Finite Group Databasebeta

← #37545894

Subgroups — #37545894

Order 23168.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041975C211yesmaximal, characteristic
2D4×Dic362C211yesmaximal, characteristic, complemented
3#26041648C211yesmaximal, characteristic, complemented
4#26041621C211yesmaximal, characteristic, complemented
5#26041615C211yesmaximal, characteristic, complemented
6#26041587C211yesmaximal, characteristic, complemented
7#26041525C211yesmaximal, characteristic, complemented
8#26041508C211yesmaximal, characteristic, complemented
9#26041485C211yesmaximal, characteristic, complemented
10#26041420C211yesmaximal, characteristic, complemented
11#26041420C211yesmaximal, characteristic, complemented
12#26041381C211yesmaximal, characteristic, complemented
13#26041289C211yesmaximal, characteristic, complemented
14C2×#19153741C211yesmaximal, characteristic, complemented
15#435×C181C211yesmaximal, characteristic, complemented
16C22×Dic362C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668438C2211yescharacteristic
21#19153816C2211yescharacteristic
22#19153815—21no—
23#19153811—21no—
24#19153808C2211yescharacteristic
25#19153798—21no—
26#19153795C2211yescharacteristic
27#19153795C2211yescharacteristic
28#19153793—21no—
29#19153793—21no—
30#19153790C2211yescharacteristic
31#19153790C2211yescharacteristic
32#19153788—21no—
33C4×Dic362C2211yescharacteristic
34C4×D724C2211yescharacteristic
35D4×Dic181—21no—
36#19153776—21no—
37#19153775C2211yescharacteristic
38C2×C4⋊Dic181C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153750C2211yescharacteristic
41#19153748C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153741C2211yescharacteristic
50#19153741C2211yescharacteristic
51#19153738C2211yescharacteristic
52C2×M16×C181C2211yescharacteristic
53C22×C8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55C4⋊C8×C181C2211yescharacteristic
56C22⋊C8×C181C2211yescharacteristic
57C22⋊C8×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362—22no—
60C2×Dic362—22no—
61C2×Dic362—21no—
62C2×Dic362—21no—
63C2×Dic362—41no—
64C2×Dic362—41no—
65C2×Dic362C2311yescharacteristic
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C22×Dic181—21no—
72C22×Dic181—21no—
73C22×Dic181—21no—
74C2×C4×D181—21no—
75#15668438—22no—
76#15668438—41no—
77#15668438—41no—
78C4⋊Dic181—21no—
79C4⋊Dic181—21no—
80C4⋊Dic181—21no—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83C4⋊Dic181C2311yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85#15668428—21no—
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96C4×Dic181—21no—
97C4×Dic181—21no—
98C2×D4×C181D411yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101M16×C181—41no—
102C2×C8×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362—42no—
118Dic362—42no—
119Dic362#4112yes—
120D724—21no—
121D724—21no—
122C22⋊D181—42no—
123C22⋊D181—81no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—21no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181—21no—
131C2×Dic181—42no—
132C2×Dic181—42no—
133C2×Dic181—81no—
134C4×D181—81no—
135C23×C181C2×D411yescharacteristic
136C23×C181C2×D411yescharacteristic
137D4×C181—22no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181C2×D411yescharacteristic
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2411yescharacteristic
147C8×C181—21no—
148C8×C181—21no—
149C8×C181—21no—
150C8×C181—21no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181#13712yes—
157C22×C181C22×D411yescharacteristic
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—41no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—21no—
164C4×C181—21no—
165C4×C181C2×#4111yescharacteristic
166C4×C181H(5,𝔽2)11yescharacteristic
167Dic181—22no—
168Dic181—41no—
169Dic181—41no—
170Dic181—41no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—22no—
175C2×C181—41no—
176C2×C181#54611yescharacteristic
177C2×C181C2×#13711yescharacteristic
178C2×C181#57711yescharacteristic
179D181—81no—
180C181#332011yescharacteristic, complemented, Sylow
181#3320—1811nomaximal, Sylow
182#553—1811no—
183D4×Q8—1811no—
184#491—1811no—
185#485—1811no—
186#484—1811no—
187#479—1811no—
188#468—1811no—
189#465—1811no—
190#461—1811no—
191#448—1811no—
192#448—1811no—
193#440—1811no—
194#435D18111yescharacteristic, complemented
195#419—1811no—
196C2×#103—1811no—
197C22×Q8—1811no—
198C22×Q8—1811no—
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×SD16—1811no—
202C2×M16C2×D18111yescharacteristic
203C22×C8C2×D18111yescharacteristic
204C4⋊Q8—3621no—
205C4⋊Q8—1811no—
206#126—3621no—
207#125—1811no—
208#123—3621no—
209C22⋊Q8—3621no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212C22⋊Q8—1811no—
213#121—3621no—
214#121—1811no—
215#121—1811no—
216C4×Q8—1811no—
217C4×D4—3621no—
218C4×D4—3621no—
219C4×D4—1811no—
220C4×D4C2×D18111yescharacteristic
221#117—1811no—
222C2×C4⋊C4—1811no—
223#107—1811no—
224#107—1811no—
225#106—1811no—
226C4⋊C8C2×D18111yescharacteristic
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#102—1811no—
237C22⋊C8C2×D18111yescharacteristic
238C22⋊C8C2×D18111yescharacteristic
239#97C2×D18111yescharacteristic
240C2×Q8—3622no—
241C2×Q8—3622no—
242C2×Q8—3621no—
243C2×Q8—3621no—
244C2×Q8—1811no—
245C2×Q8—1811no—
246C2×Q8—1811no—
247C2×Q8—7241no—
248C2×Q8—7241no—
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4D72411yescharacteristic
253C22×C4—3621no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4C22×D18111yescharacteristic
258C22×C4C22×D18111yescharacteristic
259SD16—3622no—
260SD16—7241no—
261SD16—7241no—
262M16—41no—
263C2×C8—41no—
264C2×C8C22×D18111yescharacteristic
265C2×C8C22×D18111yescharacteristic
266C2×C8C22×D18111yescharacteristic
267C2×C8C22×D18111yescharacteristic
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—1811no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4D72411yescharacteristic
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4D72411yescharacteristic
288C22⋊C4D72411yescharacteristic
289C42—3621no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—3621no—
293C23C2×D72411yescharacteristic
294C23C2×D72411yescharacteristic
295Q8—3621no—
296Q8—3621no—
297Q8—3621no—
298Q8—3621no—
299Q8—3621no—
300Q8—1812no—
301Q8—7242no—
302Q8—7242no—
303D4—22no—
304D4—3621no—
305D4—3621no—
306D4—14481no—
307D4—7242no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—14481no—
320C2×C4—14481no—
321C2×C4—7242no—
322C2×C4—7242no—
323C2×C4C2×D72411yescharacteristic
324C2×C4C2×D72411yescharacteristic
325C2×C4C2×D72411yescharacteristic
326C2×C4C2×D72411yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—7241no—
338C22—7241no—
339C22—7241no—
340C22#1915384912yes—
341C22C22×D72411yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—3622no—
347C4—7241no—
348C4—7241no—
349C4—7241no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4#41×D18111yescharacteristic
354C4#1915386111yescharacteristic
355C2—22no—
356C2—41no—
357C2—14481no—
358C2#2604192411yescharacteristic
359C2C2×#1915384911yescharacteristic
360C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#435×C181C2
socleC22×C181C22×D4
prime radical#26041975C2
hypercenter#435D181
hypocenterC181#3320
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. D4×Dic362C2semidirect
3. #26041648C2semidirect
4. #26041621C2semidirect
5. #26041615C2semidirect
6. #26041587C2semidirect
7. #26041525C2semidirect
8. #26041508C2semidirect
9. #26041485C2semidirect
10. #26041420C2semidirect
11. #26041420C2semidirect
12. #26041381C2semidirect
13. #26041289C2semidirect
14. C2×#19153741C2semidirect
15. #435×C181C2semidirect
180. C181#3320semidirect
194. #435D181semidirect

Maximal subgroups

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

subgroupindex
#260419752
D4×Dic3622
#260416482
#260416212
#260416152
#260415872
#260415252
#260415082
#260414852
#260414202
#260413812
#260412892
C2×#191537412
#435×C1812
#3320181