Finite Group Databasebeta

← #37545231

Subgroups — #37545231

Order 23168.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2Q8×C22⋊D181C211yesmaximal, characteristic, complemented
3#26041923C211yesmaximal, characteristic
4#26041670C211yesmaximal, characteristic, complemented
5#26041522C211yesmaximal, characteristic, complemented
6#26041520C211yesmaximal, characteristic, complemented
7#26041499C211yesmaximal, characteristic, complemented
8#26041482C211yesmaximal, characteristic, complemented
9#26041438C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041430C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041380C211yesmaximal, characteristic, complemented
14#26041350C211yesmaximal, characteristic, complemented
15C22×SD16×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic, complemented
18#19153845—24no—
19C2×Dic724C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668437C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668432C2211yescharacteristic, complemented
25#19153814—21no—
26#19153814C2211yescharacteristic, complemented
27#19153803C2211yescharacteristic
28#19153797—21no—
29#19153795—21no—
30#19153795C2211yescharacteristic
31#19153795C2211yescharacteristic
32#19153792—21no—
33#19153792C2211yescharacteristic, complemented
34#19153788—21no—
35#19153787C2211yescharacteristic
36#19153785—21no—
37Q8×Dic181C2211yescharacteristic, complemented
38D4×Dic181C2211yescharacteristic
39C4×C22⋊D181—21no—
40C4×C22⋊D181C2211yescharacteristic
41C2×C22⋊Dic181—21no—
42#19153750C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153741C2211yescharacteristic, complemented
46#19153740C2211yescharacteristic, complemented
47#19153740C2211yescharacteristic, complemented
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50C22⋊#13089678C2211yescharacteristic
51#19153728C2211yescharacteristic, complemented
52#19153726C2211yescharacteristic, complemented
53C22×Q8×C181C2211yescharacteristic
54C22×D4×C181C2211yescharacteristic
55C2×SD16×C181—24no—
56C2×SD16×C181C2211yescharacteristic
57C2×SD16×C181C2211yescharacteristic
58C2×SD16×C181C2211yescharacteristic
59C2×SD16×C181C2211yescharacteristic
60C22×C8×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—41no—
63C2×Dic362—21no—
64C2×Dic362C2311yescharacteristic
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—41no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181D411yescharacteristic
71C22×Dic181—21no—
72C22×Dic181—21no—
73C2×C4×D181—21no—
74C2×C4×D181C2311yescharacteristic
75Dic724—22no—
76#15668439—22no—
77#15668439—41no—
78#15668437—22no—
79#15668437—41no—
80#15668432—22no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C4⋊Dic181—21no—
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90#15668428—21no—
91#15668428D411yescharacteristic
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181D411yescharacteristic
99#15668426—21no—
100#15668426D411yescharacteristic
101C4×Dic181—21no—
102C4×Dic181C2311yescharacteristic
103C24×C181—21no—
104C2×Q8×C181—21no—
105C2×Q8×C181—22no—
106C2×Q8×C181—41no—
107C2×Q8×C181C2311yescharacteristic
108C2×Q8×C181C2311yescharacteristic
109C2×D4×C181—22no—
110C2×D4×C181—21no—
111C2×D4×C181—41no—
112C2×D4×C181C2311yescharacteristic
113C2×D4×C181C2311yescharacteristic
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116SD16×C181—22no—
117SD16×C181—22no—
118SD16×C181—22no—
119SD16×C181D414yescomplemented
120C2×C8×C181—22no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C22×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129C22⋊D181—22no—
130C22⋊D181—42no—
131C22⋊D181—81no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—42no—
137C2×Dic181—81no—
138C2×Dic181C2×D411yescharacteristic
139C2×Dic181C2×D411yescharacteristic
140C2×Dic181C2×D411yescharacteristic
141C4×D181—21no—
142C4×D181—21no—
143C4×D181—42no—
144#13089678—21no—
145#13089678—21no—
146C23×C181—21no—
147C23×C181—21no—
148C23×C181—42no—
149C23×C181—42no—
150C23×C181—81no—
151C23×C181C2×D411yescharacteristic
152Q8×C181—21no—
153Q8×C181—21no—
154Q8×C181—21no—
155Q8×C181—42no—
156Q8×C181C2×D412yes—
157D4×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—42no—
161D4×C181C2×D412yes—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—42no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181C2×D412yes—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—22no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—84no—
180C22×C181#13712yescomplemented
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—21no—
185C4×C181—21no—
186C4×C181—22no—
187C4×C181—41no—
188C4×C181C22×D411yescharacteristic
189C4×C181H(5,𝔽2)11yescharacteristic
190Dic181—21no—
191Dic181—21no—
192Dic181—21no—
193Dic181—41no—
194Dic181—41no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—22no—
198C2×C181—42no—
199C2×C181—81no—
200C2×C181D4211yescharacteristic
201C2×C181C2×#13711yescharacteristic
202C2×C181#57711yescharacteristic
203D181—41no—
204C181#326611yescharacteristic, complemented, Sylow
205#3266—1811nomaximal, Sylow
206C2×#137—1811no—
207C22×SD16D18111yescharacteristic, complemented
208D4×Q8—1811no—
209#546—1811no—
210#497—1811no—
211#468—1811no—
212#467—1811no—
213#464—1811no—
214#461—1811no—
215#452—1811no—
216#451—1811no—
217#450—1811no—
218#448—1811no—
219#440—1811no—
220#435—1811no—
221C2×#41—1811no—
222C22×Q8—1811no—
223C22×Q8C2×D18111yescharacteristic
224C22×D4C2×D18111yescharacteristic
225#137—3624no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×Q16—1811no—
229C2×SD16—24no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×SD16C2×D18111yescharacteristic
233C2×SD16C2×D18111yescharacteristic
234C2×SD16C2×D18111yescharacteristic
235C2×SD16C2×D18111yescharacteristic
236C2×M16—1811no—
237C22×C8C2×D18111yescharacteristic
238C4⋊Q8—3621no—
239C4⋊Q8—1811no—
240#124—1811no—
241#123—3621no—
242C22⋊Q8—3621no—
243C22⋊Q8—3621no—
244C22⋊Q8—1811no—
245C22⋊Q8—1811no—
246C22⋊Q8—1811no—
247#121—3621no—
248#121—1811no—
249#120—3621no—
250C4×Q8—1811no—
251C4×D4—3621no—
252C4×D4—1811no—
253C4×D4—1811no—
254C2×C22⋊C4—3621no—
255#107—1811no—
256C4⋊C8—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#103—1811no—
261#102—1811no—
262#102—1811no—
263C22⋊C8—1811no—
264C22⋊C8—1811no—
265#97—1811no—
266C24—21no—
267#41—3622no—
268#41—7241no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3621no—
273C2×Q8—3622no—
274C2×Q8—1811no—
275C2×Q8—1811no—
276C2×Q8—7241no—
277C2×Q8C22×D18111yescharacteristic
278C2×Q8C22×D18111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—1811no—
285C2×D4C22×D18111yescharacteristic
286C2×D4C22×D18111yescharacteristic
287C22×C4—21no—
288C22×C4—3621no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4—1811no—
292C22×C4C22×D18111yescharacteristic
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—3622no—
300SD16—7241no—
301SD16C22⋊D18114yescomplemented
302M16—3622no—
303C2×C8—22no—
304C2×C8—1811no—
305C2×C8—1811no—
306C2×C8C22×D18111yescharacteristic
307C2×C8C22×D18111yescharacteristic
308C4⋊C4—3621no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4—1811no—
316C4⋊C4—1811no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—1811no—
325C22⋊C4—1811no—
326C42—3621no—
327C42—1811no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1811no—
334C23D4×D18111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—3621no—
340Q8—3621no—
341Q8—3621no—
342Q8—3621no—
343Q8—7242no—
344Q8C2×C22⋊D18112yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3622no—
350D4—14481no—
351D4—7242no—
352D4C2×C22⋊D18112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—14481no—
365C2×C4—1811no—
366C2×C4—1811no—
367C2×C4—1811no—
368C2×C4—7242no—
369C2×C4—7242no—
370C2×C4D4×D18111yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—21no—
373C8—3621no—
374C8—3621no—
375C8C2×C22⋊D18112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3621no—
386C22—3621no—
387C22—3621no—
388C22#1915384512yescomplemented
389C22C2×D4×D18111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—3621no—
394C4—3621no—
395C4—3621no—
396C4—7241no—
397C4—7241no—
398C4—7241no—
399C4—7241no—
400C4C22×C22⋊D18111yescharacteristic
401C4#1915386111yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—7241no—
406C2D4×C22⋊D18111yescharacteristic
407C2C2×#1915384511yescharacteristic
408C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×SD16×C181C2
socleC22×C181C22×D4
prime radical#26041923C2
hypercenterC22×SD16D181
hypocenterC181#3266
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. Q8×C22⋊D181C2semidirect
4. #26041670C2semidirect
5. #26041522C2semidirect
6. #26041520C2semidirect
7. #26041499C2semidirect
8. #26041482C2semidirect
9. #26041438C2semidirect
10. #26041435C2semidirect
11. #26041430C2semidirect
12. #26041416C2semidirect
13. #26041380C2semidirect
14. #26041350C2semidirect
15. C22×SD16×C181C2semidirect
17. C2×Q8×D181C22semidirect
19. C2×Dic724C22semidirect
20. C2×#15668439C22semidirect
21. C2×#15668439C22semidirect
24. C2×#15668432C22semidirect
26. #19153814C22semidirect
33. #19153792C22semidirect
37. Q8×Dic181C22semidirect
42. #19153750C22semidirect
43. #19153745C22semidirect
44. #19153741C22semidirect
45. #19153741C22semidirect
46. #19153740C22semidirect
47. #19153740C22semidirect
51. #19153728C22semidirect
52. #19153726C22semidirect
119. SD16×C181D4semidirect
180. C22×C181#137semidirect
204. C181#3266semidirect
207. C22×SD16D181semidirect
301. SD16C22⋊D181semidirect
388. C22#19153845semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
Q8×C22⋊D1812
#260419232
#260416702
#260415222
#260415202
#260414992
#260414822
#260414382
#260414352
#260414302
#260414162
#260413802
#260413502
C22×SD16×C1812
#3266181