Finite Group Databasebeta

← #24989225

Subgroups — #24989225

Order 11008.

1134 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1134 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
1C22×#13063301C211yesmaximal, characteristic
2#19134519C211yesmaximal, characteristic, complemented
3#19132924C212yesmaximal, complemented
4#19132921C212yesmaximal
5C43×#1784C211yesmaximal, characteristic, complemented
6C23×C4×D43C411yescharacteristic
7C2×#13063301—24no—
8C2×#13063301—24no—
9C2×#13063301—22no—
10C2×#13063301—22no—
11C22×C4⋊Dic43C411yescharacteristic
12C22×Dic43⋊C4C411yescharacteristic
13C22×Dic43⋊C4C2211yescharacteristic
14C22×(C2×D43)⋊C4C411yescharacteristic
15C22×(C2×D43)⋊C4C2211yescharacteristic
16C2×#13063244—22no—
17#14582863C2212yes—
18#14582624—24no—
19#14582622—24no—
20C23×Q8×C43C2211yescharacteristic
21C2×#101×C43—22no—
22C43×#407C2212yes—
23C24×D43C2×C411yescharacteristic
24C23×Dic43C2×C411yescharacteristic
25C23×Dic43C2×C411yescharacteristic
26C23×Dic43C2×C411yescharacteristic
27C22×C4×D43—22no—
28C22×C4×D43—22no—
29C22×C4×D43—22no—
30C22×C4×D43—44no—
31C2×#10621877—22no—
32#13063301—216no—
33#13063301—216no—
34C2×C4⋊Dic43—22no—
35C2×C4⋊Dic43—22no—
36C2×C4⋊Dic43—22no—
37C2×Dic43⋊C4—22no—
38C2×Dic43⋊C4—21no—
39C2×Dic43⋊C4—22no—
40C2×Dic43⋊C4—21no—
41C2×Dic43⋊C4—22no—
42C2×Dic43⋊C4—22no—
43C2×Dic43⋊C4D414yes—
44C2×(C2×D43)⋊C4—22no—
45C2×(C2×D43)⋊C4—22no—
46C2×(C2×D43)⋊C4—21no—
47C2×(C2×D43)⋊C4—22no—
48C2×(C2×D43)⋊C4—21no—
49C2×(C2×D43)⋊C4—22no—
50C2×(C2×D43)⋊C4D414yes—
51#13063244—44no—
52C22⋊(C43⋊C8)—22no—
53C22⋊(C43⋊C8)—22no—
54C22×Q8×C43—22no—
55C22×Q8×C43—22no—
56C22×Q8×C43—44no—
57C22×Q8×C43D412yes—
58C22×Q8×C43D412yes—
59C23×C4×C43C2×C411yescharacteristic
60C23×C4×C43C2×C411yescharacteristic
61C23×C4×C43C2311yescharacteristic
62C2×M16×C43—22no—
63#101×C43—44no—
64C22⋊C8×C43—22no—
65C22⋊C8×C43—22no—
66C24×C43C22×C411yescharacteristic
67C23×D43—21no—
68C23×D43—21no—
69C23×D43—24no—
70C23×D43—42no—
71C23×D43—42no—
72C23×D43C22⋊C412yes—
73C2×Q8×C43—24no—
74C2×Q8×C43—24no—
75C2×Q8×C43—24no—
76C2×Q8×C43—24no—
77C2×Q8×C43—42no—
78C2×Q8×C43—42no—
79C2×Q8×C43—48no—
80C2×Q8×C43—48no—
81C22×C4×C43—24no—
82C22×C4×C43—21no—
83C22×C4×C43—21no—
84C22×C4×C43—21no—
85C22×C4×C43—21no—
86C22×C4×C43—42no—
87C22×C4×C43—42no—
88C22×C4×C43C22⋊C412yes—
89C22×C4×C43C22⋊C412yes—
90C22×C4×C43C2×D411yescharacteristic
91C22×C4×C43C2×D412yes—
92C22×C4×C43C2×D412yes—
93C22×C4×C43C2×D411yescharacteristic
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—21no—
97C22×Dic43—21no—
98C22×Dic43—21no—
99C22×Dic43—21no—
100C22×Dic43—42no—
101C22×Dic43—42no—
102C22×Dic43—42no—
103C22×Dic43C22⋊C412yes—
104C22×Dic43C22⋊C412yes—
105C22×Dic43C22⋊C412yes—
106C2×C4×D43—28no—
107C2×C4×D43—48no—
108C2×C4×D43—44no—
109C2×C4×D43—48no—
110C2×C4×D43—44no—
111M16×C43—44no—
112C2×C8×C43—42no—
113(C2×D43)⋊C4—28no—
114(C2×D43)⋊C4—24no—
115(C2×D43)⋊C4—24no—
116C4⋊Dic43—28no—
117Dic43⋊C4—28no—
118Dic43⋊C4—24no—
119Dic43⋊C4—24no—
120#10621877—44no—
121C2×C43⋊C8—42no—
122C23×C43—22no—
123C23×C43—22no—
124C23×C43—22no—
125C23×C43C2×C22⋊C411yescharacteristic
126C23×C43C2×C22⋊C411yescharacteristic
127C23×C43C2×C22⋊C411yescharacteristic
128C22×D43—24no—
129C22×D43—24no—
130C22×D43—22no—
131C22×D43—22no—
132C22×D43—42no—
133C22×D43—416no—
134C22×D43—42no—
135C22×D43—42no—
136C22×D43—42no—
137C22×D43—42no—
138C22×D43—42no—
139C22×D43#10114yescomplemented
140Q8×C43—48no—
141Q8×C43—48no—
142C2×C4×C43—24no—
143C2×C4×C43—24no—
144C2×C4×C43—22no—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—22no—
149C2×C4×C43—22no—
150C2×C4×C43—22no—
151C2×C4×C43—42no—
152C2×C4×C43—42no—
153C2×C4×C43—42no—
154C2×C4×C43—42no—
155C2×C4×C43#9914yes—
156C2×C4×C43#12012yes—
157C2×C4×C43#12012yes—
158C2×Dic43—22no—
159C2×Dic43—22no—
160C2×Dic43—22no—
161C2×Dic43—24no—
162C2×Dic43—42no—
163C2×Dic43—42no—
164C2×Dic43—42no—
165C2×Dic43—42no—
166C2×Dic43—42no—
167C2×Dic43—42no—
168C2×Dic43—42no—
169C2×Dic43—42no—
170C4×D43—416no—
171C8×C43—82no—
172C43⋊C8—82no—
173#8550—431nomaximal, Sylow
174C22×C43—28no—
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—21no—
178C22×C43—21no—
179C22×C43—21no—
180C22×C43—21no—
181C22×C43#37911yescharacteristic
182C22×C43C2×#9912yes—
183C22×C43C2×#10112yes—
184C22×C43#41312yes—
185C2×D43—24no—
186C2×D43—48no—
187C2×D43—44no—
188C2×D43—48no—
189C2×D43—42no—
190C2×D43—42no—
191C2×D43—44no—
192C4×C43—22no—
193C4×C43—22no—
194C4×C43—24no—
195C4×C43—42no—
196Dic43—42no—
197Dic43—42no—
198Dic43—42no—
199C22×C22⋊Q8—431no—
200#1784—431no—
201#1784D4311yescharacteristic, complemented
202#1463—432no—
203#1462—432no—
204C2×C43—22no—
205C2×C43—22no—
206C2×C43—22no—
207C2×C43#178111yescharacteristic
208C2×C43#178411yescharacteristic
209C2×C43#178511yescharacteristic
210D43—44no—
211C23×Q8C2×D4311yescharacteristic
212C24×C4—431no—
213C2×C22⋊Q8—864no—
214C2×C22⋊Q8—864no—
215C2×C22⋊Q8—862no—
216C2×C22⋊Q8—862no—
217C22×C4⋊C4—431no—
218C22×C4⋊C4—431no—
219C22×C4⋊C4—431no—
220C22×C22⋊C4—431no—
221C22×C22⋊C4—431no—
222C2×#101—22no—
223C2×#101—862no—
224#407—432no—
225#407C2×D4312yes—
226#324—864no—
227#323—864no—
228C43#855011yescharacteristic, complemented, Sylow
229C25—431no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—44no—
233C22×Q8D17212yes—
234C22×Q8C22⋊D4312yes—
235C23×C4—431no—
236C23×C4—431no—
237C23×C4—431no—
238C23×C4—862no—
239C23×C4—862no—
240C23×C4—862no—
241C23×C4—1724no—
242C23×C4C4×D4311yescharacteristic
243C23×C4C4×D4311yescharacteristic
244C23×C4C22×D4311yescharacteristic
245C2×M16—22no—
246C2×M16—862no—
247C22⋊Q8—8616no—
248C22⋊Q8—8616no—
249C2×C4⋊C4—434no—
250C2×C4⋊C4—862no—
251C2×C4⋊C4—862no—
252C2×C4⋊C4—862no—
253C2×C4⋊C4—862no—
254C2×C4⋊C4—862no—
255C2×C4⋊C4—862no—
256C2×C4⋊C4—861no—
257C2×C4⋊C4—862no—
258C2×C4⋊C4—861no—
259C2×C22⋊C4—434no—
260C2×C22⋊C4—862no—
261C2×C22⋊C4—862no—
262C2×C22⋊C4—862no—
263C2×C22⋊C4—861no—
264C2×C22⋊C4—862no—
265C2×C22⋊C4—861no—
266#101—44no—
267#101—1724no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—862no—
271C22⋊C8—862no—
272C24—432no—
273C24—864no—
274C24—861no—
275C24—861no—
276C24—1722no—
277C24—1722no—
278C24C2×C4×D4311yescharacteristic
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—48no—
286C2×Q8—48no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—432no—
295C22×C4—432no—
296C22×C4—432no—
297C22×C4—868no—
298C22×C4—861no—
299C22×C4—861no—
300C22×C4—861no—
301C22×C4—861no—
302C22×C4—861no—
303C22×C4—861no—
304C22×C4—1728no—
305C22×C4—1728no—
306C22×C4—1724no—
307C22×C4—1722no—
308C22×C4—1724no—
309C22×C4—1722no—
310C22×C4—1722no—
311C22×C4(C2×D43)⋊C412yes—
312C22×C4(C2×D43)⋊C412yes—
313C22×C4C2×D17211yescharacteristic
314C22×C4D4×D4312yes—
315C22×C4D4×D4312yes—
316C22×C4C2×C22⋊D4311yescharacteristic
317M16—44no—
318M16—1724no—
319C2×C8—42no—
320C2×C8—1722no—
321C4⋊C4—868no—
322C4⋊C4—868no—
323C4⋊C4—864no—
324C4⋊C4—864no—
325C22⋊C4—868no—
326C22⋊C4—864no—
327C22⋊C4—864no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—434no—
332C23—864no—
333C23—864no—
334C23—862no—
335C23—862no—
336C23—1722no—
337C23—1722no—
338C23—1722no—
339C23—17216no—
340C23—1722no—
341C23—1722no—
342C23—1722no—
343C23C22⋊C4×D4311yescharacteristic
344C23C22⋊C4×D4311yescharacteristic
345C23C2×(C2×D43)⋊C411yescharacteristic
346Q8—48no—
347Q8—48no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—864no—
362C2×C4—862no—
363C2×C4—862no—
364C2×C4—862no—
365C2×C4—17216no—
366C2×C4—1722no—
367C2×C4—1722no—
368C2×C4—1722no—
369C2×C4—1722no—
370C2×C4—1722no—
371C2×C4—1722no—
372C2×C4—1722no—
373C2×C4—1722no—
374C2×C4#1306323914yes—
375C2×C4#1306329312yes—
376C2×C4#1306329512yes—
377C8—82no—
378C8—3442no—
379C22—28no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—864no—
387C22—1724no—
388C22—1728no—
389C22—1724no—
390C22—1728no—
391C22—1722no—
392C22—1722no—
393C22#1458275411yescharacteristic
394C22C2×#1306323912yes—
395C22#101×D4312yes—
396C22#1458288212yes—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—42no—
401C4—1722no—
402C4—1722no—
403C4—1722no—
404C2—22no—
405C2—22no—
406C2—22no—
407C2—1724no—
408C2#1913450911yescharacteristic
409C2#1913451711yescharacteristic
410C2#1913452211yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1784C2
socleC22×C43#379
prime radicalC24×D43C2×C4
cosocleC23×C4×C43C23
hypercenter#1784D43
hypocenterC43#8550
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #19134519C2semidirect
3. #19132924C2semidirect
5. C43×#1784C2semidirect
139. C22×D43#101semidirect
201. #1784D43semidirect
228. C43#8550semidirect

Maximal subgroups

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

subgroupindex
C22×#130633012
#191345192
#191329242
#191329212
C43×#17842
#855043