Finite Group Databasebeta

← #28374549

Subgroups — #28374549

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
2C2×SD16×D109C211yesmaximal, characteristic, complemented
3#20295689C211yesmaximal, characteristic, complemented
4#20295673C211yesmaximal, characteristic
5#20295417C211yesmaximal, characteristic, complemented
6#20295268C211yesmaximal, characteristic, complemented
7#20295247C211yesmaximal, characteristic, complemented
8#20295230C211yesmaximal, characteristic, complemented
9#20295186C211yesmaximal, characteristic, complemented
10#20295183C211yesmaximal, characteristic, complemented
11#20295178C211yesmaximal, characteristic, complemented
12#20295164C211yesmaximal, characteristic, complemented
13#20295125C211yesmaximal, characteristic, complemented
14#20295095C211yesmaximal, characteristic, complemented
15#468×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×Dic218C2211yescharacteristic, complemented
18C2×Q8×D109C2211yescharacteristic
19C2×D4×D109C2211yescharacteristic
20#16812732—24no—
21C2×Dic436C2211yescharacteristic, complemented
22C2×Dic436C2211yescharacteristic, complemented
23C2×#13182019C2211yescharacteristic, complemented
24C2×#13182018C2211yescharacteristic
25C2×#13182018C2211yescharacteristic
26C2×#13182017C2211yescharacteristic
27C2×#13182017C2211yescharacteristic
28C2×#13182016C2211yescharacteristic
29SD16×D109—24no—
30C2×C8×D109C2211yescharacteristic
31#16812698—21no—
32#16812697C2211yescharacteristic, complemented
33#16812686C2211yescharacteristic
34#16812681—21no—
35#16812679C2211yescharacteristic
36#16812679C2211yescharacteristic
37#16812677—21no—
38#16812676—21no—
39#16812670—21no—
40#16812667—21no—
41#16812664C2211yescharacteristic, complemented
42#16812661C2211yescharacteristic
43C4×C22⋊D109C2211yescharacteristic
44#16812659—21no—
45C22⋊C4×D109—21no—
46#16812628C2211yescharacteristic, complemented
47#16812622C2211yescharacteristic, complemented
48#16812622C2211yescharacteristic, complemented
49#16812622C2211yescharacteristic, complemented
50#16812619C2211yescharacteristic
51C22⋊#11724784C2211yescharacteristic, complemented
52#16812611C2211yescharacteristic
53#16812609C2211yescharacteristic, complemented
54C2×SD16×C109C2211yescharacteristic
55C2×M16×C109C2211yescharacteristic, complemented
56C22⋊Q8×C109C2211yescharacteristic, complemented
57#121×C109C2211yescharacteristic
58#107×C109C2211yescharacteristic, complemented
59#103×C109C2211yescharacteristic, complemented
60#102×C109C2211yescharacteristic
61C23×D109—21no—
62#13182029—22no—
63#13182029—41no—
64C2×Dic218—22no—
65C2×Dic218—21no—
66C2×Dic218—21no—
67C2×Dic218—41no—
68C2×Dic218C2311yescharacteristic
69C2×Dic218C2311yescharacteristic
70C2×Dic218C2311yescharacteristic
71Q8×D109—22no—
72Q8×D109—41no—
73C2×D436C2311yescharacteristic
74C2×C22⋊D109—21no—
75C2×C22⋊D109—21no—
76C2×C22⋊D109D411yescharacteristic
77D4×D109—22no—
78D4×D109—41no—
79C22×Dic109—21no—
80C2×C4×D109—21no—
81C2×C4×D109—21no—
82C2×C4×D109—21no—
83C2×C4×D109C2311yescharacteristic
84Dic436—22no—
85Dic436—22no—
86#13182019—41no—
87#13182018—22no—
88#13182018D414yescomplemented
89#13182017—22no—
90#13182017—41no—
91#13182016—22no—
92C2×#11724784C2311yescharacteristic
93C2×#11724784C2311yescharacteristic
94C8×D109—22no—
95C4⋊Dic109—21no—
96C4⋊Dic109—21no—
97#13182008—21no—
98#13182008—21no—
99#13182008—21no—
100#13182008D411yescharacteristic
101C22⋊Dic109—21no—
102C22⋊Dic109—21no—
103C22⋊Dic109D411yescharacteristic
104#13182006—21no—
105#13182006—21no—
106#13182006—21no—
107#13182006D411yescharacteristic
108C4×Dic109—21no—
109C4×Dic109C2311yescharacteristic
110C2×Q8×C109C2311yescharacteristic
111C2×D4×C109—21no—
112C2×D4×C109C2311yescharacteristic
113C22×C4×C109C2311yescharacteristic
114SD16×C109—22no—
115M16×C109—22no—
116C2×C8×C109C2311yescharacteristic
117C2×C8×C109C2311yescharacteristic
118C4⋊C4×C109—21no—
119C4⋊C4×C109C2311yescharacteristic
120C4⋊C4×C109C2311yescharacteristic
121C22⋊C4×C109—21no—
122C22⋊C4×C109—21no—
123C23×C109—21no—
124C23×C109C2×D411yescharacteristic
125C22×D109—21no—
126C22×D109—42no—
127C22×D109—42no—
128C22×D109—81no—
129C22×D109C2×D411yescharacteristic
130Q8×C109—21no—
131Q8×C109—21no—
132D4×C109—21no—
133D4×C109—21no—
134D4×C109—81no—
135C2×C4×C109—21no—
136C2×C4×C109—21no—
137C2×C4×C109—21no—
138C2×C4×C109—21no—
139C2×C4×C109—21no—
140C2×C4×C109C2×D411yescharacteristic
141C2×C4×C109C2411yescharacteristic
142C22⋊D109—22no—
143C22⋊D109—42no—
144C22⋊D109—42no—
145C2×Dic109—21no—
146C2×Dic109—21no—
147C2×Dic109—21no—
148C2×Dic109—42no—
149C2×Dic109C2×D411yescharacteristic
150C2×Dic109C2×D411yescharacteristic
151C2×Dic109C2×D411yescharacteristic
152D436—21no—
153D436C2×D412yes—
154C4×D109—21no—
155C4×D109—21no—
156C4×D109—42no—
157C4×D109—42no—
158C4×D109—81no—
159Dic218—21no—
160Dic218—21no—
161Dic218—21no—
162Dic218—21no—
163Dic218—21no—
164Dic218—42no—
165Dic218—42no—
166Dic218C2×D412yes—
167C8×C109—21no—
168C8×C109C2×D412yes—
169#11724784—21no—
170#11724784—21no—
171C22×C109—21no—
172C22×C109—21no—
173C22×C109—21no—
174C22×C109—41no—
175C22×C109—41no—
176C22×C109—41no—
177C22×C109C22×D411yescharacteristic
178C2×D109—21no—
179C2×D109—21no—
180C2×D109—22no—
181C2×D109—41no—
182C2×D109—41no—
183C2×D109—84no—
184C2×D109#13712yescomplemented
185C4×C109—21no—
186C4×C109—41no—
187C4×C109—41no—
188C4×C109—41no—
189C4×C109C22×D411yescharacteristic
190C4×C109H(5,𝔽2)11yescharacteristic
191Dic109—22no—
192Dic109—21no—
193Dic109—21no—
194Dic109—21no—
195Dic109—41no—
196Dic109—41no—
197C2×C109—41no—
198C2×C109—81no—
199C2×C109D4211yescharacteristic
200C2×C109C2×#13711yescharacteristic
201C2×C109#57711yescharacteristic
202D109—22no—
203D109—42no—
204#3266—1091nomaximal, Sylow
205C109#326611yescharacteristic, complemented, Sylow
206C2×#137—1091no—
207C22×SD16—1091no—
208D4×Q8—1091no—
209#546—1091no—
210#497—1091no—
211#468D10911yescharacteristic, complemented
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#435—1091no—
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—2184no—
235C2×SD16C2×D10911yescharacteristic
236C2×M16C2×D10911yescharacteristic, complemented
237C22×C8—1091no—
238C4⋊Q8—1091no—
239C4⋊Q8—2181no—
240#124—1091no—
241#123—2181no—
242C22⋊Q8—1091no—
243C22⋊Q8—1091no—
244C22⋊Q8—2181no—
245C22⋊Q8—2181no—
246C22⋊Q8C2×D10911yescharacteristic, complemented
247#121—2181no—
248#121C2×D10911yescharacteristic
249#120—2181no—
250C4×Q8—1091no—
251C4×D4—1091no—
252C4×D4—1091no—
253C4×D4—2181no—
254C2×C22⋊C4—2181no—
255#107C2×D10911yescharacteristic, complemented
256C4⋊C8—1091no—
257#103—1091no—
258#103—1091no—
259#103—1091no—
260#103C2×D10911yescharacteristic, complemented
261#102—1091no—
262#102C2×D10911yescharacteristic
263C22⋊C8—1091no—
264C22⋊C8—1091no—
265#97—1091no—
266C24—2181no—
267#41—4361no—
268#41—2182no—
269C2×Q8—1091no—
270C2×Q8—1091no—
271C2×Q8—1091no—
272C2×Q8—4361no—
273C2×Q8—4361no—
274C2×Q8—2181no—
275C2×Q8—2182no—
276C2×Q8—2182no—
277C2×Q8—2181no—
278C2×Q8C22×D10911yescharacteristic
279C2×D4—21no—
280C2×D4—1091no—
281C2×D4—1091no—
282C2×D4—4361no—
283C2×D4—2181no—
284C2×D4—2182no—
285C2×D4—2181no—
286C2×D4C22×D10911yescharacteristic
287C22×C4—1091no—
288C22×C4—2181no—
289C22×C4—2181no—
290C22×C4—2181no—
291C22×C4—2181no—
292C22×C4C22×D10911yescharacteristic
293Q16—4361no—
294Q16—2182no—
295Q16—2182no—
296SD16—22no—
297SD16—1094no—
298SD16—4361no—
299SD16—2182no—
300SD16—2182no—
301SD16—2182no—
302M16—22no—
303C2×C8—1091no—
304C2×C8—1091no—
305C2×C8—2182no—
306C2×C8C22×D10911yescharacteristic
307C2×C8C22×D10911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—1091no—
310C4⋊C4—2181no—
311C4⋊C4—2181no—
312C4⋊C4—2181no—
313C4⋊C4—2181no—
314C4⋊C4—2181no—
315C4⋊C4C22×D10911yescharacteristic
316C4⋊C4C22×D10911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1091no—
320C22⋊C4—1091no—
321C22⋊C4—2181no—
322C22⋊C4—2181no—
323C22⋊C4—2181no—
324C22⋊C4—2181no—
325C22⋊C4—2181no—
326C42—1091no—
327C42—2181no—
328C23—21no—
329C23—8721no—
330C23—1091no—
331C23—4362no—
332C23—4362no—
333C23—2181no—
334C23D4×D10911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1092no—
338Q8—4362no—
339Q8—4362no—
340Q8—2181no—
341Q8—2181no—
342Q8—2181no—
343Q8—2181no—
344Q8—2181no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—1092no—
349D4—4362no—
350D4—4362no—
351D4—2182no—
352D4—2181no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—8721no—
359C2×C4—1091no—
360C2×C4—1091no—
361C2×C4—1091no—
362C2×C4—4362no—
363C2×C4—4362no—
364C2×C4—4362no—
365C2×C4—2181no—
366C2×C4—2181no—
367C2×C4—2181no—
368C2×C4—2181no—
369C2×C4—2181no—
370C2×C4D4×D10911yescharacteristic
371C2×C4C23×D10911yescharacteristic
372C8—21no—
373C8—2181no—
374C8—2181no—
375C8D4×D10912yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8724no—
383C22—1092no—
384C22—4361no—
385C22—4361no—
386C22—2182no—
387C22—2181no—
388C22—2181no—
389C22C2×D4×D10911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—4361no—
395C4—4361no—
396C4—2181no—
397C4—2181no—
398C4—2181no—
399C4—2182no—
400C4C2×D4×D10911yescharacteristic
401C4#1681274511yescharacteristic
402C2—41no—
403C2—81no—
404C2—4362no—
405C2—2182no—
406C2#2029566011yescharacteristic
407C2#137×D10911yescharacteristic
408C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#468×C109C2
socleC22×C109C22×D4
prime radical#20295673C2
hypercenter#468D109
hypocenterC109#3266
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×SD16×D109C2semidirect
3. #20295689C2semidirect
5. #20295417C2semidirect
6. #20295268C2semidirect
7. #20295247C2semidirect
8. #20295230C2semidirect
9. #20295186C2semidirect
10. #20295183C2semidirect
11. #20295178C2semidirect
12. #20295164C2semidirect
13. #20295125C2semidirect
14. #20295095C2semidirect
15. #468×C109C2semidirect
17. C22×Dic218C22semidirect
21. C2×Dic436C22semidirect
22. C2×Dic436C22semidirect
23. C2×#13182019C22semidirect
32. #16812697C22semidirect
41. #16812664C22semidirect
46. #16812628C22semidirect
47. #16812622C22semidirect
48. #16812622C22semidirect
49. #16812622C22semidirect
51. C22⋊#11724784C22semidirect
53. #16812609C22semidirect
55. C2×M16×C109C22semidirect
56. C22⋊Q8×C109C22semidirect
58. #107×C109C22semidirect
59. #103×C109C22semidirect
88. #13182018D4semidirect
184. C2×D109#137semidirect
205. C109#3266semidirect
211. #468D109semidirect
236. C2×M16C2×D109semidirect
246. C22⋊Q8C2×D109semidirect
255. #107C2×D109semidirect
260. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×SD16×D1092
#202956892
#202956732
#202954172
#202952682
#202952472
#202952302
#202951862
#202951832
#202951782
#202951642
#202951252
#202950952
#468×C1092
#3266109