Finite Group Databasebeta

← #26005157

Subgroups — #26005157

Order 11008.

6930 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19148421C218yesmaximal, complemented
2#19148414C218yesmaximal, complemented
3#19148405C214yesmaximal, complemented
4#19148398C2112yesmaximal, complemented
5#41×Dic86C2116yesmaximal, complemented
6#19148000C216yesmaximal, complemented
7C2×#14583456C216yesmaximal, complemented
8#19147798C212yesmaximal
9C2×C4×#41×C43C211yesmaximal, characteristic
10Q8×Dic86C22116yescomplemented
11#14583628C22116yescomplemented
12#14583625C22132yescomplemented
13#14583608C22196yescomplemented
14#14583606C22148yescomplemented
15D4×Dic86C22148yescomplemented
16#14583530C22124yes—
17#14583527C22148yescomplemented
18#14583525C22124yes—
19#14583456C2214yes—
20#14583456C22124yes—
21#14583456C22112yes—
22#14583456C22112yescomplemented
23#14583456C22112yescomplemented
24#14583456C22124yescomplemented
25#14583452C22148yescomplemented
26C2×#13063325C22112yescomplemented
27C2×#13063317C22112yescomplemented
28C2×#13063304C22124yes—
29#14583382C22116yes—
30#14583377C2214yes—
31#14583374C22112yes—
32#41×Dic43C22116yes—
33C2×C4×Dic86C22124yescomplemented
34C2×#13063284C22112yes—
35C22×#41×C43C2211yescharacteristic
36C4×#41×C43C2214yes—
37C4×#41×C43C2214yes—
38C4×#41×C43C2218yes—
39C2×C4×Q8×C43C2212yes—
40C2×C4×D4×C43C2216yes—
41C2×#117×C43C2213yes—
42C22×C42×C43C2213yes—
43C22×Dic86C23124yes—
44#13063325C23124yes—
45#13063325C23124yes—
46#13063325C2318yes—
47#13063325C23124yes—
48#13063324C23196yes—
49#13063317C23148yes—
50#13063317C2318yes—
51#13063317C23124yes—
52#13063316C23196yes—
53#13063304C23148yes—
54#13063304C23148yes—
55#13063304C23116yes—
56#13063304C23148yes—
57#13063302C231192yes—
58Q8×Dic43C23132yes—
59C4×Dic86C23148yes—
60C4×Dic86C23148yes—
61C4×Dic86C23116yes—
62C4×Dic86C23148yes—
63D4×Dic43C23196yes—
64#13063284C2318yes—
65#13063284C23124yes—
66#13063284C23124yes—
67#13063284C23124yes—
68C2×C4⋊Dic43C23124yes—
69C2×C4⋊Dic43C23124yes—
70C2×Dic43⋊C4C23148yes—
71C2×C22⋊Dic43C23124yes—
72C2×C4×Dic43C23124yes—
73C2×#41×C43Q8116yescomplemented
74C2×#41×C43C2314yes—
75C2×#41×C43C2314yes—
76C22×Q8×C43C2311yescharacteristic
77C22×D4×C43C2313yes—
78C23×C4×C43C2316yes—
79C23×C4×C43C2313yes—
80C4×Q8×C43C2318yes—
81C4×Q8×C43C2318yes—
82C4×D4×C43C23124yes—
83C4×D4×C43C23124yes—
84#117×C43C23112yes—
85#117×C43C23112yes—
86C2×C4⋊C4×C43C2316yes—
87C2×C22⋊C4×C43C2316yes—
88C2×C42×C43C2311yescharacteristic
89C2×C42×C43C2313yes—
90C2×C42×C43C2316yes—
91C2×C42×C43C23112yes—
92C2×C42×C43C2316yes—
93C2×C42×C43C2316yes—
94C24×C43C2413yes—
95#41×C43C2×Q8132yescomplemented
96#41×C43C2×Q8132yescomplemented
97C2×Q8×C43C2×Q818yescomplemented
98C2×Q8×C43C2414yes—
99C2×D4×C43C2×Q8124yescomplemented
100C2×D4×C43C24112yes—
101C22×C4×C43—412no—
102C22×C4×C43C2×Q8124yescomplemented
103C22×C4×C43C2412yes—
104C22×C4×C43C24112yes—
105C22×C4×C43C2416yes—
106C22×C4×C43C24112yes—
107C22×C4×C43C2411yescharacteristic
108C22×C4×C43C2413yes—
109C22×C4×C43C2416yes—
110C22×C4×C43C2416yes—
111C22×Dic43C24148yes—
112C2×Dic86—448no—
113C2×Dic86C24116yes—
114C2×Dic86C24148yes—
115C4⋊C4×C43C24124yes—
116C22⋊C4×C43C24124yes—
117C42×C43C2412yes—
118C42×C43C2411yescharacteristic
119C42×C43C2411yescharacteristic
120C42×C43C2416yes—
121C42×C43C2416yes—
122C42×C43C24112yes—
123C42×C43C24112yes—
124C22⋊Dic43C24116yes—
125C22⋊Dic43C24148yes—
126C4⋊Dic43C24116yes—
127C4⋊Dic43C24116yes—
128C4⋊Dic43C24148yes—
129C4⋊Dic43C24148yes—
130Dic43⋊C4C24132yes—
131Dic43⋊C4C24196yes—
132C4×Dic43C24116yes—
133C4×Dic43C24148yes—
134C23×C43—46no—
135C23×C43C22×Q8112yes—
136C23×C43C2511yescharacteristic
137C23×C43C2516yes—
138Q8×C43C22×Q8116yes—
139D4×C43C22×Q8148yes—
140C2×C4×C43—224no—
141C2×C4×C43—24no—
142C2×C4×C43—46no—
143C2×C4×C43—412no—
144C2×C4×C43C22×Q814yes—
145C2×C4×C43C22×Q8112yes—
146C2×C4×C43C22×Q8124yes—
147C2×C4×C43C22×Q8124yes—
148C2×C4×C43C2512yes—
149C2×C4×C43C25112yes—
150C2×C4×C43C2516yes—
151C2×C4×C43C2511yescharacteristic
152C2×C4×C43C2512yes—
153C2×C4×C43C2511yescharacteristic
154C2×Dic43—448no—
155C2×Dic43C25132yes—
156Dic86—232no—
157Dic86—448no—
158#62813—431nomaximal, Sylow
159C22×C43—22no—
160C22×C43—212no—
161C22×C43—412no—
162C22×C43C23×Q812yes—
163C22×C43C23×Q8112yes—
164C22×C43C2611yescharacteristic
165C4×C43—212no—
166C4×C43—22no—
167C4×C43C23×Q812yes—
168C4×C43C23×Q812yes—
169C4×C43C23×Q8112yes—
170C4×C43#58514yes—
171Dic43—232no—
172#3504—438no—
173#3503—438no—
174#3502—434no—
175#3501—4312no—
176Q8×#41—4316no—
177#3458—436no—
178C2×#533—436no—
179#3428—432no—
180C2×C4×#41D4311yescharacteristic
181C2×C43—22no—
182C2×C43—212no—
183C2×C43C24×Q811yescharacteristic
184C2×C43C2×#58512yes—
185C22×#41C2×D4311yescharacteristic
186Q82—4316no—
187#557—4332no—
188#557—4316no—
189#554—4348no—
190#554—4396no—
191D4×Q8—4348no—
192#544—4348no—
193#544—4324no—
194#543—4324no—
195#533—4348no—
196#533—434no—
197#533—4312no—
198#533—4324no—
199#533—4324no—
200#533—4312no—
201#533—4312no—
202C2×C4⋊Q8—4312no—
203C2×#125—4312no—
204C2×C22⋊Q8—4324no—
205#520—4316no—
206#519—434no—
207#518—4312no—
208C4×#41—4316no—
209C4×#41C2×D4318yes—
210C4×#41C2×D4314yes—
211C4×#41C2×D4314yes—
212C2×C4×Q8—4324no—
213C2×C4×Q8C2×D4312yes—
214C2×C4×D4C2×D4316yes—
215C2×#117—4312no—
216C2×#117C2×D4313yes—
217C22×C42C2×D4313yes—
218C43#6281311yescharacteristic, complemented, Sylow
219C2×#41Dic86116yescomplemented
220C2×#41C22×D4314yes—
221C2×#41C22×D4314yes—
222C22×Q8—4324no—
223C22×Q8C22×D4311yescharacteristic
224C22×D4C22×D4313yes—
225C23×C4C22×D4316yes—
226C23×C4C22×D4313yes—
227C4⋊Q8—4396no—
228C4⋊Q8—438no—
229C4⋊Q8—4324no—
230C4⋊Q8—4324no—
231C4⋊Q8—4324no—
232#125—4396no—
233#125—438no—
234#125—4324no—
235#125—4348no—
236C22⋊Q8—4316no—
237C22⋊Q8—4348no—
238C22⋊Q8—4348no—
239C22⋊Q8—4348no—
240C22⋊Q8—43192no—
241C4×Q8—4332no—
242C4×Q8—4316no—
243C4×Q8—4348no—
244C4×Q8—4348no—
245C4×Q8—4348no—
246C4×Q8C22×D4318yes—
247C4×Q8C22×D4318yes—
248C4×D4—4396no—
249C4×D4C22×D43124yes—
250C4×D4C22×D43124yes—
251#117—438no—
252#117—4324no—
253#117—4324no—
254#117—4324no—
255#117C22×D43112yes—
256#117C22×D43112yes—
257C2×C4⋊C4—4324no—
258C2×C4⋊C4—4348no—
259C2×C4⋊C4—4324no—
260C2×C4⋊C4C22×D4316yes—
261C2×C22⋊C4—4324no—
262C2×C22⋊C4C22×D4316yes—
263C2×C42—4324no—
264C2×C42C22×D4313yes—
265C2×C42C22×D4316yes—
266C2×C42C22×D43112yes—
267C2×C42C22×D4316yes—
268C2×C42C22×D4316yes—
269C2×C42C22×D4311yescharacteristic
270C24C23×D4313yes—
271#41C2×Dic86132yescomplemented
272#41C2×Dic86132yescomplemented
273C2×Q8—4316no—
274C2×Q8—4348no—
275C2×Q8—17248no—
276C2×Q8C2×Dic8618yescomplemented
277C2×Q8C23×D4314yes—
278C2×D4C2×Dic86124yescomplemented
279C2×D4C23×D43112yes—
280C22×C4—412no—
281C22×C4—4348no—
282C22×C4C2×Dic86124yescomplemented
283C22×C4C23×D4312yes—
284C22×C4C23×D43112yes—
285C22×C4C23×D4316yes—
286C22×C4C23×D43112yes—
287C22×C4C23×D4316yes—
288C22×C4C23×D4313yes—
289C22×C4C23×D4316yes—
290C22×C4C23×D4311yescharacteristic
291C4⋊C4—4316no—
292C4⋊C4—4332no—
293C4⋊C4—4316no—
294C4⋊C4—4348no—
295C4⋊C4—4396no—
296C4⋊C4—4348no—
297C4⋊C4C23×D43124yes—
298C22⋊C4—4316no—
299C22⋊C4—4348no—
300C22⋊C4C23×D43124yes—
301C42—4348no—
302C42—4316no—
303C42C23×D4311yescharacteristic
304C42C23×D4311yescharacteristic
305C42C23×D4316yes—
306C42C23×D43112yes—
307C42C23×D43112yes—
308C42C23×D4312yes—
309C42C23×D4316yes—
310C23—46no—
311C23C22×Dic86112yes—
312C23C24×D4311yescharacteristic
313C23C24×D4316yes—
314Q8—8632no—
315Q8—17248no—
316Q8C22×Dic86116yes—
317D4C22×Dic86148yes—
318C2×C4—24no—
319C2×C4—224no—
320C2×C4—412no—
321C2×C4—46no—
322C2×C4—4332no—
323C2×C4—17248no—
324C2×C4C22×Dic8614yes—
325C2×C4C22×Dic86112yes—
326C2×C4C22×Dic86124yes—
327C2×C4C22×Dic86124yes—
328C2×C4C24×D4312yes—
329C2×C4C24×D43112yes—
330C2×C4C24×D4316yes—
331C2×C4C24×D4311yescharacteristic
332C2×C4C24×D4312yes—
333C2×C4C24×D4311yescharacteristic
334C22—22no—
335C22—212no—
336C22—412no—
337C22C23×Dic8612yes—
338C22C23×Dic86112yes—
339C22C25×D4311yescharacteristic
340C4—22no—
341C4—212no—
342C4—8632no—
343C4C23×Dic8612yes—
344C4C23×Dic8612yes—
345C4C23×Dic86112yes—
346C4#1458373514yes—
347C2—22no—
348C2—212no—
349C2C24×Dic8611yescharacteristic
350C2C2×#1458373512yes—
subgroupG/H
centerC2×C4C24×D43
commutator, socle, cosocleC22×C43C26
FrattiniC22C25×D43
FittingC2×C4×#41×C43C2
prime radicalC22×#41×C43C22
hypercenterC2×C4×#41D43
hypocenterC43#62813
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #19148421C2semidirect
2. #19148414C2semidirect
3. #19148405C2semidirect
4. #19148398C2semidirect
5. #41×Dic86C2semidirect
6. #19148000C2semidirect
7. C2×#14583456C2semidirect
10. Q8×Dic86C22semidirect
11. #14583628C22semidirect
12. #14583625C22semidirect
13. #14583608C22semidirect
14. #14583606C22semidirect
15. D4×Dic86C22semidirect
17. #14583527C22semidirect
22. #14583456C22semidirect
23. #14583456C22semidirect
24. #14583456C22semidirect
25. #14583452C22semidirect
26. C2×#13063325C22semidirect
27. C2×#13063317C22semidirect
33. C2×C4×Dic86C22semidirect
73. C2×#41×C43Q8semidirect
95. #41×C43C2×Q8semidirect
96. #41×C43C2×Q8semidirect
97. C2×Q8×C43C2×Q8semidirect
99. C2×D4×C43C2×Q8semidirect
102. C22×C4×C43C2×Q8semidirect
218. C43#62813semidirect
219. C2×#41Dic86semidirect
271. #41C2×Dic86semidirect
272. #41C2×Dic86semidirect
276. C2×Q8C2×Dic86semidirect
278. C2×D4C2×Dic86semidirect
282. C22×C4C2×Dic86semidirect

Maximal subgroups

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

subgroupindex
#191484212
#191484142
#191484052
#191483982
#41×Dic862
#191480002
C2×#145834562
#191477982
C2×C4×#41×C432
#6281343