Finite Group Databasebeta

← #25100637

Subgroups — #25100637

Order 11008.

1562 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147928C211yesmaximal, characteristic
2C2×#14582917C212yesmaximal, complemented
3C22×#13063253C212yesmaximal, complemented
4#19143014C211yesmaximal, characteristic, complemented
5#19135513C218yesmaximal, complemented
6C43×#2296C211yesmaximal, characteristic, complemented
7C22×#10621898C2212yes—
8C22×#10621877C2212yes—
9#14583480—22no—
10#14583475C414yescomplemented
11#14583468C414yescomplemented
12#14583461—22no—
13#14583449C2214yes—
14#14583411—24no—
15#14582917—28no—
16C2×#13063253—28no—
17C2×#13063253C2218yescomplemented
18C2×#13063253C2218yescomplemented
19#14582855—24no—
20C4×#10621877C2214yescomplemented
21C2×#117×C43C2211yescharacteristic
22C2×#117×C43C2211yescharacteristic
23C22×C42×C43C2211yescharacteristic
24C43×#389—22no—
25C43×#385—22no—
26C4×C4⋊C4×C43C2214yescomplemented
27C2×#10621898—22no—
28C2×#10621898—22no—
29C2×#10621898—24no—
30C2×#10621898—24no—
31C2×#10621898—24no—
32C2×#10621898—24no—
33C2×#10621898C2×C414yes—
34C2×#10621898C2314yes—
35C22×Dic86C2×C412yes—
36C22×D172C2×C412yes—
37C22×C22⋊D43—22no—
38C22×C4×D43—22no—
39C2×#10621877—22no—
40C2×#10621877—24no—
41C2×#10621877D418yescomplemented
42C2×#10621877C2314yes—
43C22×C43⋊C8—22no—
44#13063325C2×C414yescomplemented
45#13063322C2×C414yescomplemented
46#13063313C2×C418yescomplemented
47#13063309—28no—
48#13063305—28no—
49#13063300—28no—
50#13063299—28no—
51#13063295—28no—
52C4×D172—28no—
53#13063253—216no—
54#13063253—216no—
55#13063236—24no—
56C4×C43⋊C8—24no—
57C23×C4×C43—21no—
58C23×C4×C43C2311yescharacteristic
59#117×C43—24no—
60#117×C43C2×C414yes—
61#117×C43D414yes—
62C2×C4⋊C4×C43—21no—
63C2×C4⋊C4×C43—24no—
64C2×C4⋊C4×C43D412yes—
65C2×C22⋊C4×C43—21no—
66C2×C22⋊C4×C43D412yes—
67C2×C42×C43—24no—
68C2×C42×C43—21no—
69C2×C42×C43C2312yes—
70C2×C42×C43C2312yes—
71C2×C42×C43C2312yes—
72#95×C43—24no—
73C24×C43C2×D411yescharacteristic
74C23×D43—22no—
75C22×C4×C43—22no—
76C22×C4×C43—22no—
77C22×C4×C43—44no—
78C22×C4×C43—42no—
79C22×C4×C43C22×C412yes—
80C22×C4×C43C2×D412yes—
81C22×C4×C43C2×D414yes—
82C22×C4×C43C2×D414yes—
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43#4114yes—
85C22×C4×C43C2411yescharacteristic
86C2×C22⋊D43—24no—
87C2×C22⋊D43—24no—
88C2×C22⋊D43—44no—
89C2×C22⋊D43—44no—
90C22×Dic43—22no—
91#10621898—28no—
92#10621898—28no—
93#10621898—24no—
94#10621898—24no—
95#10621898—24no—
96#10621898—24no—
97#10621898—44no—
98#10621898—44no—
99#10621898—44no—
100#10621898—44no—
101C2×D172—22no—
102C2×D172—28no—
103C2×D172—24no—
104C2×D172—24no—
105C2×D172C22×C414yes—
106C2×C4×D43—24no—
107C2×C4×D43—24no—
108C2×C4×D43—44no—
109C2×C4×D43—44no—
110C2×Dic86—24no—
111C2×Dic86—24no—
112C2×Dic86—22no—
113C2×Dic86C22×C414yes—
114C4⋊C4×C43—48no—
115C4⋊C4×C43—42no—
116C4⋊C4×C43C22⋊C418yes—
117C22⋊C4×C43—42no—
118C22⋊C4×C43C22⋊C418yes—
119C42×C43—24no—
120C42×C43—24no—
121C42×C43—42no—
122C42×C43C22×C414yes—
123C42×C43C2×D414yes—
124(C2×D43)⋊C4—216no—
125C4⋊Dic43—28no—
126#10621877—28no—
127#10621877—24no—
128#10621877C2×D418yes—
129C2×C43⋊C8—28no—
130C2×C43⋊C8—24no—
131C23×C43—42no—
132C23×C43C2×C22⋊C412yes—
133C23×C43#12412yes—
134C23×C43#12712yes—
135C23×C43C22×D411yescharacteristic
136C22×D43—24no—
137C22×D43—44no—
138C22×D43—84no—
139C2×C4×C43—24no—
140C2×C4×C43—44no—
141C2×C4×C43—42no—
142C2×C4×C43—48no—
143C2×C4×C43—42no—
144C2×C4×C43—44no—
145C2×C4×C43—42no—
146C2×C4×C43C2×C22⋊C418yes—
147C2×C4×C43C2×C22⋊C412yes—
148C2×C4×C43C4×D414yes—
149C2×C4×C43C4×D414yes—
150C2×C4×C43#12114yes—
151C2×C4×C43#12114yes—
152C2×C4×C43#12412yes—
153C2×C4×C43#12712yes—
154C2×C4×C43C23×C411yescharacteristic
155C2×C4×C43C22×D412yes—
156C2×C4×C43C22×D411yescharacteristic
157C2×C4×C43C2×#4112yes—
158C22⋊D43—44no—
159C22⋊D43—44no—
160C22⋊D43—48no—
161C2×Dic43—24no—
162C2×Dic43—44no—
163D172—24no—
164D172—24no—
165D172—44no—
166D172—88no—
167C4×D43—44no—
168C4×D43—44no—
169C4×D43—48no—
170Dic86—24no—
171Dic86—24no—
172Dic86—44no—
173C43⋊C8—28no—
174#21253—431nomaximal, Sylow
175C22×C43—24no—
176C22×C43—22no—
177C22×C43—42no—
178C22×C43—42no—
179C22×C43#39014yes—
180C22×C43C22×C22⋊C411yescharacteristic
181C22×C43C2×#12411yescharacteristic
182C22×C43C2×#12711yescharacteristic
183C2×D43—48no—
184C2×D43—44no—
185C2×D43—88no—
186C4×C43—28no—
187C4×C43—44no—
188C4×C43—82no—
189C4×C43#39014yes—
190C4×C43C2×C4×D412yes—
191C4×C43C2×#12112yes—
192Dic43—44no—
193#3449—431no—
194C2×#421—432no—
195C22×#104—432no—
196#2873—431no—
197#2296D4311yescharacteristic, complemented
198#1980—438no—
199C2×C43—21no—
200C2×C43—24no—
201C2×C43—21no—
202C2×C43C2×#39011yescharacteristic
203C2×C43#290112yes—
204D43—84no—
205C22×#41—432no—
206C22×M16—432no—
207#537—862no—
208#536—434no—
209#535—434no—
210#534—862no—
211#532—434no—
212#525—864no—
213C2×#117C2×D4311yescharacteristic
214C2×#117C2×D4311yescharacteristic
215C22×C42C2×D4311yescharacteristic
216#421—868no—
217C2×#104—438no—
218C2×#104—438no—
219C2×#104—868no—
220#405—864no—
221C4×M16—434no—
222#389—22no—
223#385—22no—
224C4×C4⋊C4C2×D4314yescomplemented
225C43#2125311yescharacteristic, complemented, Sylow
226C2×#41—434no—
227C2×#41—434no—
228C2×#41—862no—
229C2×#41—862no—
230C2×#41—864no—
231C2×#41—864no—
232C2×#41—864no—
233C2×#41—864no—
234C22×Q8—432no—
235C22×D4—432no—
236C22×D4—862no—
237C23×C4—21no—
238C23×C4—862no—
239C23×C4C22×D4311yescharacteristic
240C2×M16—438no—
241C2×M16—434no—
242C2×M16—862no—
243C2×M16—864no—
244C22×C8—862no—
245C4⋊Q8—434no—
246#127—434no—
247#124—438no—
248#123—868no—
249C22⋊Q8—868no—
250#121—868no—
251#121—868no—
252#120—868no—
253C4×D4—868no—
254#117—24no—
255#117C4×D4314yes—
256#117D17214yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4C22⋊D4312yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C22⋊D4312yes—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42C22×D4312yes—
265C2×C42C22×D4312yes—
266C2×C42C22×D4312yes—
267#104—8616no—
268#104—8616no—
269#97—864no—
270C4×C8—864no—
271#95—24no—
272C24—862no—
273C24C2×C22⋊D4311yescharacteristic
274#41—868no—
275#41—868no—
276#41—864no—
277#41—864no—
278#41—864no—
279#41—864no—
280#41—1724no—
281#41—1724no—
282#41—1724no—
283#41—1724no—
284C2×Q8—434no—
285C2×Q8—862no—
286C2×Q8—864no—
287C2×Q8—864no—
288C2×D4—434no—
289C2×D4—864no—
290C2×D4—864no—
291C2×D4—868no—
292C2×D4—862no—
293C2×D4—864no—
294C2×D4—864no—
295C2×D4—1724no—
296C2×D4—1724no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—862no—
302C22×C4—864no—
303C22×C4—864no—
304C22×C4—1724no—
305C22×C4—1724no—
306C22×C4C2×C4×D4312yes—
307C22×C4C2×D17212yes—
308C22×C4D4×D4314yes—
309C22×C4#1062190214yes—
310C22×C4C2×C22⋊D4314yes—
311C22×C4C2×C22⋊D4311yescharacteristic
312C22×C4C23×D4311yescharacteristic
313M16—438no—
314M16—868no—
315M16—864no—
316C2×C8—868no—
317C2×C8—864no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—868no—
321C4⋊C4(C2×D43)⋊C418yes—
322C22⋊C4—42no—
323C22⋊C4—8616no—
324C22⋊C4(C2×D43)⋊C418yes—
325C42—24no—
326C42—24no—
327C42—42no—
328C42C2×C4×D4314yes—
329C42C2×D17214yes—
330C23—42no—
331C23—864no—
332C23—3444no—
333C23—1724no—
334C23C2×(C2×D43)⋊C412yes—
335C23#1306331412yes—
336C23#1306332112yes—
337C23C22×C22⋊D4311yescharacteristic
338Q8—864no—
339Q8—864no—
340Q8—1724no—
341D4—864no—
342D4—864no—
343D4—3448no—
344D4—1728no—
345D4—1724no—
346D4—1724no—
347D4—1724no—
348C2×C4—24no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—864no—
356C2×C4—1724no—
357C2×C4—1728no—
358C2×C4—1724no—
359C2×C4—1724no—
360C2×C4C2×(C2×D43)⋊C418yes—
361C2×C4C2×(C2×D43)⋊C412yes—
362C2×C4#1306328714yes—
363C2×C4#1306328714yes—
364C2×C4#1306330014yes—
365C2×C4#1306330014yes—
366C2×C4#1306331412yes—
367C2×C4#1306332112yes—
368C2×C4C22×C4×D4311yescharacteristic
369C2×C4C2×D4×D4312yes—
370C2×C4C22×D17211yescharacteristic
371C2×C4C2×#1062190212yes—
372C8—868no—
373C22—22no—
374C22—24no—
375C22—42no—
376C22—42no—
377C22—3448no—
378C22—1724no—
379C22—1728no—
380C22#1458280014yes—
381C22C22×(C2×D43)⋊C411yescharacteristic
382C22C2×#1306331411yescharacteristic
383C22C2×#1306332111yescharacteristic
384C4—28no—
385C4—44no—
386C4—82no—
387C4—1724no—
388C4#1458280014yes—
389C4C2×#1306328712yes—
390C4C2×#1306330012yes—
391C2—21no—
392C2—24no—
393C2—21no—
394C2—3444no—
395C2C2×#1458280011yescharacteristic
396C2#1914314312yes—
subgroupG/H
centerC2×C4C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#2296C2
socleC22×C43C22×C22⋊C4
prime radical#19147928C2
cosocleC22×C4×C43C24
hypercenter#2296D43
hypocenterC43#21253
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. C2×#14582917C2semidirect
3. C22×#13063253C2semidirect
4. #19143014C2semidirect
5. #19135513C2semidirect
6. C43×#2296C2semidirect
10. #14583475C4semidirect
11. #14583468C4semidirect
17. C2×#13063253C22semidirect
18. C2×#13063253C22semidirect
20. C4×#10621877C22semidirect
26. C4×C4⋊C4×C43C22semidirect
41. C2×#10621877D4semidirect
44. #13063325C2×C4semidirect
45. #13063322C2×C4semidirect
46. #13063313C2×C4semidirect
197. #2296D43semidirect
224. C4×C4⋊C4C2×D43semidirect
225. C43#21253semidirect

Maximal subgroups

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

subgroupindex
#191479282
C2×#145829172
C22×#130632532
#191430142
#191355132
C43×#22962
#2125343