Finite Group Databasebeta

← #37546712

Subgroups — #37546712

Order 23168.

372 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042011C211yesmaximal, characteristic, complemented
2#26042001C211yesmaximal, characteristic, complemented
3#26041664C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041650C211yesmaximal, characteristic, complemented
6#26041504C211yesmaximal, characteristic, complemented
7#26041496C211yesmaximal, characteristic, complemented
8#26041495C211yesmaximal, characteristic, complemented
9#26041487C211yesmaximal, characteristic, complemented
10#26041473C211yesmaximal, characteristic
11#26041403C211yesmaximal, characteristic, complemented
12#26041370C211yesmaximal, characteristic, complemented
13C4×#15668436C211yesmaximal, characteristic, complemented
14#26041364C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668438C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×D1448C2211yescharacteristic
22C2×#15668434C2211yescharacteristic
23#19153814C2211yescharacteristic
24#19153812C2211yescharacteristic
25#19153811—21no—
26#19153810—21no—
27#19153805C2211yescharacteristic
28#19153805C2211yescharacteristic
29#19153804C2211yescharacteristic
30#19153802—21no—
31#19153800—21no—
32#19153798—21no—
33#19153794—21no—
34Q8×Dic181C2211yescharacteristic
35C4×Dic362C2211yescharacteristic
36#19153781C2211yescharacteristic
37C4×D724C2211yescharacteristic
38D4×Dic181C2211yescharacteristic
39#19153778C2211yescharacteristic
40#19153776—21no—
41#19153773—21no—
42#19153747C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153736C2211yescharacteristic
49C4×#13089678C2211yescharacteristic
50C8×Dic181C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362D411yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68#15668439—41no—
69#15668438—41no—
70#15668437—22no—
71#15668436—22no—
72#15668436—22no—
73D1448—41no—
74#15668434—41no—
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C4⋊Dic181—21no—
78C4⋊Dic181D411yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85C22⋊Dic181—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97C4×Dic181C2311yescharacteristic
98C2×Q8×C181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181C2311yescharacteristic
101Q16×C181—41no—
102SD16×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C22⋊C4×C181—21no—
108C22⋊C4×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113Dic362—22no—
114Dic362—21no—
115Dic362—21no—
116D724—21no—
117D724—21no—
118D724—21no—
119D724—21no—
120C22⋊D181—42no—
121C22⋊D181—81no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—81no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—42no—
130C4×D181—81no—
131#13089678—21no—
132#13089678#4112yes—
133C23×C181—21no—
134Q8×C181—21no—
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—21no—
138D4×C181—21no—
139D4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—21no—
147C8×C181—21no—
148C22×C181—41no—
149C22×C181—41no—
150C22×C181—41no—
151C22×C181C22×D411yescharacteristic
152C2×D181—41no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181C2×#4111yescharacteristic
164C4×C181H(5,𝔽2)11yescharacteristic
165Dic181—21no—
166Dic181—21no—
167Dic181—21no—
168Dic181—41no—
169Dic181—41no—
170Dic181#13512yes—
171C2×C181—81no—
172C2×C181#55011yescharacteristic
173C2×C181C2×#13511yescharacteristic
174C2×C181#57711yescharacteristic
175D181—81no—
176D181—81no—
177C181#339411yescharacteristic, complemented, Sylow
178#3394—1811nomaximal, Sylow
179#559—1811no—
180#556—1811no—
181#495—1811no—
182#495—1811no—
183#492—1811no—
184#464—1811no—
185#464D18111yescharacteristic, complemented
186#463—1811no—
187#463—1811no—
188#462—1811no—
189#459—1811no—
190C8×Q8—1811no—
191C4×SD16—1811no—
192C4×SD16—1811no—
193C4×SD16—1811no—
194C2×Q16—1811no—
195C2×Q16C2×D18111yescharacteristic
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16—1811no—
199C2×SD16—1811no—
200C2×SD16C2×D18111yescharacteristic
201C2×D8—1811no—
202C2×D8—1811no—
203C4⋊Q8—1811no—
204#127—1811no—
205#126—3621no—
206#126—3621no—
207#124—3621no—
208#124—1811no—
209#124—1811no—
210#124—1811no—
211#124C2×D18111yescharacteristic
212#123—3621no—
213#123—3621no—
214C22⋊Q8—3621no—
215C4×Q8—1811no—
216C4×Q8—1811no—
217C4×Q8—1811no—
218C4×Q8C2×D18111yescharacteristic
219C4×D4—3621no—
220C4×D4—1811no—
221C4×D4—1811no—
222C4×D4—1811no—
223#117—3621no—
224#106—1811no—
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227C4⋊C8C2×D18111yescharacteristic
228#103—1811no—
229#103—1811no—
230#103C2×D18111yescharacteristic
231#102—1811no—
232#102—1811no—
233#102C2×D18111yescharacteristic
234C4×C8—1811no—
235C4×C8—1811no—
236C4×C8—1811no—
237C2×Q8—1811no—
238C2×Q8—1811no—
239C2×Q8C22×D18111yescharacteristic
240C2×Q8C22×D18111yescharacteristic
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—1811no—
244C2×D4—1811no—
245C2×D4C22×D18111yescharacteristic
246C22×C4—3621no—
247C22×C4—3621no—
248C22×C4—3621no—
249Q16—41no—
250Q16—7241no—
251SD16—41no—
252SD16—3622no—
253SD16—3622no—
254SD16—3622no—
255SD16—7241no—
256D8—7241no—
257D8—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8C22×D18111yescharacteristic
261C2×C8C22×D18111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3621no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4—1811no—
270C4⋊C4—1811no—
271C4⋊C4C22×D18111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3621no—
275C22⋊C4—3621no—
276C22⋊C4—3621no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C42—21no—
284C42—3621no—
285C42—3621no—
286C42—1811no—
287C42—1811no—
288C42C22×D18111yescharacteristic
289C23—21no—
290C23—3621no—
291C23—3621no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—3622no—
297Q8—3621no—
298Q8—3621no—
299D4—21no—
300D4—21no—
301D4—3621no—
302D4—3621no—
303D4—3621no—
304D4—3621no—
305D4—14481no—
306D4—7242no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—14481no—
313C2×C4—14481no—
314C2×C4—1811no—
315C2×C4—1811no—
316C2×C4—1811no—
317C2×C4—1811no—
318C2×C4—7242no—
319C2×C4D4×D18111yescharacteristic
320C2×C4D4×D18111yescharacteristic
321C2×C4C23×D18111yescharacteristic
322C8—21no—
323C8—21no—
324C8—3621no—
325C8—1812no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—7241no—
330C22—7241no—
331C22—7241no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22C2×D4×D18111yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3621no—
342C4—3621no—
343C4—3621no—
344C4—1812no—
345C4—7241no—
346C4—7241no—
347C4#41×D18111yescharacteristic
348C4#1915386111yescharacteristic
349C2—81no—
350C2—14481no—
351C2—14481no—
352C2#2604194711yescharacteristic
353C2#135×D18111yescharacteristic
354C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radical#26041473C2
hypercenter#464D181
hypocenterC181#3394
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #26042011C2semidirect
2. #26042001C2semidirect
3. #26041664C2semidirect
4. #26041661C2semidirect
5. #26041650C2semidirect
6. #26041504C2semidirect
7. #26041496C2semidirect
8. #26041495C2semidirect
9. #26041487C2semidirect
11. #26041403C2semidirect
12. #26041370C2semidirect
13. C4×#15668436C2semidirect
14. #26041364C2semidirect
15. #464×C181C2semidirect
177. C181#3394semidirect
185. #464D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
#260420012
#260416642
#260416612
#260416502
#260415042
#260414962
#260414952
#260414872
#260414732
#260414032
#260413702
C4×#156684362
#260413642
#464×C1812
#3394181