Finite Group Databasebeta

← #25100486

Subgroups — #25100486

Order 11008.

1790 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D344C211yesmaximal, characteristic
2C2×#14582903C212yesmaximal, complemented
3C2×#14582854C211yesmaximal, characteristic, complemented
4#19137175C212yesmaximal, complemented
5#19135469C214yesmaximal, complemented
6#19135459C214yesmaximal, complemented
7C2×C43×#427C211yesmaximal, characteristic, complemented
8C23×D172C2212yes—
9C22×D344—24no—
10C22×D344—24no—
11C22×D344C418yescomplemented
12C22×D344C2212yes—
13C22×D344C2212yes—
14C22×#10621877C2211yescharacteristic
15C2×#13063284C2211yescharacteristic
16#14582903—28no—
17C2×#13063245C2218yescomplemented
18#14582854—24no—
19C2×#13063235C2212yescomplemented
20C2×C8×Dic43C2212yescomplemented
21#14582801—24no—
22#14582800C2218yescomplemented
23#14582754—24no—
24C23×C8×C43C2211yescharacteristic
25C2×#117×C43C2212yes—
26C43×#427—24no—
27C2×#107×C43C2212yescomplemented
28C2×#106×C43C2212yescomplemented
29C24×D43—22no—
30C22×D172—24no—
31C22×D172—24no—
32C22×D172—24no—
33C22×D172—24no—
34C22×D172—22no—
35C22×D172—24no—
36C22×D172C2×C418yes—
37C22×D172C2314yes—
38C2×D344—232no—
39C2×D344—24no—
40C2×D344—24no—
41C2×D344—24no—
42C2×D344—24no—
43C2×D344C2×C418yes—
44C2×D344C2×C418yes—
45C2×#10621877—22no—
46C2×#10621877—22no—
47C2×#10621877D414yes—
48C22×C43⋊C8C2312yes—
49#13063284—22no—
50#13063284—22no—
51C2×C4⋊Dic43D412yes—
52C2×C22⋊Dic43D412yes—
53C2×(C2×D43)⋊C4—216no—
54C2×C4×Dic43C2312yes—
55#13063245—216no—
56#13063235—24no—
57C8×Dic43—24no—
58C23×C4×C43C2311yescharacteristic
59C22×C8×C43C2×C414yes—
60C22×C8×C43D414yes—
61C22×C8×C43D414yes—
62C22×C8×C43C2311yescharacteristic
63C22×C8×C43C2311yescharacteristic
64#117×C43—24no—
65#117×C43—24no—
66C2×C4⋊C4×C43C2314yes—
67C2×C22⋊C4×C43—22no—
68C2×C42×C43—22no—
69#107×C43—24no—
70#106×C43—24no—
71C24×C43C2×D411yescharacteristic
72C23×D43—24no—
73C23×D43—28no—
74C23×D43—44no—
75C23×D43—44no—
76C23×D43—88no—
77C22×C4×C43—24no—
78C22×C4×C43C22×C412yes—
79C22×C4×C43C2×D412yes—
80C22×C4×C43C2×D412yes—
81C22×C4×C43C2×D412yes—
82C22×C4×C43C2×D411yescharacteristic
83C22×C4×C43#4112yes—
84C22×C4×C43#4112yes—
85C22×C4×C43C2411yescharacteristic
86C22×Dic43C2×D414yes—
87C2×D172—28no—
88C2×D172—24no—
89C2×D172—216no—
90C2×D172—216no—
91C2×D172—24no—
92C2×D172—24no—
93C2×D172—24no—
94C2×D172—24no—
95C2×D172—44no—
96C2×D172—48no—
97C2×D172—48no—
98C2×D172—44no—
99C2×D172C22×C418yes—
100C2×C8×C43C22⋊C4116yescomplemented
101C2×C8×C43C22×C412yes—
102C2×C8×C43C22×C412yes—
103C2×C8×C43C2×D412yes—
104C2×C8×C43C2×D412yes—
105C2×C8×C43C2×D412yes—
106C2×C8×C43C2×D412yes—
107C4⋊C4×C43—24no—
108C4⋊C4×C43—24no—
109C22⋊C4×C43—44no—
110C42×C43—44no—
111C22⋊Dic43—24no—
112(C2×D43)⋊C4—816no—
113C4⋊Dic43—24no—
114C4×Dic43—22no—
115C4×Dic43—22no—
116#10621877—28no—
117C2×C43⋊C8—22no—
118C2×C43⋊C8—22no—
119C2×C43⋊C8C2×D414yes—
120D344—216no—
121D344—216no—
122C23×C43—24no—
123C23×C43C4×D412yes—
124C23×C43#12112yes—
125C23×C43#12712yes—
126C23×C43C22×D411yescharacteristic
127C22×D43—28no—
128C22×D43—44no—
129C22×D43—44no—
130C22×D43—416no—
131C22×D43—416no—
132C22×D43—816no—
133C22×D43—816no—
134C22×D43—816no—
135C2×C4×C43—44no—
136C2×C4×C43—48no—
137C2×C4×C43C2×C22⋊C414yes—
138C2×C4×C43C4×D414yes—
139C2×C4×C43C4×D412yes—
140C2×C4×C43#12114yes—
141C2×C4×C43#12112yes—
142C2×C4×C43#12414yes—
143C2×C4×C43#12712yes—
144C2×C4×C43C23×C411yescharacteristic
145C2×C4×C43C22×D411yescharacteristic
146C2×C4×C43C22×D411yescharacteristic
147C2×C4×C43C22×D411yescharacteristic
148C2×C4×C43C2×#4111yescharacteristic
149C2×C4×C43C2×#4111yescharacteristic
150C2×Dic43—24no—
151C2×Dic43—28no—
152D172—28no—
153D172—28no—
154D172—44no—
155D172—44no—
156D172—416no—
157C8×C43C2×C22⋊C414yes—
158C8×C43C2×C22⋊C414yes—
159C43⋊C8—24no—
160#21233—431nomaximal, Sylow
161C22×C43—24no—
162C22×C43—24no—
163C22×C43—24no—
164C22×C43#39014yes—
165C22×C43C2×C4×D411yescharacteristic
166C22×C43C2×#12111yescharacteristic
167C22×C43C2×#12711yescharacteristic
168C22×C43#57614yes—
169C2×D43—416no—
170C2×D43—48no—
171C2×D43—832no—
172C2×D43—88no—
173C2×D43—88no—
174C4×C43—84no—
175C4×C43#39014yes—
176C4×C43C22×C22⋊C411yescharacteristic
177C4×C43C2×C4×D411yescharacteristic
178C4×C43C2×#12111yescharacteristic
179C4×C43C2×#12411yescharacteristic
180Dic43—44no—
181C23×D8—431no—
182C2×#427D4311yescharacteristic, complemented
183C2×#418—432no—
184C2×#405—431no—
185#2309—432no—
186#1972—434no—
187#1970—434no—
188C2×C43—24no—
189C2×C43C2×#39011yescharacteristic
190C2×C43#294812yes—
191C2×C43#305812yes—
192C2×C43#315512yes—
193D43—88no—
194C23×D4—432no—
195C22×D8—432no—
196C22×D8—432no—
197C22×D8—438no—
198C22×D8—864no—
199C22×D8—864no—
200C22×M16—431no—
201C23×C8C2×D4311yescharacteristic
202C2×#117—431no—
203C2×#117C2×D4312yes—
204#427—24no—
205C2×#107C2×D4312yescomplemented
206C2×#106C2×D4312yescomplemented
207#418—868no—
208C2×#102—438no—
209#405—864no—
210C2×#97—432no—
211C2×C4×C8—432no—
212#390—438no—
213#390—864no—
214#379—864no—
215C43#2123311yescharacteristic, complemented, Sylow
216C25—862no—
217C22×D4—438no—
218C22×D4—434no—
219C22×D4—864no—
220C22×D4—864no—
221C22×D4—862no—
222C22×D4—864no—
223C22×D4—864no—
224C22×D4—864no—
225C23×C4C22×D4311yescharacteristic
226C2×D8—438no—
227C2×D8—438no—
228C2×D8—8632no—
229C2×D8—864no—
230C2×D8—864no—
231C2×D8—864no—
232C2×D8—864no—
233C2×M16—434no—
234C2×M16—862no—
235C2×M16—862no—
236C22×C8—432no—
237C22×C8C4×D4314yes—
238C22×C8D17214yes—
239C22×C8C22⋊D4314yes—
240C22×C8C22×D4311yescharacteristic
241C22×C8C22×D4311yescharacteristic
242#117—24no—
243#117—24no—
244#117—862no—
245#117—862no—
246C2×C4⋊C4—432no—
247C2×C4⋊C4C22×D4314yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—432no—
250C2×C22⋊C4—8616no—
251C2×C42—22no—
252C2×C42—432no—
253#107—24no—
254#106—24no—
255#102—8616no—
256#97—864no—
257C4×C8—864no—
258C24—868no—
259C24—864no—
260C24—3448no—
261C24—1724no—
262C24—1724no—
263C24D4×D4311yescharacteristic
264C2×D4—438no—
265C2×D4—8616no—
266C2×D4—8616no—
267C2×D4—868no—
268C2×D4—864no—
269C2×D4—864no—
270C2×D4—864no—
271C2×D4—864no—
272C2×D4—864no—
273C2×D4—1724no—
274C2×D4—1728no—
275C2×D4—1728no—
276C2×D4—1724no—
277C22×C4—24no—
278C22×C4—434no—
279C22×C4C2×C4×D4312yes—
280C22×C4C2×D17212yes—
281C22×C4D4×D4311yescharacteristic
282C22×C4D4×D4312yes—
283C22×C4#1062190212yes—
284C22×C4#1062190212yes—
285C22×C4C2×C22⋊D4312yes—
286C22×C4C23×D4311yescharacteristic
287D8—8616no—
288D8—8616no—
289M16—868no—
290C2×C8—434no—
291C2×C8—862no—
292C2×C8—862no—
293C2×C8(C2×D43)⋊C4116yescomplemented
294C2×C8C2×C4×D4312yes—
295C2×C8C2×C4×D4312yes—
296C2×C8C2×D17212yes—
297C2×C8C2×D17212yes—
298C2×C8C2×C22⋊D4312yes—
299C2×C8C2×C22⋊D4312yes—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—864no—
303C22⋊C4—44no—
304C22⋊C4—864no—
305C22⋊C4—34416no—
306C42—44no—
307C42—862no—
308C42—862no—
309C23—24no—
310C23—868no—
311C23—34416no—
312C23—34416no—
313C23—34416no—
314C23—17216no—
315C23—1724no—
316C23—1724no—
317C23—17216no—
318C23#1306328712yes—
319C23#1306330012yes—
320C23#1306332112yes—
321C23C2×D4×D4311yescharacteristic
322D4—868no—
323D4—868no—
324D4—17216no—
325D4—1724no—
326D4—1724no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—864no—
330C2×C4—868no—
331C2×C4C2×(C2×D43)⋊C414yes—
332C2×C4#1306328712yes—
333C2×C4#1306328714yes—
334C2×C4#1306330012yes—
335C2×C4#1306330014yes—
336C2×C4#1306331414yes—
337C2×C4#1306332112yes—
338C2×C4C22×C4×D4311yescharacteristic
339C2×C4C2×D4×D4311yescharacteristic
340C2×C4C22×C22⋊D4311yescharacteristic
341C2×C4C22×D17211yescharacteristic
342C2×C4C2×#1062190211yescharacteristic
343C2×C4C2×#1062190211yescharacteristic
344C8—864no—
345C8C2×(C2×D43)⋊C414yes—
346C8C2×(C2×D43)⋊C414yes—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—34432no—
351C22—3448no—
352C22—3448no—
353C22—17216no—
354C22—1728no—
355C22#1458280014yes—
356C22C2×#1306328711yescharacteristic
357C22C2×#1306330011yescharacteristic
358C22C2×#1306332111yescharacteristic
359C22#1458370114yes—
360C4—84no—
361C4—1724no—
362C4#1458280014yes—
363C4C22×(C2×D43)⋊C411yescharacteristic
364C4C2×#1306328711yescharacteristic
365C4C2×#1306330011yescharacteristic
366C4C2×#1306331411yescharacteristic
367C2—24no—
368C2—3448no—
369C2C2×#1458280011yescharacteristic
370C2#1914344612yes—
371C2#1914424412yes—
372C2#1914508212yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#427C2
socleC23×C43C22×D4
prime radicalC23×D344C2
cosocleC22×C4×C43C24
hypercenterC2×#427D43
hypocenterC43#21233
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#14582903C2semidirect
3. C2×#14582854C2semidirect
4. #19137175C2semidirect
5. #19135469C2semidirect
6. #19135459C2semidirect
7. C2×C43×#427C2semidirect
11. C22×D344C4semidirect
17. C2×#13063245C22semidirect
19. C2×#13063235C22semidirect
20. C2×C8×Dic43C22semidirect
22. #14582800C22semidirect
27. C2×#107×C43C22semidirect
28. C2×#106×C43C22semidirect
100. C2×C8×C43C22⋊C4semidirect
182. C2×#427D43semidirect
205. C2×#107C2×D43semidirect
206. C2×#106C2×D43semidirect
215. C43#21233semidirect
293. C2×C8(C2×D43)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×D3442
C2×#145829032
C2×#145828542
#191371752
#191354692
#191354592
C2×C43×#4272
#2123343