Finite Group Databasebeta

← #37543268

Subgroups — #37543268

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×#19153848C211yesmaximal, characteristic, complemented
2C2×#19153839C211yesmaximal, characteristic
3#26041854C211yesmaximal, characteristic, complemented
4#26041761C211yesmaximal, characteristic, complemented
5#26041503C211yesmaximal, characteristic, complemented
6#26041501C211yesmaximal, characteristic, complemented
7#26041497C211yesmaximal, characteristic, complemented
8#26041495C211yesmaximal, characteristic, complemented
9#26041487C211yesmaximal, characteristic, complemented
10#26041483C211yesmaximal, characteristic, complemented
11#26041481C211yesmaximal, characteristic, complemented
12#26041304C211yesmaximal, characteristic, complemented
13#26041279C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#462×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19#19153848—24no—
20#19153839—24no—
21C2×#15668439C2211yescharacteristic
22C2×#15668438C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×D1448C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153814C2211yescharacteristic
30#19153805C2211yescharacteristic
31#19153804C2211yescharacteristic
32#19153801—21no—
33#19153801—21no—
34#19153794—21no—
35#19153794—21no—
36Q8×Dic181C2211yescharacteristic
37C4×D724—21no—
38C4×D724C2211yescharacteristic
39#19153778—21no—
40#19153778C2211yescharacteristic
41#19153773—21no—
42#19153773C2211yescharacteristic
43C4⋊C4×D181—21no—
44C4⋊C4×D181C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic
46#19153745C2211yescharacteristic
47#19153745C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153740C2211yescharacteristic
50#19153740C2211yescharacteristic
51#19153739C2211yescharacteristic
52#19153738C2211yescharacteristic
53#19153736C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C2×Q16×C181C2211yescharacteristic
56C4⋊Q8×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic
58C4⋊C8×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#103×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—22no—
63#15668450—22no—
64#15668450D414yes—
65C2×Dic362—21no—
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181—22no—
69C2×D724—21no—
70C2×D724—21no—
71C2×D724C2311yescharacteristic
72C2×D724C2311yescharacteristic
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77C2×C4×D181—21no—
78C2×C4×D181—21no—
79C2×C4×D181C2311yescharacteristic
80#15668439—22no—
81#15668438—22no—
82#15668436—22no—
83#15668436—22no—
84#15668436—22no—
85D1448—22no—
86#15668432—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C8×D181—22no—
90C4⋊Dic181—21no—
91C4⋊Dic181C2311yescharacteristic
92#15668428—21no—
93#15668428—21no—
94#15668428—21no—
95#15668428D411yescharacteristic
96#15668428D411yescharacteristic
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426D411yescharacteristic
104#15668426D411yescharacteristic
105C4×Dic181—21no—
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110Q16×C181—22no—
111Q16×C181—22no—
112C2×C8×C181C2311yescharacteristic
113C2×C8×C181C2311yescharacteristic
114C4⋊C4×C181—21no—
115C4⋊C4×C181—21no—
116C4⋊C4×C181—21no—
117C4⋊C4×C181C2311yescharacteristic
118C42×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181—21no—
121C22×D181—21no—
122C22×D181C2×D411yescharacteristic
123Dic362—21no—
124Dic362—21no—
125Dic362—42no—
126D724—24no—
127D724—21no—
128D724—21no—
129D724—21no—
130D724—42no—
131D724C2×D412yes—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181C2×D411yescharacteristic
135C2×Dic181C2×D411yescharacteristic
136C2×Dic181C2×D411yescharacteristic
137C4×D181—24no—
138C4×D181—42no—
139C4×D181—42no—
140C4×D181—81no—
141C4×D181—81no—
142C4×D181—81no—
143C4×D181C2×D412yes—
144C4×D181#4112yes—
145#13089678—21no—
146#13089678—21no—
147Q8×C181—21no—
148Q8×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151Q8×C181—21no—
152Q8×C181C2×D412yes—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2411yescharacteristic
160C8×C181—21no—
161C8×C181—21no—
162C22×C181C22×D411yescharacteristic
163C2×D181—21no—
164C2×D181—21no—
165C2×D181—22no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181#12112yes—
172C4×C181—22no—
173C4×C181—21no—
174C4×C181—21no—
175C4×C181—41no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181C22×D411yescharacteristic
179C4×C181C2×#4111yescharacteristic
180Dic181—21no—
181Dic181—21no—
182Dic181—41no—
183Dic181—41no—
184Dic181#12112yes—
185C2×C181C2×#12111yescharacteristic
186C2×C181#57711yescharacteristic
187C2×C181#57811yescharacteristic
188D181—22no—
189D181—42no—
190D181—81no—
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#463—1811no—
201#462—1811no—
202#462D18111yescharacteristic, complemented
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×Q16C2×D18111yescharacteristic
215C2×Q16C2×D18111yescharacteristic
216C2×SD16—1811no—
217C2×SD16—1811no—
218C2×SD16—1811no—
219C2×SD16—1811no—
220C2×D8—1811no—
221C2×M16—1811no—
222C22×C8—1811no—
223C4⋊Q8—1811no—
224C4⋊Q8C2×D18111yescharacteristic
225#124—1811no—
226#124—1811no—
227#123—3621no—
228#123—3621no—
229C22⋊Q8—3621no—
230C22⋊Q8—3621no—
231C4×Q8—1811no—
232C4×Q8C2×D18111yescharacteristic
233C4×D4—3621no—
234C4×D4—3621no—
235C4×D4—1811no—
236C4×D4—1811no—
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#103C2×D18111yescharacteristic
250#103C2×D18111yescharacteristic
251#102—1811no—
252#102—1811no—
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×Q8C22×D18111yescharacteristic
262C2×Q8C22×D18111yescharacteristic
263C2×Q8C22×D18111yescharacteristic
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—1811no—
267C2×D4—1811no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4—3621no—
271C22×C4—3621no—
272C22×C4—3621no—
273C22×C4—3621no—
274C22×C4—1811no—
275Q16—22no—
276Q16—22no—
277Q16—3622no—
278SD16—3622no—
279SD16—3622no—
280SD16—3622no—
281SD16—3622no—
282D8—3622no—
283M16—3622no—
284C2×C8—3622no—
285C2×C8—1811no—
286C2×C8—1811no—
287C2×C8C22×D18111yescharacteristic
288C2×C8C22×D18111yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
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—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—1811no—
307C22⋊C4—1811no—
308C42—21no—
309C42—3621no—
310C42—1811no—
311C42C22×D18111yescharacteristic
312C23—3621no—
313C23—3621no—
314C23—1811no—
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—3621no—
321Q8—3621no—
322Q8—7242no—
323Q8D4×D18112yes—
324D4—3624no—
325D4—3621no—
326D4—3621no—
327D4—3621no—
328D4—1812no—
329D4—7242no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—3624no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—14481no—
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—3621no—
355C22—3621no—
356C22—3622no—
357C22—1812no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22—7241no—
363C22C2×D4×D18111yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3621no—
371C4—3621no—
372C4—1812no—
373C4—7241no—
374C4—7241no—
375C4C2×D4×D18111yescharacteristic
376C4#41×D18111yescharacteristic
377C2—3622no—
378C2—14481no—
379C2—7242no—
380C2#121×D18111yescharacteristic
381C2#2604208811yescharacteristic
382C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#462×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153839C2
hypercenter#462D181
hypocenterC181#3046
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
3. #26041854C2semidirect
4. #26041761C2semidirect
5. #26041503C2semidirect
6. #26041501C2semidirect
7. #26041497C2semidirect
8. #26041495C2semidirect
9. #26041487C2semidirect
10. #26041483C2semidirect
11. #26041481C2semidirect
12. #26041304C2semidirect
13. #26041279C2semidirect
14. #103×D181C2semidirect
15. #462×C181C2semidirect
191. C181#3046semidirect
202. #462D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×#191538392
#260418542
#260417612
#260415032
#260415012
#260414972
#260414952
#260414872
#260414832
#260414812
#260413042
#260412792
#103×D1812
#462×C1812
#3046181