Finite Group Databasebeta

← #28374547

Subgroups — #28374547

Order 13952.

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×#16812732C211yesmaximal, characteristic, complemented
2C22×#13182018C211yesmaximal, characteristic, complemented
3D4×Dic218C211yesmaximal, characteristic, complemented
4#20295672C211yesmaximal, characteristic
5#20295420C211yesmaximal, characteristic, complemented
6#20295272C211yesmaximal, characteristic, complemented
7#20295267C211yesmaximal, characteristic, complemented
8#20295249C211yesmaximal, characteristic, complemented
9#20295232C211yesmaximal, characteristic, complemented
10#20295190C211yesmaximal, characteristic, complemented
11#20295184C211yesmaximal, characteristic, complemented
12#20295179C211yesmaximal, characteristic, complemented
13#20295167C211yesmaximal, characteristic, complemented
14#20295128C211yesmaximal, characteristic, complemented
15#435×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×Dic218C2211yescharacteristic
18C22×Dic218C2211yescharacteristic, complemented
19C22×D436C2211yescharacteristic
20#16812732—24no—
21C2×Dic436C2211yescharacteristic, complemented
22C2×Dic436C2211yescharacteristic, complemented
23C2×Dic436C2211yescharacteristic, complemented
24C2×#13182018—24no—
25C2×#13182018C2211yescharacteristic
26C2×#13182018C2211yescharacteristic
27C2×#13182018C2211yescharacteristic
28C2×#13182018C2211yescharacteristic
29C2×#13182018C2211yescharacteristic
30C2×#13182018C2211yescharacteristic
31#16812699C2211yescharacteristic, complemented
32#16812698—21no—
33#16812687C2211yescharacteristic
34#16812684—21no—
35#16812678C2211yescharacteristic
36#16812678C2211yescharacteristic
37#16812678C2211yescharacteristic, complemented
38#16812676—21no—
39#16812676—21no—
40#16812673C2211yescharacteristic
41#16812672—21no—
42#16812667—21no—
43C4×Dic218C2211yescharacteristic, complemented
44C4×D436C2211yescharacteristic
45D4×Dic109—21no—
46C22⋊C4×D109—21no—
47#16812633C2211yescharacteristic, complemented
48#16812624C2211yescharacteristic, complemented
49#16812624C2211yescharacteristic, complemented
50#16812624C2211yescharacteristic, complemented
51#16812624C2211yescharacteristic, complemented
52#16812621C2211yescharacteristic
53#16812621C2211yescharacteristic
54C2×M16×C109C2211yescharacteristic, complemented
55C22×C8×C109C2211yescharacteristic
56C4×D4×C109C2211yescharacteristic
57C4⋊C8×C109C2211yescharacteristic, complemented
58C22⋊C8×C109C2211yescharacteristic
59C22⋊C8×C109C2211yescharacteristic, complemented
60#97×C109C2211yescharacteristic, complemented
61C23×D109—21no—
62#13182029—22no—
63#13182029—41no—
64C2×Dic218—22no—
65C2×Dic218—22no—
66C2×Dic218—21no—
67C2×Dic218—21no—
68C2×Dic218—41no—
69C2×Dic218—41no—
70C2×Dic218C2311yescharacteristic
71C2×Dic218C2311yescharacteristic
72C2×Dic218C2311yescharacteristic
73C2×Dic218C2311yescharacteristic
74C2×D436—22no—
75C2×D436—21no—
76C2×D436—41no—
77C2×D436C2311yescharacteristic
78C2×D436C2311yescharacteristic
79C2×C22⋊D109—21no—
80C2×C22⋊D109—21no—
81C22×Dic109—21no—
82C22×Dic109—21no—
83C2×C4×D109—21no—
84C2×C4×D109—21no—
85Dic436—22no—
86Dic436—22no—
87Dic436—41no—
88#13182018—22no—
89#13182018—22no—
90#13182018—22no—
91#13182018—22no—
92#13182018—41no—
93#13182018D414yescomplemented
94C4⋊Dic109—21no—
95C4⋊Dic109—21no—
96C4⋊Dic109C2311yescharacteristic
97C4⋊Dic109C2311yescharacteristic
98#13182008—21no—
99#13182008—21no—
100#13182008—21no—
101#13182008—21no—
102C22⋊Dic109—21no—
103C22⋊Dic109—21no—
104C22⋊Dic109—21no—
105#13182006—21no—
106#13182006—21no—
107#13182006—21no—
108#13182006—21no—
109C4×Dic109—21no—
110C2×D4×C109D411yescharacteristic
111C22×C4×C109C2311yescharacteristic
112C22×C4×C109C2311yescharacteristic
113M16×C109—22no—
114C2×C8×C109—22no—
115C2×C8×C109C2311yescharacteristic
116C2×C8×C109C2311yescharacteristic
117C2×C8×C109C2311yescharacteristic
118C2×C8×C109C2311yescharacteristic
119C4⋊C4×C109D411yescharacteristic
120C22⋊C4×C109D411yescharacteristic
121C22⋊C4×C109D411yescharacteristic
122C42×C109C2311yescharacteristic
123C23×C109C2×D411yescharacteristic
124C23×C109C2×D411yescharacteristic
125C22×D109—21no—
126C22×D109—21no—
127C22×D109—42no—
128C22×D109—42no—
129C22×D109—81no—
130D4×C109—22no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109—21no—
134C2×C4×C109—21no—
135C2×C4×C109C2×D411yescharacteristic
136C2×C4×C109C2×D411yescharacteristic
137C2×C4×C109C2×D411yescharacteristic
138C2×C4×C109C2×D411yescharacteristic
139C2×C4×C109C2411yescharacteristic
140C22⋊D109—42no—
141C22⋊D109—81no—
142C2×Dic109—21no—
143C2×Dic109—21no—
144C2×Dic109—21no—
145C2×Dic109—21no—
146C2×Dic109—21no—
147C2×Dic109—21no—
148C2×Dic109—42no—
149C2×Dic109—42no—
150D436—21no—
151D436—21no—
152D436—21no—
153D436—42no—
154D436C2×D412yes—
155C4×D109—42no—
156C4×D109—81no—
157Dic218—21no—
158Dic218—21no—
159Dic218—21no—
160Dic218—21no—
161Dic218—21no—
162Dic218—21no—
163Dic218—21no—
164Dic218—42no—
165Dic218—42no—
166Dic218C2×D412yes—
167C8×C109—21no—
168C8×C109—21no—
169C8×C109—21no—
170C8×C109C2×D412yes—
171C22×C109—21no—
172C22×C109—21no—
173C22×C109—21no—
174C22×C109—21no—
175C22×C109—21no—
176C22×C109#13712yescomplemented
177C22×C109C22×D411yescharacteristic
178C2×D109—22no—
179C2×D109—41no—
180C2×D109—41no—
181C2×D109—41no—
182C2×D109—41no—
183C2×D109—41no—
184C2×D109—84no—
185C4×C109—21no—
186C4×C109—21no—
187C4×C109—21no—
188C4×C109—21no—
189C4×C109C22×D411yescharacteristic
190C4×C109H(5,𝔽2)11yescharacteristic
191Dic109—22no—
192Dic109—41no—
193Dic109—41no—
194Dic109—41no—
195Dic109—41no—
196Dic109—41no—
197C2×C109—22no—
198C2×C109—41no—
199C2×C109D4211yescharacteristic
200C2×C109C2×#13711yescharacteristic
201C2×C109#57711yescharacteristic
202D109—42no—
203D109—81no—
204#3266—1091nomaximal, Sylow
205C109#326611yescharacteristic, complemented, Sylow
206C2×#137—1091no—
207C22×SD16—1091no—
208D4×Q8—1091no—
209#546—1091no—
210#497—1091no—
211#468—1091no—
212#467—1091no—
213#464—1091no—
214#461—1091no—
215#452—1091no—
216#451—1091no—
217#450—1091no—
218#448—1091no—
219#440—1091no—
220#435D10911yescharacteristic, complemented
221C2×#41—1091no—
222C22×Q8—1091no—
223C22×Q8—1091no—
224C22×D4—1091no—
225#137—2184no—
226C2×Q16—1091no—
227C2×Q16—1091no—
228C2×Q16—1091no—
229C2×SD16—1091no—
230C2×SD16—1091no—
231C2×SD16—1091no—
232C2×SD16—1091no—
233C2×SD16—1091no—
234C2×SD16—1091no—
235C2×SD16—2184no—
236C2×M16C2×D10911yescharacteristic, complemented
237C22×C8C2×D10911yescharacteristic
238C4⋊Q8—1091no—
239C4⋊Q8—2181no—
240#124—1091no—
241#123—2181no—
242C22⋊Q8—1091no—
243C22⋊Q8—1091no—
244C22⋊Q8—1091no—
245C22⋊Q8—2181no—
246C22⋊Q8—2181no—
247#121—1091no—
248#121—2181no—
249#120—2181no—
250C4×Q8—1091no—
251C4×D4—1091no—
252C4×D4—2181no—
253C4×D4C2×D10911yescharacteristic
254C2×C22⋊C4—2181no—
255#107—1091no—
256C4⋊C8C2×D10911yescharacteristic, complemented
257#103—1091no—
258#103—1091no—
259#103—1091no—
260#103—1091no—
261#102—1091no—
262#102—1091no—
263C22⋊C8C2×D10911yescharacteristic
264C22⋊C8C2×D10911yescharacteristic, complemented
265#97C2×D10911yescharacteristic, complemented
266C24—2181no—
267#41—4361no—
268#41—2182no—
269C2×Q8—1091no—
270C2×Q8—1091no—
271C2×Q8—1091no—
272C2×Q8—1091no—
273C2×Q8—4361no—
274C2×Q8—4361no—
275C2×Q8—2182no—
276C2×Q8—2182no—
277C2×Q8—2181no—
278C2×Q8—2181no—
279C2×D4—1091no—
280C2×D4—1091no—
281C2×D4—4361no—
282C2×D4—2182no—
283C2×D4—2181no—
284C2×D4—2181no—
285C2×D4—2181no—
286C2×D4D43611yescharacteristic
287C22×C4—2181no—
288C22×C4—2181no—
289C22×C4—2181no—
290C22×C4—2181no—
291C22×C4C22×D10911yescharacteristic
292C22×C4C22×D10911yescharacteristic
293Q16—4361no—
294Q16—2182no—
295Q16—2182no—
296SD16—1094no—
297SD16—4361no—
298SD16—2182no—
299SD16—2182no—
300SD16—2182no—
301SD16—2182no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D10911yescharacteristic
305C2×C8C22×D10911yescharacteristic
306C2×C8C22×D10911yescharacteristic
307C2×C8C22×D10911yescharacteristic
308C4⋊C4—1091no—
309C4⋊C4—1091no—
310C4⋊C4—2181no—
311C4⋊C4—2181no—
312C4⋊C4—2181no—
313C4⋊C4—2181no—
314C4⋊C4—2181no—
315C4⋊C4—2181no—
316C4⋊C4D43611yescharacteristic
317C22⋊C4—2181no—
318C22⋊C4—2181no—
319C22⋊C4—2181no—
320C22⋊C4—2181no—
321C22⋊C4—2181no—
322C22⋊C4—2181no—
323C22⋊C4—2181no—
324C22⋊C4D43611yescharacteristic
325C22⋊C4D43611yescharacteristic
326C42—2181no—
327C42C22×D10911yescharacteristic
328C23—8721no—
329C23—4362no—
330C23—4362no—
331C23—2181no—
332C23—2181no—
333C23C2×D43611yescharacteristic
334C23C2×D43611yescharacteristic
335Q8—1092no—
336Q8—4362no—
337Q8—4362no—
338Q8—2181no—
339Q8—2181no—
340Q8—2181no—
341Q8—2181no—
342Q8—2181no—
343Q8—2181no—
344Q8—2181no—
345D4—22no—
346D4—8721no—
347D4—1092no—
348D4—4362no—
349D4—4362no—
350D4—2181no—
351D4—2181no—
352D4—2181no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8721no—
358C2×C4—4362no—
359C2×C4—4362no—
360C2×C4—4362no—
361C2×C4—2181no—
362C2×C4—2181no—
363C2×C4—2181no—
364C2×C4—2181no—
365C2×C4—2181no—
366C2×C4—2181no—
367C2×C4C2×D43611yescharacteristic
368C2×C4C2×D43611yescharacteristic
369C2×C4C2×D43611yescharacteristic
370C2×C4C2×D43611yescharacteristic
371C2×C4C23×D10911yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D10912yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8724no—
382C22—4361no—
383C22—4361no—
384C22—4361no—
385C22—4361no—
386C22—4361no—
387C22—2182no—
388C22#1681273212yescomplemented
389C22C22×D43611yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—4361no—
395C4—4361no—
396C4—4361no—
397C4—4361no—
398C4—4361no—
399C4—2182no—
400C4C2×D4×D10911yescharacteristic
401C4#1681274511yescharacteristic
402C2—22no—
403C2—41no—
404C2—8721no—
405C2—4362no—
406C2D4×D43611yescharacteristic
407C2C2×#1681273211yescharacteristic
408C2#2029583811yescharacteristic
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#435×C109C2
socleC22×C109C22×D4
prime radical#20295672C2
hypercenter#435D109
hypocenterC109#3266
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C22×#13182018C2semidirect
3. D4×Dic218C2semidirect
5. #20295420C2semidirect
6. #20295272C2semidirect
7. #20295267C2semidirect
8. #20295249C2semidirect
9. #20295232C2semidirect
10. #20295190C2semidirect
11. #20295184C2semidirect
12. #20295179C2semidirect
13. #20295167C2semidirect
14. #20295128C2semidirect
15. #435×C109C2semidirect
18. C22×Dic218C22semidirect
21. C2×Dic436C22semidirect
22. C2×Dic436C22semidirect
23. C2×Dic436C22semidirect
31. #16812699C22semidirect
37. #16812678C22semidirect
43. C4×Dic218C22semidirect
47. #16812633C22semidirect
48. #16812624C22semidirect
49. #16812624C22semidirect
50. #16812624C22semidirect
51. #16812624C22semidirect
54. C2×M16×C109C22semidirect
57. C4⋊C8×C109C22semidirect
59. C22⋊C8×C109C22semidirect
60. #97×C109C22semidirect
93. #13182018D4semidirect
176. C22×C109#137semidirect
205. C109#3266semidirect
220. #435D109semidirect
236. C2×M16C2×D109semidirect
256. C4⋊C8C2×D109semidirect
264. C22⋊C8C2×D109semidirect
265. #97C2×D109semidirect
388. C22#16812732semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C22×#131820182
D4×Dic2182
#202956722
#202954202
#202952722
#202952672
#202952492
#202952322
#202951902
#202951842
#202951792
#202951672
#202951282
#435×C1092
#3266109