Finite Group Databasebeta

← #25112527

Subgroups — #25112527

Order 11008.

1338 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583704C211yesmaximal, characteristic
2#19143511C212yesmaximal, complemented
3#19143072C212yesmaximal, complemented
4#19135503C212yesmaximal, complemented
5#19135076C214yesmaximal, complemented
6#19134566C212yesmaximal, complemented
7#19134395C211yesmaximal, characteristic, complemented
8C43×#1813C211yesmaximal, characteristic, complemented
9C2×#13063371C2212yes—
10#14583704—24no—
11#14583704—24no—
12C2×#13063353C414yescomplemented
13C2×#13063353C2212yes—
14C2×#13063347C412yescomplemented
15C2×#13063347C2211yescharacteristic
16C22×D344C412yescomplemented
17C22×D344C2211yescharacteristic
18#14583665C2214yescomplemented
19#14583020—24no—
20#14583015—24no—
21#14582915C2214yescomplemented
22#14582905C2214yescomplemented
23#14582883C2211yescharacteristic
24#14582882—24no—
25C2×#13063241C2214yescomplemented
26#100×D43—24no—
27#14582873C2211yescharacteristic
28#14582854C2212yescomplemented
29#14582670C2212yescomplemented
30C2×#131×C43C2211yescharacteristic
31C2×#108×C43C2212yescomplemented
32C43×#413C2212yes—
33C43×#338C2212yescomplemented
34#13063371—28no—
35#13063371—44no—
36C2×#10621902—22no—
37C2×#10621898C2×C414yes—
38C2×#10621898C2312yes—
39C22×D172C2×C414yes—
40C22×D172C2312yes—
41C2×D4×D43—22no—
42C2×D4×D43—24no—
43#13063353—28no—
44#13063353—216no—
45#13063347—24no—
46#13063347—24no—
47#13063347D414yes—
48#13063347D414yes—
49C2×Dic172C2×C411yescharacteristic
50C2×#10621873C2×C414yes—
51C2×#10621873C2×C412yes—
52C2×D344—22no—
53C2×D344—22no—
54C2×D344—24no—
55C2×D344—24no—
56C2×D344C2×C412yes—
57C2×D344C2×C414yes—
58C2×D344C2×C412yes—
59C2×D344C2×C411yescharacteristic
60#13063333—28no—
61#13063331—44no—
62C2×#10621877C2312yes—
63M16×D43—28no—
64#13063284C2312yes—
65#13063251—28no—
66#13063247—44no—
67#13063244D412yes—
68#13063242D412yes—
69#13063241—28no—
70#13063240—22no—
71#13063235—22no—
72C8×Dic43—22no—
73C2×#41×C43C2311yescharacteristic
74#131×C43—22no—
75#131×C43—22no—
76C2×M16×C43C2×C412yes—
77C2×M16×C43C2314yes—
78C2×M16×C43C2311yescharacteristic
79C22×C8×C43C2×C412yes—
80C22×C8×C43C2311yescharacteristic
81#108×C43—22no—
82#108×C43—22no—
83#101×C43—22no—
84#100×C43—22no—
85C23×D43—24no—
86C23×D43—22no—
87#41×C43—22no—
88#41×C43—22no—
89C2×Q8×C43C2×D411yescharacteristic
90C2×D4×C43C2×D411yescharacteristic
91C2×D4×C43#4112yes—
92C22×C4×C43C22×C412yes—
93C22×C4×C43C2411yescharacteristic
94C2×C22⋊D43—24no—
95C2×C22⋊D43—22no—
96#10621902—48no—
97D4×D43—48no—
98D4×D43—416no—
99#10621898—24no—
100#10621898—28no—
101#10621898—44no—
102#10621898C2×D418yes—
103C2×D172—24no—
104C2×D172—22no—
105C2×D172—24no—
106C2×D172—24no—
107C2×D172—28no—
108C2×D172—44no—
109C2×D172C22×C414yes—
110C2×D172C22×C412yes—
111C2×C4×D43—22no—
112C2×C4×D43—24no—
113C2×Dic86C22×C412yes—
114M16×C43—22no—
115M16×C43—24no—
116M16×C43—24no—
117M16×C43—24no—
118C2×C8×C43—22no—
119C2×C8×C43—22no—
120C2×C8×C43—24no—
121C2×C8×C43C22×C412yes—
122C2×C8×C43C22×C411yescharacteristic
123C2×C8×C43C22×C411yescharacteristic
124C2×C8×C43C2×D412yes—
125C2×C8×C43C2×D412yes—
126C22⋊Dic43—22no—
127C4⋊Dic43—22no—
128C4×Dic43—22no—
129#10621877—22no—
130#10621877C2×D414yes—
131C2×C43⋊C8—22no—
132Dic172C22⋊C412yes—
133Dic172C22⋊C412yes—
134D344—28no—
135D344—22no—
136D344—22no—
137D344—24no—
138D344C22⋊C412yes—
139D344C22⋊C412yes—
140#10621873—28no—
141#10621873C22⋊C418yes—
142#10621872—48no—
143C8×D43—48no—
144C23×C43C4×D412yes—
145C23×C43C2×#4111yescharacteristic
146C22×D43—22no—
147C22×D43—24no—
148C22×D43—48no—
149C22×D43—44no—
150C22×D43—164no—
151C22×D43—162no—
152Q8×C43—22no—
153D4×C43—24no—
154D4×C43—22no—
155C2×C4×C43—22no—
156C2×C4×C43—22no—
157C2×C4×C43C4×D412yes—
158C2×C4×C43C23×C411yescharacteristic
159C2×C4×C43C22×D412yes—
160C2×C4×C43C22×D411yescharacteristic
161C2×C4×C43H(5,𝔽2)12yes—
162C22⋊D43—28no—
163C22⋊D43—48no—
164C2×Dic43—22no—
165C2×Dic43—42no—
166D172—24no—
167D172—22no—
168D172—24no—
169D172—48no—
170D172—44no—
171D172C2×C22⋊C414yes—
172C4×D43—28no—
173C4×D43—48no—
174Dic86—22no—
175Dic86C2×C22⋊C414yes—
176C8×C43—22no—
177C8×C43—44no—
178C8×C43C2×C22⋊C412yes—
179C8×C43C2×C22⋊C411yescharacteristic
180C8×C43C2×C22⋊C411yescharacteristic
181C43⋊C8—24no—
182#22458—431nomaximal, Sylow
183C22×C43—22no—
184C22×C43—41no—
185C22×C43—42no—
186C22×C43C2×C4×D411yescharacteristic
187C22×C43#54612yes—
188C2×D43—24no—
189C2×D43—44no—
190C2×D43—82no—
191C2×D43—84no—
192C2×D43—168no—
193C2×D43—164no—
194C4×C43—22no—
195C4×C43C22×C22⋊C411yescharacteristic
196C4×C43#51811yescharacteristic
197C4×C43D4212yes—
198Dic43—24no—
199Dic43—82no—
200C2×#576—431no—
201#2957—432no—
202#2888—432no—
203#1978—432no—
204#1903—434no—
205#1813D4311yescharacteristic, complemented
206#1795—432no—
207#1757—431no—
208C2×C43—22no—
209C2×C43—21no—
210C2×C43—42no—
211C2×C43D4×C22⋊C411yescharacteristic
212D43—44no—
213D43—84no—
214C2×H(5,𝔽2)—432no—
215#576—864no—
216#576—864no—
217C2×AGL(1,ℤ/8)—432no—
218C2×AGL(1,ℤ/8)—434no—
219C2×#135—431no—
220C2×#135—432no—
221C22×D8—431no—
222C22×D8—432no—
223#568—434no—
224C2×#131C2×D4311yescharacteristic
225#444—864no—
226#443—864no—
227C2×#108C2×D4312yescomplemented
228#421—434no—
229#418—434no—
230#413—431no—
231#413—864no—
232#413C2×D4312yes—
233C2×#100—434no—
234C2×#100—864no—
235#410—431no—
236#405—432no—
237#344—432no—
238#338C2×D4312yescomplemented
239C43#2245811yescharacteristic, complemented, Sylow
240H(5,𝔽2)—868no—
241H(5,𝔽2)—1724no—
242C2×#41—434no—
243C2×#41—432no—
244C2×#41—862no—
245C2×#41C22×D4311yescharacteristic
246C22×D4—434no—
247C22×D4—432no—
248C22×D4—862no—
249C22×D4—864no—
250AGL(1,ℤ/8)—8616no—
251AGL(1,ℤ/8)—868no—
252#135—434no—
253#135—434no—
254#135—864no—
255#135—864no—
256C2×Q16—431no—
257C2×SD16—432no—
258C2×SD16—434no—
259C2×D8—431no—
260C2×D8—432no—
261C2×D8—434no—
262C2×D8—432no—
263C2×D8—864no—
264C2×D8—864no—
265C2×D8—862no—
266C2×D8—862no—
267#131—22no—
268#131—22no—
269#131—868no—
270#131—1724no—
271C2×M16—432no—
272C2×M16—868no—
273C2×M16C4×D4312yes—
274C2×M16C22×D4314yes—
275C2×M16C22×D4311yescharacteristic
276C22×C8C4×D4312yes—
277C22×C8C22×D4311yescharacteristic
278#117—432no—
279#108—22no—
280#108—22no—
281#104—868no—
282#102—1724no—
283#101—22no—
284#101—432no—
285#100—22no—
286#100—432no—
287#100—868no—
288#99—862no—
289#97—862no—
290C4×C8—862no—
291C24—864no—
292C24—862no—
293#41—22no—
294#41—22no—
295#41—438no—
296#41—868no—
297#41—864no—
298#41—1724no—
299#41—1728no—
300C2×Q8—432no—
301C2×Q8D4×D4311yescharacteristic
302C2×D4—432no—
303C2×D4—434no—
304C2×D4—868no—
305C2×D4—862no—
306C2×D4—864no—
307C2×D4—864no—
308C2×D4—864no—
309C2×D4—862no—
310C2×D4—864no—
311C2×D4—1724no—
312C2×D4—1728no—
313C2×D4—17216no—
314C2×D4D4×D4311yescharacteristic
315C2×D4#1062190212yes—
316C22×C4—862no—
317C22×C4—864no—
318C22×C4C2×C4×D4312yes—
319C22×C4C23×D4311yescharacteristic
320Q16—432no—
321Q16—432no—
322SD16—438no—
323SD16—868no—
324D8—432no—
325D8—432no—
326D8—868no—
327D8—862no—
328D8—862no—
329D8—864no—
330M16—22no—
331M16—24no—
332M16—24no—
333M16—24no—
334M16—434no—
335M16—862no—
336M16—1728no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—862no—
341C2×C8—1728no—
342C2×C8C2×C4×D4312yes—
343C2×C8C2×C4×D4311yescharacteristic
344C2×C8C2×C4×D4311yescharacteristic
345C2×C8D4×D4312yes—
346C2×C8D4×D4312yes—
347C4⋊C4—862no—
348C22⋊C4—862no—
349C42—862no—
350C23—862no—
351C23—864no—
352C23—1728no—
353C23—1724no—
354C23—6882no—
355C23—6884no—
356C23#1306328712yes—
357C23C2×#1062190211yescharacteristic
358Q8—22no—
359Q8—434no—
360Q8—862no—
361D4—24no—
362D4—22no—
363D4—434no—
364D4—868no—
365D4—864no—
366D4—862no—
367D4—864no—
368D4—1728no—
369D4—1728no—
370D4—1724no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—862no—
374C2×C4—868no—
375C2×C4—1728no—
376C2×C4—1722no—
377C2×C4#1306328712yes—
378C2×C4C22×C4×D4311yescharacteristic
379C2×C4C2×D4×D4312yes—
380C2×C4C2×D4×D4311yescharacteristic
381C2×C4#1306337112yes—
382C8—22no—
383C8—44no—
384C8—864no—
385C8C22⋊C4×D4311yescharacteristic
386C8C22⋊C4×D4311yescharacteristic
387C8C22⋊C4×D4312yes—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—864no—
392C22—3444no—
393C22—3442no—
394C22—1724no—
395C22—6884no—
396C22—6888no—
397C22C2×#1306328711yescharacteristic
398C22#1458354012yes—
399C4—22no—
400C4—864no—
401C4—3442no—
402C4C2×C22⋊C4×D4311yescharacteristic
403C4#1458337311yescharacteristic
404C4#1458353312yes—
405C2—22no—
406C2—21no—
407C2—42no—
408C2—3444no—
409C2—1724no—
410C2#1913741511yescharacteristic
subgroupG/H
centerC2#19137415
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1813C2
socleC2×C43D4×C22⋊C4
prime radicalC2×#14583704C2
cosocleC22×C4×C43C24
hypercenter#1813D43
hypocenterC43#22458
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1813

Complements

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

NH ≅ G/Ntype
2. #19143511C2semidirect
3. #19143072C2semidirect
4. #19135503C2semidirect
5. #19135076C2semidirect
6. #19134566C2semidirect
7. #19134395C2semidirect
8. C43×#1813C2semidirect
12. C2×#13063353C4semidirect
14. C2×#13063347C4semidirect
16. C22×D344C4semidirect
18. #14583665C22semidirect
21. #14582915C22semidirect
22. #14582905C22semidirect
25. C2×#13063241C22semidirect
28. #14582854C22semidirect
29. #14582670C22semidirect
31. C2×#108×C43C22semidirect
33. C43×#338C22semidirect
205. #1813D43semidirect
227. C2×#108C2×D43semidirect
238. #338C2×D43semidirect
239. C43#22458semidirect

Maximal subgroups

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

subgroupindex
C2×#145837042
#191435112
#191430722
#191355032
#191350762
#191345662
#191343952
C43×#18132
#2245843