1306335611yescharacteristic
298C8—82no—
299C8—3442no—
300C22—21no—
301C22—21no—
302C22—864no—
303C22—1724no—
304C22—1724no—
305C22—1722no—
306C22—1722no—
307C22#1458275411yescharacteristic
308C22#1458289311yescharacteristic
309C22#1458289811yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—1722no—
315C4—1724no—
316C4#101×D4312yes—
317C4#1458304212yes—
318C4#1458305912yes—
319C2—21no—
320C2—21no—
321C2—1724no—
322C2#1913451711yescharacteristic
323C2#1913453311yescharacteristic
324C2#1913456611yescharacteristic
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#8398
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. #19133281C2semidirect
4. #19132919C2semidirect
5. C43×#1801C2semidirect
130. D172#101semidirect
158. #1801D43semidirect
185. C43#8398semidirect

Maximal subgroups

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

subgroupindex
C2×#145836092
#191332812
#191330192
#191329192
C43×#18012
#839843