Finite Group Databasebeta

← #37545889

Subgroups — #37545889

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#26041982C211yesmaximal, characteristic
2Q8×D724C211yesmaximal, characteristic, complemented
3#26041649C211yesmaximal, characteristic, complemented
4#26041623C211yesmaximal, characteristic, complemented
5#26041616C211yesmaximal, characteristic, complemented
6#26041526C211yesmaximal, characteristic, complemented
7#26041509C211yesmaximal, characteristic, complemented
8#26041485C211yesmaximal, characteristic, complemented
9#26041419C211yesmaximal, characteristic, complemented
10#26041419C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041349C211yesmaximal, characteristic, complemented
13#26041286C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#479×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×#15668432C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153816—21no—
24#19153816C2211yescharacteristic
25#19153811—21no—
26#19153801—21no—
27#19153796—21no—
28#19153796—21no—
29#19153796C2211yescharacteristic
30#19153796C2211yescharacteristic
31#19153791—21no—
32#19153791C2211yescharacteristic
33#19153791C2211yescharacteristic
34C4×D724—21no—
35C4×D724C2211yescharacteristic
36C4×D724C2211yescharacteristic
37#19153778—21no—
38#19153774C2211yescharacteristic
39C4⋊C4×D181C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153741C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153727C2211yescharacteristic
52#125×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362C2311yescharacteristic
62C2×Dic362C2311yescharacteristic
63Q8×D181—22no—
64Q8×D181—22no—
65Q8×D181—41no—
66Q8×D181—41no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724D411yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2311yescharacteristic
77#15668438—41no—
78#15668438—41no—
79#15668436—22no—
80#15668432—41no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C8×D181—41no—
84C4⋊Dic181—21no—
85C4⋊Dic181—21no—
86C4⋊Dic181—21no—
87C4⋊Dic181—21no—
88C4⋊Dic181D411yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426D411yescharacteristic
98#15668426D411yescharacteristic
99C4×Dic181—21no—
100C2×Q8×C181C2311yescharacteristic
101C2×C8×C181C2311yescharacteristic
102C2×C8×C181C2311yescharacteristic
103C4⋊C4×C181—21no—
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113C22×D181C2×D411yescharacteristic
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362—21no—
118Dic362—42no—
119Dic362—42no—
120D724—22no—
121D724—21no—
122D724—21no—
123D724—42no—
124D724—81no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—21no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—21no—
133C4×D181—21no—
134C4×D181—42no—
135C4×D181—42no—
136C4×D181—81no—
137C4×D181—81no—
138#13089678—21no—
139#13089678—21no—
140Q8×C181—21no—
141Q8×C181#4112yes—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C8×C181—21no—
150C8×C181—21no—
151C22×C181C22×D411yescharacteristic
152C2×D181—21no—
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—21no—
157C2×D181—41no—
158C2×D181—41no—
159C2×D181—41no—
160C2×D181#13712yes—
161C4×C181—22no—
162C4×C181—21no—
163C4×C181—21no—
164C4×C181—41no—
165C4×C181—41no—
166C4×C181—41no—
167C4×C181C2×#4111yescharacteristic
168C4×C181H(5,𝔽2)11yescharacteristic
169Dic181—21no—
170Dic181—21no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181#54611yescharacteristic
175C2×C181C2×#13711yescharacteristic
176C2×C181#57711yescharacteristic
177D181—22no—
178D181—41no—
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#479D18111yescharacteristic, complemented
188#468—1811no—
189#465—1811no—
190#461—1811no—
191#448—1811no—
192#448—1811no—
193#440—1811no—
194#435—1811no—
195#419—1811no—
196C2×#103—1811no—
197C22×Q8—1811no—
198C22×Q8—1811no—
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×SD16—1811no—
202C2×M16—1811no—
203C22×C8—1811no—
204C4⋊Q8—3621no—
205C4⋊Q8—1811no—
206#126—3621no—
207#125C2×D18111yescharacteristic
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×Q8C2×D18111yescharacteristic
217C4×D4—3621no—
218C4×D4—3621no—
219C4×D4—1811no—
220C4×D4—1811no—
221#117—1811no—
222C2×C4⋊C4—1811no—
223#107—1811no—
224#107C2×D18111yescharacteristic
225#106C2×D18111yescharacteristic
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#103C2×D18111yescharacteristic
235#103C2×D18111yescharacteristic
236#102—1811no—
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239#97—1811no—
240C2×Q8—3621no—
241C2×Q8—3621no—
242C2×Q8—3622no—
243C2×Q8—3622no—
244C2×Q8—1811no—
245C2×Q8—1811no—
246C2×Q8—7241no—
247C2×Q8—7241no—
248C2×Q8C22×D18111yescharacteristic
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4—1811no—
253C22×C4—3621no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4—1811no—
258C22×C4—1811no—
259SD16—3622no—
260SD16—7241no—
261SD16—7241no—
262M16—7241no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8—7241no—
266C2×C8C22×D18111yescharacteristic
267C2×C8C22×D18111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—1811no—
277C4⋊C4—1811no—
278C4⋊C4C22×D18111yescharacteristic
279C4⋊C4C22×D18111yescharacteristic
280C4⋊C4C22×D18111yescharacteristic
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—1811no—
288C22⋊C4—1811no—
289C42—21no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—3621no—
293C23—1811no—
294C23—1811no—
295Q8—21no—
296Q8—3621no—
297Q8—3621no—
298Q8—3621no—
299Q8—3621no—
300Q8—7242no—
301Q8—7242no—
302Q8#1566845012yes—
303D4—3622no—
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—1811no—
322C2×C4—1811no—
323C2×C4—7242no—
324C2×C4—7242no—
325C2×C4D4×D18111yescharacteristic
326C2×C4D4×D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—21no—
329C8—21no—
330C8—3621no—
331C8—3621no—
332C22—3621no—
333C22—3621no—
334C22—3621no—
335C22—3621no—
336C22—3621no—
337C22—1812no—
338C22—7241no—
339C22—7241no—
340C22—7241no—
341C22C2×D4×D18111yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3621no—
349C4—3621no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4C2×#1566845011yescharacteristic
354C4#1915386211yescharacteristic
355C2—3622no—
356C2—14481no—
357C2—7241no—
358C2#2604192011yescharacteristic
359C2#137×D18111yescharacteristic
360C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#479×C181C2
socleC22×C181C22×D4
prime radical#26041982C2
hypercenter#479D181
hypocenterC181#3320
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. Q8×D724C2semidirect
3. #26041649C2semidirect
4. #26041623C2semidirect
5. #26041616C2semidirect
6. #26041526C2semidirect
7. #26041509C2semidirect
8. #26041485C2semidirect
9. #26041419C2semidirect
10. #26041419C2semidirect
11. #26041377C2semidirect
12. #26041349C2semidirect
13. #26041286C2semidirect
14. #103×D181C2semidirect
15. #479×C181C2semidirect
180. C181#3320semidirect
187. #479D181semidirect

Maximal subgroups

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

subgroupindex
#260419822
Q8×D7242
#260416492
#260416232
#260416162
#260415262
#260415092
#260414852
#260414192
#260413772
#260413492
#260412862
#103×D1812
#479×C1812
#3320181