Finite Group Databasebeta

← #37545885

Subgroups — #37545885

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#26041976C211yesmaximal, characteristic
2Q8×C22⋊D181C211yesmaximal, characteristic, complemented
3#26041646C211yesmaximal, characteristic, complemented
4#26041621C211yesmaximal, characteristic, complemented
5#26041613C211yesmaximal, characteristic, complemented
6#26041584C211yesmaximal, characteristic, complemented
7#26041523C211yesmaximal, characteristic, complemented
8#26041505C211yesmaximal, characteristic, complemented
9#26041482C211yesmaximal, characteristic, complemented
10#26041416C211yesmaximal, characteristic, complemented
11#26041380C211yesmaximal, characteristic, complemented
12#26041353C211yesmaximal, characteristic, complemented
13#26041288C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#448×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×#15668437C2211yescharacteristic
18C2×#15668433C2211yescharacteristic
19C2×C8×D181C2211yescharacteristic
20#19153814—21no—
21#19153814C2211yescharacteristic
22#19153809—21no—
23#19153807C2211yescharacteristic
24#19153797—21no—
25#19153795—21no—
26#19153795C2211yescharacteristic
27#19153792—21no—
28#19153792C2211yescharacteristic
29#19153787—21no—
30#19153787C2211yescharacteristic
31Q8×Dic181C2211yescharacteristic
32D4×Dic181C2211yescharacteristic
33C4×C22⋊D181—21no—
34C4×C22⋊D181C2211yescharacteristic
35#19153776—21no—
36#19153775C2211yescharacteristic
37C4⋊C4×D181C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153748C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic
52C22×Q8×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61Q8×D181—22no—
62Q8×D181—41no—
63C2×C22⋊D181—21no—
64C2×C22⋊D181D411yescharacteristic
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181C2311yescharacteristic
69#15668437—41no—
70#15668433—41no—
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C8×D181—41no—
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428D411yescharacteristic
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181D411yescharacteristic
91#15668426—21no—
92#15668426D411yescharacteristic
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C2×Q8×C181—21no—
97C2×Q8×C181—22no—
98C2×Q8×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×Q8×C181C2311yescharacteristic
101C2×D4×C181—21no—
102C2×D4×C181C2311yescharacteristic
103C22×C4×C181—21no—
104C22×C4×C181C2311yescharacteristic
105SD16×C181—22no—
106SD16×C181—41no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181C2×D411yescharacteristic
112Dic362—21no—
113Dic362—21no—
114Dic362—42no—
115C22⋊D181—22no—
116C22⋊D181—42no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—81no—
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C4×D181—21no—
126C4×D181—21no—
127C4×D181—42no—
128C4×D181—81no—
129#13089678—21no—
130#13089678—21no—
131C23×C181—21no—
132C23×C181C2×D411yescharacteristic
133Q8×C181—21no—
134Q8×C181—21no—
135Q8×C181—21no—
136Q8×C181—42no—
137Q8×C181#4112yes—
138D4×C181—21no—
139D4×C181—21no—
140D4×C181—81no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—42no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C8×C181—21no—
150C8×C181—21no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—41no—
155C22×C181—41no—
156C22×C181—41no—
157C22×C181C22×D411yescharacteristic
158C2×D181—21no—
159C2×D181—21no—
160C2×D181#13712yes—
161C4×C181—21no—
162C4×C181—22no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181C2×#4111yescharacteristic
166C4×C181H(5,𝔽2)11yescharacteristic
167Dic181—21no—
168Dic181—21no—
169Dic181—21no—
170Dic181—41no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—41no—
175C2×C181—81no—
176C2×C181#54611yescharacteristic
177C2×C181C2×#13711yescharacteristic
178C2×C181#57711yescharacteristic
179D181—22no—
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#448D18111yescharacteristic, complemented
193#440—1811no—
194#435—1811no—
195#419—1811no—
196C2×#103—1811no—
197C22×Q8—1811no—
198C22×Q8C2×D18111yescharacteristic
199C2×SD16—1811no—
200C2×SD16C2×D18111yescharacteristic
201C2×SD16C2×D18111yescharacteristic
202C2×M16—1811no—
203C22×C8—1811no—
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#121C2×D18111yescharacteristic
216C4×Q8—1811no—
217C4×D4—3621no—
218C4×D4—3621no—
219C4×D4—1811no—
220C4×D4—1811no—
221#117—1811no—
222C2×C4⋊C4—1811no—
223#107—1811no—
224#107—1811no—
225#106—1811no—
226C4⋊C8—1811no—
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⋊C8C2×D18111yescharacteristic
239#97—1811no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—3621no—
244C2×Q8—3622no—
245C2×Q8—1811no—
246C2×Q8—7241no—
247C2×Q8C22×D18111yescharacteristic
248C2×Q8C22×D18111yescharacteristic
249C2×D4—21no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4C22×D18111yescharacteristic
253C22×C4—21no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4—1811no—
258C22×C4C22×D18111yescharacteristic
259SD16—22no—
260SD16—41no—
261SD16—7241no—
262M16—7241no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8—7241no—
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⋊C4C22×D18111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—1811no—
288C22⋊C4—1811no—
289C42—3621no—
290C42—3621no—
291C42—1811no—
292C23—21no—
293C23—1811no—
294C23D4×D18111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—3621no—
300Q8—3621no—
301Q8—7242no—
302Q8#1566844812yes—
303D4—21no—
304D4—21no—
305D4—81no—
306D4—3622no—
307D4—7242no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—14481no—
321C2×C4—14481no—
322C2×C4—1811no—
323C2×C4—1811no—
324C2×C4—1811no—
325C2×C4—7242no—
326C2×C4D4×D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—21no—
329C8—21no—
330C8—3621no—
331C8—3621no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3621no—
339C22—3621no—
340C22—1812no—
341C22C2×D4×D18111yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—3621no—
347C4—3621no—
348C4—3621no—
349C4—7241no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4C2×#1566844811yescharacteristic
354C4#1915386111yescharacteristic
355C2—41no—
356C2—81no—
357C2—3622no—
358C2#2604191611yescharacteristic
359C2#137×D18111yescharacteristic
360C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#448×C181C2
socleC22×C181C22×D4
prime radical#26041976C2
hypercenter#448D181
hypocenterC181#3320
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊D181C2semidirect
3. #26041646C2semidirect
4. #26041621C2semidirect
5. #26041613C2semidirect
6. #26041584C2semidirect
7. #26041523C2semidirect
8. #26041505C2semidirect
9. #26041482C2semidirect
10. #26041416C2semidirect
11. #26041380C2semidirect
12. #26041353C2semidirect
13. #26041288C2semidirect
14. #103×D181C2semidirect
15. #448×C181C2semidirect
180. C181#3320semidirect
192. #448D181semidirect

Maximal subgroups

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

subgroupindex
#260419762
Q8×C22⋊D1812
#260416462
#260416212
#260416132
#260415842
#260415232
#260415052
#260414822
#260414162
#260413802
#260413532
#260412882
#103×D1812
#448×C1812
#3320181