Finite Group Databasebeta

← #24988348

Subgroups — #24988348

Order 11008.

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

normal subgroups only

This group has 998 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×#14583554C211yesmaximal, characteristic
2#19134587C211yesmaximal, characteristic, complemented
3#19133275C212yesmaximal, complemented
4#19133021C212yesmaximal, complemented
5C43×#1460C211yesmaximal, characteristic, complemented
6C22×#10621898C412yes—
7#14583554—24no—
8#14583554—24no—
9C2×#13063325C2211yescharacteristic
10C2×#13063307—22no—
11C2×#13063304C412yes—
12C2×#13063296—21no—
13C2×C4×D172C2211yescharacteristic
14C2×C4⋊C4×D43—21no—
15#14582923C2212yes—
16C2×#13063242—21no—
17#14582668—21no—
18#14582647—24no—
19#14582628—24no—
20C2×C4×D4×C43C2211yescharacteristic
21C43×#431C2212yes—
22C43×#323—22no—
23C2×#10621898—24no—
24C2×#10621898—24no—
25C2×#10621898—48no—
26C22×Dic86C2×C412yes—
27C22×D172C2×C411yescharacteristic
28C22×C22⋊D43—22no—
29C22×C4×D43—22no—
30C22×C4×D43—21no—
31C2×#10621877—22no—
32#13063325—22no—
33#13063325D414yes—
34#13063307—216no—
35#13063304—24no—
36#13063304—24no—
37#13063296—28no—
38C4×D172—22no—
39C4×D172D414yes—
40C2×C4⋊Dic43—21no—
41C2×C4⋊Dic43—21no—
42C2×C4⋊Dic43C2×C411yescharacteristic
43C2×Dic43⋊C4—22no—
44C4⋊C4×D43—28no—
45C2×C22⋊Dic43—22no—
46C2×(C2×D43)⋊C4—21no—
47C4⋊(C43⋊C8)—22no—
48C4⋊(C43⋊C8)—22no—
49#13063242—42no—
50C22×D4×C43D411yescharacteristic
51C23×C4×C43C2×C412yes—
52C4×D4×C43—22no—
53C4×D4×C43—22no—
54C2×C4⋊C4×C43D411yescharacteristic
55C2×C22⋊C4×C43D412yes—
56C2×C42×C43C2311yescharacteristic
57C4⋊C8×C43—22no—
58C22⋊C8×C43—22no—
59C4×C8×C43—22no—
60C24×C43C22⋊C412yes—
61C23×D43—21no—
62C2×D4×C43—22no—
63C2×D4×C43—22no—
64C2×D4×C43—21no—
65C2×D4×C43—21no—
66C22×C4×C43—22no—
67C22×C4×C43—22no—
68C22×C4×C43—44no—
69C22×C4×C43C22⋊C412yes—
70C22×C4×C43C22×C411yescharacteristic
71C22×C4×C43C2×D411yescharacteristic
72C22×C4×C43C2×D411yescharacteristic
73C2×C22⋊D43—28no—
74C2×C22⋊D43—48no—
75C22×Dic43—21no—
76C22×Dic43—22no—
77#10621898—48no—
78#10621898—416no—
79#10621898—48no—
80C2×D172—21no—
81C2×D172—24no—
82C2×D172C22⋊C412yes—
83C2×C4×D43—28no—
84C2×C4×D43—24no—
85C2×C4×D43—48no—
86C2×C4×D43—42no—
87C2×C4×D43—41no—
88C2×C4×D43—41no—
89C2×Dic86—22no—
90C2×Dic86—44no—
91C2×Dic86C22⋊C414yes—
92C2×C8×C43—42no—
93C4⋊C4×C43—22no—
94C22⋊C4×C43—24no—
95C42×C43C2×D412yes—
96C42×C43C2×D412yes—
97C22⋊Dic43—28no—
98(C2×D43)⋊C4—24no—
99C4⋊Dic43—21no—
100C4⋊Dic43—24no—
101C4⋊Dic43—24no—
102C4⋊Dic43C22⋊C412yes—
103Dic43⋊C4—28no—
104#10621877—42no—
105#10621877—42no—
106C2×C43⋊C8—42no—
107C23×C43—22no—
108C23×C43—22no—
109C23×C43—22no—
110C23×C43—42no—
111C23×C43—42no—
112C23×C43C2×C22⋊C411yescharacteristic
113C22×D43—24no—
114C22×D43—22no—
115C22×D43—42no—
116C22×D43—41no—
117C22×D43—41no—
118D4×C43—42no—
119D4×C43—42no—
120C2×C4×C43—22no—
121C2×C4×C43—22no—
122C2×C4×C43—41no—
123C2×C4×C43—44no—
124C2×C4×C43—44no—
125C2×C4×C43—42no—
126C2×C4×C43C2×C22⋊C411yescharacteristic
127C2×C4×C43C2×C22⋊C411yescharacteristic
128C2×C4×C43#12012yes—
129C2×C4×C43#12012yes—
130C2×C4×C43#13512yes—
131C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C43#13711yescharacteristic
133C22⋊D43—416no—
134C2×Dic43—22no—
135C2×Dic43—24no—
136C2×Dic43—44no—
137C2×Dic43—41no—
138C2×Dic43—41no—
139D172—24no—
140D172—22no—
141D172#10014yescomplemented
142C4×D43—416no—
143C4×D43—44no—
144C4×D43—44no—
145Dic86—44no—
146Dic86—44no—
147C8×C43—82no—
148C43⋊C8—82no—
149#8393—431nomaximal, Sylow
150C22×C43—21no—
151C22×C43—21no—
152C22×C43—42no—
153C22×C43—42no—
154C22×C43—42no—
155C22×C43—82no—
156C22×C43—82no—
157C22×C43#37911yescharacteristic
158C22×C43#41611yescharacteristic
159C22×C43#41711yescharacteristic
160C2×D43—24no—
161C2×D43—44no—
162C2×D43—44no—
163C2×D43—42no—
164C2×D43—42no—
165C4×C43—22no—
166C4×C43—21no—
167C4×C43—42no—
168C4×C43C2×#10012yes—
169C4×C43#44912yes—
170C4×C43#45212yes—
171Dic43—44no—
172Dic43—42no—
173C2×#547—431no—
174#1800—431no—
175#1548—432no—
176#1486—432no—
177#1460D4311yescharacteristic, complemented
178C2×C43—21no—
179C2×C43—21no—
180C2×C43—82no—
181C2×C43#178311yescharacteristic
182C2×C43#178811yescharacteristic
183C2×C43#179611yescharacteristic
184D43—44no—
185C22×#41—432no—
186#547—864no—
187#547—864no—
188C2×C4⋊Q8—431no—
189C2×#123—862no—
190C2×C22⋊Q8—432no—
191C2×#121—861no—
192C2×C4×D4—431no—
193C2×C4×D4C2×D4311yescharacteristic
194C22×C4⋊C4—861no—
195#431C2×D4312yes—
196#423—432no—
197C2×#100—861no—
198#343—861no—
199#333—864no—
200D4⋊C8—864no—
201#323—22no—
202C43#839311yescharacteristic, complemented, Sylow
203C2×#41—864no—
204C2×#41—864no—
205C2×#41—1728no—
206C22×Q8—432no—
207C22×D4—431no—
208C22×D4—862no—
209C22×D4D17211yescharacteristic
210C23×C4—862no—
211C23×C4—861no—
212C23×C4C4×D4312yes—
213C2×M16—862no—
214C4⋊Q8—434no—
215C4⋊Q8—862no—
216#123—8616no—
217C22⋊Q8—864no—
218C22⋊Q8—864no—
219#121—868no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—434no—
223C4×D4—862no—
224C2×C4⋊C4—431no—
225C2×C4⋊C4—868no—
226C2×C4⋊C4—862no—
227C2×C4⋊C4—861no—
228C2×C4⋊C4—861no—
229C2×C4⋊C4D17211yescharacteristic
230C2×C22⋊C4—862no—
231C2×C22⋊C4—861no—
232C2×C22⋊C4C22⋊D4312yes—
233C2×C42C22×D4311yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—862no—
236C4⋊C8—862no—
237#100—1722no—
238C22⋊C8—22no—
239C4×C8—22no—
240C24—861no—
241C24(C2×D43)⋊C412yes—
242#41—1728no—
243#41—17216no—
244#41—1728no—
245C2×Q8—434no—
246C2×Q8—862no—
247C2×Q8—1724no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—432no—
253C2×D4—864no—
254C2×D4—868no—
255C2×D4—861no—
256C2×D4—1728no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4—868no—
261C22×C4—861no—
262C22×C4—862no—
263C22×C4—864no—
264C22×C4—1728no—
265C22×C4—1722no—
266C22×C4—1721no—
267C22×C4—1721no—
268C22×C4(C2×D43)⋊C412yes—
269C22×C4C2×C4×D4311yescharacteristic
270C22×C4C2×D17211yescharacteristic
271C22×C4C2×C22⋊D4311yescharacteristic
272M16—1722no—
273M16—1722no—
274C2×C8—42no—
275C2×C8—1722no—
276C4⋊C4—22no—
277C4⋊C4—432no—
278C4⋊C4—868no—
279C4⋊C4—861no—
280C4⋊C4—864no—
281C4⋊C4—864no—
282C22⋊C4—24no—
283C22⋊C4—868no—
284C22⋊C4—864no—
285C42D4×D4312yes—
286C42D4×D4312yes—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—864no—
293C23—862no—
294C23—1722no—
295C23—1721no—
296C23—1721no—
297C23C2×(C2×D43)⋊C411yescharacteristic
298Q8—1724no—
299Q8—1724no—
300D4—42no—
301D4—42no—
302D4—434no—
303D4—864no—
304D4—862no—
305D4—17216no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—862no—
313C2×C4—864no—
314C2×C4—17216no—
315C2×C4—1724no—
316C2×C4—1721no—
317C2×C4—1721no—
318C2×C4—1724no—
319C2×C4—1724no—
320C2×C4C22⋊C4×D4311yescharacteristic
321C2×C4C22⋊C4×D4311yescharacteristic
322C2×C4#1306329312yes—
323C2×C4#1306329512yes—
324C2×C4#1306334712yes—
325C2×C4#1306335311yescharacteristic
326C2×C4#1306335811yescharacteristic
327C8—82no—
328C8—3442no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22—82no—
336C22—864no—
337C22—1724no—
338C22—1724no—
339C22—1722no—
340C22—1722no—
341C22#1458275411yescharacteristic
342C22#1458289611yescharacteristic
343C22#1458290111yescharacteristic
344C4—21no—
345C4—22no—
346C4—42no—
347C4—1724no—
348C4—1722no—
349C4#100×D4312yes—
350C4#1458304612yes—
351C4#1458306312yes—
352C2—21no—
353C2—21no—
354C2—82no—
355C2—1724no—
356C2#1913451411yescharacteristic
357C2#1913453611yescharacteristic
358C2#1913456811yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1460C2
socleC22×C43#379
prime radicalC2×#14583554C2
cosocleC2×C42×C43C23
hypercenter#1460D43
hypocenterC43#8393
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #19134587C2semidirect
3. #19133275C2semidirect
4. #19133021C2semidirect
5. C43×#1460C2semidirect
141. D172#100semidirect
177. #1460D43semidirect
202. C43#8393semidirect

Maximal subgroups

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

subgroupindex
C2×#145835542
#191345872
#191332752
#191330212
C43×#14602
#839343