Finite Group Databasebeta

← #24988326

Subgroups — #24988326

Order 11008.

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

normal subgroups only

This group has 966 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
1C2×Q8×D172C211yesmaximal, characteristic
2#19133268C212yesmaximal, complemented
3#19133019C212yesmaximal
4#19132919C211yesmaximal, characteristic, complemented
5C43×#1801C211yesmaximal, characteristic, complemented
6C22×Q8×D43—21no—
7Q8×D172—24no—
8Q8×D172—24no—
9C2×#13063325C412yes—
10C2×#13063325C2211yescharacteristic
11C2×#13063305—22no—
12C2×#13063305—21no—
13C2×C4×D172C412yes—
14C2×C4×D172C2211yescharacteristic
15#14582952C2212yes—
16#14582646—24no—
17#14582626—24no—
18#14582625—22no—
19C2×C4×Q8×C43C2211yescharacteristic
20C43×#423C2212yes—
21C2×#101×C43—21no—
22C43×#343—21no—
23C22×Dic86—22no—
24C22×Dic86—21no—
25C2×Q8×D43—28no—
26C2×Q8×D43—44no—
27C2×Q8×D43—42no—
28C2×Q8×D43—42no—
29C22×D172C2×C411yescharacteristic
30C22×C4×D43—22no—
31C22×C4×D43—21no—
32#13063325—22no—
33#13063325—24no—
34#13063325—24no—
35#13063325D414yes—
36#13063305—28no—
37#13063305—216no—
38C4×D172—24no—
39C4×D172—22no—
40C4×D172—24no—
41C4×D172D414yes—
42C2×C4⋊Dic43—22no—
43C2×C4⋊Dic43—21no—
44C2×C4⋊Dic43C2×C412yes—
45C2×C4⋊Dic43C2×C411yescharacteristic
46C2×(C2×D43)⋊C4—22no—
47C2×(C2×D43)⋊C4—21no—
48C4⋊(C43⋊C8)—22no—
49C22⋊(C43⋊C8)—22no—
50C4×C43⋊C8—22no—
51C22×Q8×C43D411yescharacteristic
52C2×M16×C43—22no—
53C4×Q8×C43—22no—
54C4×Q8×C43—22no—
55C2×C4⋊C4×C43D412yes—
56C2×C4⋊C4×C43D411yescharacteristic
57C2×C42×C43C2×C412yes—
58C2×C42×C43C2311yescharacteristic
59C4⋊C8×C43—22no—
60C4⋊C8×C43—22no—
61#101×C43—42no—
62C23×D43—21no—
63C2×Q8×C43—22no—
64C2×Q8×C43—22no—
65C2×Q8×C43D812yes—
66C2×Q8×C43SD1612yes—
67C22×C4×C43C22⋊C412yes—
68C22×C4×C43C22⋊C412yes—
69C22×C4×C43C22×C411yescharacteristic
70C22×C4×C43C2×D411yescharacteristic
71C22×C4×C43C2×D411yescharacteristic
72C22×Dic43—22no—
73C22×Dic43—21no—
74Q8×D43—416no—
75Q8×D43—416no—
76C2×D172—21no—
77C2×D172—24no—
78C2×D172C22⋊C412yes—
79C2×C4×D43—28no—
80C2×C4×D43—24no—
81C2×C4×D43—44no—
82C2×C4×D43—44no—
83C2×C4×D43—41no—
84C2×C4×D43—41no—
85C2×C4×D43—42no—
86C2×Dic86—28no—
87C2×Dic86—24no—
88C2×Dic86—44no—
89C2×Dic86—44no—
90C2×Dic86—41no—
91C2×Dic86—41no—
92C2×Dic86—42no—
93M16×C43—42no—
94M16×C43—42no—
95C2×C8×C43—42no—
96C4⋊C4×C43—22no—
97C4⋊C4×C43—24no—
98C42×C43—22no—
99C42×C43—22no—
100C42×C43C2×D412yes—
101C42×C43C2×D412yes—
102(C2×D43)⋊C4—28no—
103(C2×D43)⋊C4—24no—
104C4⋊Dic43—28no—
105C4⋊Dic43—21no—
106C4⋊Dic43—22no—
107C4⋊Dic43—24no—
108C4⋊Dic43C22⋊C414yes—
109C4⋊Dic43C22⋊C412yes—
110C2×C43⋊C8—42no—
111C23×C43C2×C22⋊C411yescharacteristic
112C22×D43—24no—
113C22×D43—22no—
114C22×D43—42no—
115C22×D43—41no—
116C22×D43—41no—
117Q8×C43—24no—
118Q8×C43—24no—
119C2×C4×C43—22no—
120C2×C4×C43—22no—
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—41no—
124C2×C4×C43—42no—
125C2×C4×C43#10214yes—
126C2×C4×C43C2×C22⋊C411yescharacteristic
127C2×C4×C43C2×C22⋊C411yescharacteristic
128C2×C4×C43#12012yes—
129C2×C4×C43#12012yes—
130C2×C4×C43C2×D811yescharacteristic
131C2×C4×C43C2×SD1611yescharacteristic
132C2×C4×C43AGL(1,ℤ/8)12yes—
133C2×Dic43—22no—
134C2×Dic43—24no—
135C2×Dic43—41no—
136C2×Dic43—42no—
137C2×Dic43—41no—
138C2×Dic43—42no—
139D172—24no—
140D172—22no—
141D172#10114yescomplemented
142C4×D43—416no—
143C4×D43—44no—
144C4×D43—44no—
145Dic86—416no—
146Dic86—44no—
147Dic86—44no—
148C8×C43—82no—
149C43⋊C8—82no—
150#8388—431nomaximal, Sylow
151C22×C43—21no—
152C22×C43—21no—
153C22×C43#37911yescharacteristic
154C22×C43C2×#10211yescharacteristic
155C22×C43#41811yescharacteristic
156C2×D43—24no—
157C2×D43—44no—
158C2×D43—44no—
159C2×D43—42no—
160C2×D43—42no—
161C4×C43—22no—
162C4×C43—21no—
163C4×C43—24no—
164C4×C43—42no—
165C4×C43C2×#10112yes—
166C4×C43#44712yes—
167C4×C43#45012yes—
168Dic43—42no—
169Dic43—44no—
170C2×D4×Q8—431no—
171#1801D4311yescharacteristic, complemented
172#1546—432no—
173#1486—432no—
174#1461—431no—
175C2×C43—21no—
176C2×C43—21no—
177C2×C43#178411yescharacteristic
178C2×C43#178611yescharacteristic
179C2×C43#179611yescharacteristic
180D43—44no—
181C23×Q8—861no—
182D4×Q8—864no—
183D4×Q8—864no—
184C2×C4⋊Q8—431no—
185C2×C4⋊Q8—432no—
186C2×C22⋊Q8—862no—
187C2×C22⋊Q8—861no—
188C2×C4×Q8C2×D4311yescharacteristic
189C2×C4×D4—431no—
190C2×C4×D4—432no—
191#431—432no—
192#423C2×D4312yes—
193C2×#101—21no—
194#343—21no—
195#332—864no—
196D4⋊C8—864no—
197#324—862no—
198C43#838811yescharacteristic, complemented, Sylow
199C22×Q8—868no—
200C22×Q8—862no—
201C22×Q8—861no—
202C22×Q8—1724no—
203C22×Q8—1722no—
204C22×Q8—1722no—
205C22×Q8C22⋊D4311yescharacteristic
206C22×D4—431no—
207C23×C4—862no—
208C23×C4—861no—
209C2×M16—22no—
210C4⋊Q8—434no—
211C4⋊Q8—864no—
212C4⋊Q8—864no—
213C4⋊Q8—862no—
214C22⋊Q8—868no—
215C22⋊Q8—8616no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×D4—434no—
219C4×D4—864no—
220C4×D4—864no—
221C4×D4—862no—
222C2×C4⋊C4—431no—
223C2×C4⋊C4—432no—
224C2×C4⋊C4—862no—
225C2×C4⋊C4—861no—
226C2×C4⋊C4D17212yes—
227C2×C4⋊C4C22⋊D4311yescharacteristic
228C2×C22⋊C4—862no—
229C2×C22⋊C4—861no—
230C2×C42C4×D4312yes—
231C2×C42C22×D4311yescharacteristic
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—862no—
235#101—42no—
236C22⋊C8—862no—
237C4×C8—862no—
238C24—861no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—868no—
242C2×Q8—864no—
243C2×Q8—1724no—
244C2×Q8—17216no—
245C2×Q8—17216no—
246C2×Q8—1724no—
247C2×Q8—1721no—
248C2×Q8—1721no—
249C2×Q8—1722no—
250C2×Q8#1062188212yes—
251C2×Q8#1062188412yes—
252C2×D4—432no—
253C2×D4—864no—
254C2×D4—861no—
255C22×C4—868no—
256C22×C4—862no—
257C22×C4—861no—
258C22×C4—864no—
259C22×C4—1724no—
260C22×C4—1724no—
261C22×C4—1721no—
262C22×C4—1721no—
263C22×C4—1722no—
264C22×C4(C2×D43)⋊C412yes—
265C22×C4(C2×D43)⋊C412yes—
266C22×C4C2×C4×D4311yescharacteristic
267C22×C4C2×D17211yescharacteristic
268C22×C4C2×C22⋊D4311yescharacteristic
269M16—42no—
270M16—42no—
271C2×C8—42no—
272C2×C8—1722no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—432no—
276C4⋊C4—434no—
277C4⋊C4—868no—
278C4⋊C4—861no—
279C4⋊C4—862no—
280C4⋊C4—864no—
281C22⋊C4—868no—
282C22⋊C4—864no—
283C42—22no—
284C42—22no—
285C42D4×D4312yes—
286C42D4×D4312yes—
287C23—864no—
288C23—862no—
289C23—1722no—
290C23—1721no—
291C23—1721no—
292C23C2×(C2×D43)⋊C411yescharacteristic
293Q8—24no—
294Q8—24no—
295Q8—17216no—
296Q8—1724no—
297Q8—1724no—
298D4—434no—
299D4—864no—
300D4—862no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—864no—
308C2×C4—862no—
309C2×C4—17216no—
310C2×C4—1722no—
311C2×C4—1721no—
312C2×C4—1722no—
313C2×C4—1721no—
314C2×C4—1724no—
315C2×C4—1724no—
316C2×C4#1306324514yes—
317C2×C4C22⋊C4×D4311yescharacteristic
318C2×C4C22⋊C4×D4311yescharacteristic
319C2×C4#1306329312yes—
320C2×C4#1306329512yes—
321C2×C4C2×#1062188211yescharacteristic
322C2×C4C2×#1062188411yescharacteristic
323C2×C4#1306335012yes—
324C8—82no—
325C8—3442no—
326C22—21no—
327C22—21no—
328C22—864no—
329C22—1724no—
330C22—1724no—
331C22—1722no—
332C22—1722no—
333C22#1458275411yescharacteristic
334C22C2×#1306324511yescharacteristic
335C22#1458290311yescharacteristic
336C4—21no—
337C4—24no—
338C4—22no—
339C4—42no—
340C4—1724no—
341C4—1722no—
342C4#101×D4312yes—
343C4#1458303212yes—
344C4#1458304912yes—
345C2—21no—
346C2—21no—
347C2—1724no—
348C2#1913451711yescharacteristic
349C2#1913452511yescharacteristic
350C2#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#8388
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. #19133268C2semidirect
4. #19132919C2semidirect
5. C43×#1801C2semidirect
141. D172#101semidirect
171. #1801D43semidirect
198. C43#8388semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D1722
#191332682
#191330192
#191329192
C43×#18012
#838843