Finite Group Databasebeta

← #37543272

Subgroups — #37543272

Order 23168.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2C2×#19153835C211yesmaximal, characteristic
3#26041849C211yesmaximal, characteristic, complemented
4#26041758C211yesmaximal, characteristic, complemented
5#26041500C211yesmaximal, characteristic, complemented
6#26041498C211yesmaximal, characteristic, complemented
7#26041492C211yesmaximal, characteristic, complemented
8#26041490C211yesmaximal, characteristic, complemented
9#26041488C211yesmaximal, characteristic, complemented
10#26041484C211yesmaximal, characteristic, complemented
11#26041482C211yesmaximal, characteristic, complemented
12#26041304C211yesmaximal, characteristic, complemented
13#26041277C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#463×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19#19153845—24no—
20#19153835—24no—
21C2×Dic724C2211yescharacteristic
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153816C2211yescharacteristic
30#19153814C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153798—21no—
33#19153798—21no—
34#19153793—21no—
35#19153793—21no—
36C4×Dic362C2211yescharacteristic
37#19153781C2211yescharacteristic
38D4×Dic181C2211yescharacteristic
39C4×C22⋊D181—21no—
40#19153776—21no—
41#19153773C2211yescharacteristic
42#19153772—21no—
43C2×#15668428—21no—
44C4⋊C4×D181C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic
46#19153745C2211yescharacteristic
47#19153745C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153741C2211yescharacteristic
50#19153740C2211yescharacteristic
51#19153739C2211yescharacteristic
52#19153739C2211yescharacteristic
53#19153737C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C2×D8×C181C2211yescharacteristic
56#124×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58C4⋊C8×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#102×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—22no—
63#15668448—22no—
64#15668448D414yes—
65C2×Dic362—21no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69Q8×D181—22no—
70Q8×D181—22no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C22×Dic181—21no—
74C22×Dic181—21no—
75C22×Dic181—21no—
76C2×C4×D181—21no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79Dic724—22no—
80Dic724—22no—
81#15668439—22no—
82#15668437—22no—
83#15668437—22no—
84#15668437—22no—
85#15668432—22no—
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C8×D181—22no—
89C4⋊Dic181—21no—
90C4⋊Dic181—21no—
91C4⋊Dic181C2311yescharacteristic
92#15668428—21no—
93#15668428—21no—
94#15668428—21no—
95#15668428—21no—
96#15668428—21no—
97#15668428D411yescharacteristic
98#15668428D411yescharacteristic
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102#15668426D411yescharacteristic
103#15668426D411yescharacteristic
104C4×Dic181—21no—
105C4×Dic181—21no—
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181C2311yescharacteristic
109C2×D4×C181C2311yescharacteristic
110C22×C4×C181—21no—
111SD16×C181—22no—
112D8×C181—22no—
113C2×C8×C181C2311yescharacteristic
114C2×C8×C181C2311yescharacteristic
115C4⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181C2×D411yescharacteristic
121Dic362—21no—
122Dic362—21no—
123Dic362—21no—
124Dic362—21no—
125Dic362—21no—
126Dic362—42no—
127Dic362C2×D412yes—
128C22⋊D181—24no—
129C22⋊D181—42no—
130C2×Dic181—21no—
131C2×Dic181—24no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—42no—
136C2×Dic181—81no—
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C2×Dic181C2×D411yescharacteristic
140C4×D181—42no—
141C4×D181—81no—
142C4×D181C2×D412yes—
143C4×D181#4112yes—
144#13089678—21no—
145#13089678—21no—
146C23×C181—21no—
147C23×C181—21no—
148Q8×C181—21no—
149Q8×C181—21no—
150D4×C181—21no—
151D4×C181—21no—
152D4×C181—21no—
153D4×C181C2×D412yes—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—81no—
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2411yescharacteristic
160C8×C181—21no—
161C8×C181—21no—
162C22×C181—22no—
163C22×C181—41no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181C22×D411yescharacteristic
169C2×D181—21no—
170C2×D181—21no—
171C2×D181#12112yes—
172C4×C181—21no—
173C4×C181—21no—
174C4×C181—41no—
175C4×C181—41no—
176C4×C181C22×D411yescharacteristic
177C4×C181C2×#4111yescharacteristic
178Dic181—22no—
179Dic181—21no—
180Dic181—21no—
181Dic181—41no—
182Dic181—41no—
183Dic181—41no—
184Dic181#12112yes—
185C2×C181—42no—
186C2×C181—81no—
187C2×C181C2×#12111yescharacteristic
188C2×C181#57711yescharacteristic
189C2×C181#57811yescharacteristic
190D181—22no—
191C181#304611yescharacteristic, complemented, Sylow
192#3046—1811nomaximal, Sylow
193C2×#137—1811no—
194C2×#135—1811no—
195#536—1811no—
196#520—1811no—
197#464—1811no—
198#464—1811no—
199#463—1811no—
200#463D18111yescharacteristic, complemented
201#462—1811no—
202#462—1811no—
203#461—1811no—
204#461—1811no—
205#424—1811no—
206#417—1811no—
207C2×#103—1811no—
208C2×#41—1811no—
209C2×#41—1811no—
210C22×Q8—1811no—
211#137—3624no—
212#135—3624no—
213C2×Q16—1811no—
214C2×Q16—1811no—
215C2×Q16—1811no—
216C2×SD16—1811no—
217C2×SD16—1811no—
218C2×SD16—1811no—
219C2×SD16C2×D18111yescharacteristic
220C2×D8C2×D18111yescharacteristic
221C2×M16—1811no—
222C22×C8—1811no—
223C4⋊Q8—1811no—
224C4⋊Q8—1811no—
225#124—1811no—
226#124C2×D18111yescharacteristic
227#123—3621no—
228#123—3621no—
229C22⋊Q8—3621no—
230C22⋊Q8—3621no—
231C4×Q8—1811no—
232C4×Q8—1811no—
233C4×D4—3621no—
234C4×D4—3621no—
235C4×D4—1811no—
236C4×D4C2×D18111yescharacteristic
237#117—3621no—
238#117—1811no—
239C2×C4⋊C4—3621no—
240C2×C4⋊C4—1811no—
241C4⋊C8—1811no—
242C4⋊C8—1811no—
243C4⋊C8—1811no—
244C4⋊C8C2×D18111yescharacteristic
245#103—1811no—
246#103—1811no—
247#103—1811no—
248#103—1811no—
249#103—1811no—
250#103C2×D18111yescharacteristic
251#102—1811no—
252#102C2×D18111yescharacteristic
253#41—3622no—
254#41—3622no—
255#41—3622no—
256#41—1814no—
257C2×Q8—3621no—
258C2×Q8—3622no—
259C2×Q8—3622no—
260C2×Q8—1811no—
261C2×Q8—1811no—
262C2×Q8—1811no—
263C2×Q8C22×D18111yescharacteristic
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4C22×D18111yescharacteristic
267C2×D4C22×D18111yescharacteristic
268C22×C4—21no—
269C22×C4—3621no—
270C22×C4—3621no—
271C22×C4—3621no—
272C22×C4—3621no—
273C22×C4—3621no—
274C22×C4—1811no—
275Q16—3622no—
276Q16—3622no—
277Q16—3622no—
278SD16—22no—
279SD16—3622no—
280SD16—3622no—
281SD16—3622no—
282D8—22no—
283M16—3622no—
284C2×C8—3622no—
285C2×C8—1811no—
286C2×C8—1811no—
287C2×C8C22×D18111yescharacteristic
288C2×C8C22×D18111yescharacteristic
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—3621no—
292C4⋊C4—3621no—
293C4⋊C4—3621no—
294C4⋊C4—3621no—
295C4⋊C4—3621no—
296C4⋊C4—1811no—
297C4⋊C4—1811no—
298C4⋊C4—1811no—
299C4⋊C4C22×D18111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—1811no—
307C22⋊C4—1811no—
308C42—3621no—
309C42—3621no—
310C42—1811no—
311C42C22×D18111yescharacteristic
312C23—21no—
313C23—21no—
314C23—1811no—
315Q8—21no—
316Q8—21no—
317Q8—3621no—
318Q8—3621no—
319Q8—3621no—
320Q8—3621no—
321Q8—3621no—
322Q8—1812no—
323Q8—7242no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—3624no—
328D4—7242no—
329D4D4×D18112yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—3621no—
334C2×C4—3624no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—14481no—
339C2×C4—14481no—
340C2×C4—1811no—
341C2×C4—1811no—
342C2×C4—1811no—
343C2×C4—1812no—
344C2×C4—1812no—
345C2×C4—7242no—
346C2×C4—7242no—
347C2×C4D4×D18111yescharacteristic
348C2×C4D4×D18111yescharacteristic
349C2×C4C23×D18111yescharacteristic
350C8—21no—
351C8—21no—
352C8—3621no—
353C8—3621no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3621no—
361C22—3621no—
362C22—1812no—
363C22C2×D4×D18111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3622no—
369C4—3621no—
370C4—3621no—
371C4—1812no—
372C4—7241no—
373C4—7241no—
374C4—7241no—
375C4C2×D4×D18111yescharacteristic
376C4#41×D18111yescharacteristic
377C2—42no—
378C2—81no—
379C2—3622no—
380C2#121×D18111yescharacteristic
381C2#2604208911yescharacteristic
382C2#2604209311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#463×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153835C2
hypercenter#463D181
hypocenterC181#3046
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
3. #26041849C2semidirect
4. #26041758C2semidirect
5. #26041500C2semidirect
6. #26041498C2semidirect
7. #26041492C2semidirect
8. #26041490C2semidirect
9. #26041488C2semidirect
10. #26041484C2semidirect
11. #26041482C2semidirect
12. #26041304C2semidirect
13. #26041277C2semidirect
14. #103×D181C2semidirect
15. #463×C181C2semidirect
191. C181#3046semidirect
200. #463D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
C2×#191538352
#260418492
#260417582
#260415002
#260414982
#260414922
#260414902
#260414882
#260414842
#260414822
#260413042
#260412772
#103×D1812
#463×C1812
#3046181