Finite Group Databasebeta

← #28374546

Subgroups — #28374546

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×#16812728C211yesmaximal, characteristic, complemented
2C22×#13182017C211yesmaximal, characteristic, complemented
3#20295689C211yesmaximal, characteristic, complemented
4#20295670C211yesmaximal, characteristic
5#20295417C211yesmaximal, characteristic, complemented
6#20295271C211yesmaximal, characteristic, complemented
7#20295265C211yesmaximal, characteristic, complemented
8#20295247C211yesmaximal, characteristic, complemented
9#20295229C211yesmaximal, characteristic, complemented
10#20295189C211yesmaximal, characteristic, complemented
11#20295184C211yesmaximal, characteristic, complemented
12#20295163C211yesmaximal, characteristic, complemented
13#20295123C211yesmaximal, characteristic, complemented
14#20295094C211yesmaximal, characteristic, complemented
15#450×C109C211yesmaximal, characteristic, complemented
16C2×#13182028C2211yescharacteristic
17C22×Dic218C2211yescharacteristic
18C2×Q8×D109C2211yescharacteristic, complemented
19#16812728—24no—
20C2×Dic436C2211yescharacteristic, complemented
21C2×Dic436C2211yescharacteristic, complemented
22C2×#13182019C2211yescharacteristic, complemented
23C2×#13182017—24no—
24C2×#13182017C2211yescharacteristic
25C2×#13182017C2211yescharacteristic
26C2×#13182017C2211yescharacteristic
27C2×#13182017C2211yescharacteristic
28C2×#13182012C2211yescharacteristic, complemented
29C22×#11724784C2211yescharacteristic
30#16812698—21no—
31#16812697C2211yescharacteristic, complemented
32#16812686C2211yescharacteristic
33#16812680—21no—
34#16812679C2211yescharacteristic, complemented
35#16812678C2211yescharacteristic
36#16812677—21no—
37#16812676—21no—
38#16812671—21no—
39#16812670C2211yescharacteristic
40#16812668—21no—
41#16812664C2211yescharacteristic, complemented
42D4×Dic109C2211yescharacteristic
43C4×C22⋊D109C2211yescharacteristic
44#16812659—21no—
45C2×C22⋊Dic109—21no—
46#16812632C2211yescharacteristic, complemented
47#16812628C2211yescharacteristic, complemented
48#16812624C2211yescharacteristic, complemented
49#16812624C2211yescharacteristic, complemented
50#16812623C2211yescharacteristic, complemented
51#16812622C2211yescharacteristic, complemented
52#16812611C2211yescharacteristic, complemented
53#16812609C2211yescharacteristic, complemented
54C22×D4×C109C2211yescharacteristic
55C2×SD16×C109C2211yescharacteristic
56C2×SD16×C109C2211yescharacteristic
57C22⋊Q8×C109C2211yescharacteristic
58#102×C109C2211yescharacteristic
59#102×C109C2211yescharacteristic
60C22⋊C8×C109C2211yescharacteristic
61#13182028—22no—
62#13182028—41no—
63C2×Dic218—22no—
64C2×Dic218—21no—
65C2×Dic218—21no—
66C2×Dic218—41no—
67C2×Dic218C2311yescharacteristic
68C2×Dic218C2311yescharacteristic
69C2×Dic218C2311yescharacteristic
70Q8×D109—22no—
71Q8×D109—41no—
72C2×C22⋊D109—21no—
73C2×C22⋊D109—21no—
74C2×C22⋊D109D411yescharacteristic
75C22×Dic109—21no—
76C22×Dic109—21no—
77C22×Dic109—21no—
78C2×C4×D109—21no—
79C2×C4×D109C2311yescharacteristic
80Dic436—22no—
81Dic436—41no—
82#13182019—22no—
83#13182017—22no—
84#13182017—22no—
85#13182017—22no—
86#13182017D414yescomplemented
87#13182012—22no—
88C2×#11724784—22no—
89C2×#11724784C2311yescharacteristic
90C2×#11724784C2311yescharacteristic
91C4⋊Dic109—21no—
92C4⋊Dic109C2311yescharacteristic
93#13182008—21no—
94#13182008—21no—
95#13182008—21no—
96#13182008—21no—
97#13182008D411yescharacteristic
98C22⋊Dic109—21no—
99C22⋊Dic109—21no—
100C22⋊Dic109—21no—
101C22⋊Dic109—21no—
102C22⋊Dic109—21no—
103C22⋊Dic109D411yescharacteristic
104#13182006—21no—
105#13182006D411yescharacteristic
106C4×Dic109—21no—
107C4×Dic109C2311yescharacteristic
108C24×C109—21no—
109C2×Q8×C109C2311yescharacteristic
110C2×D4×C109—22no—
111C2×D4×C109—21no—
112C2×D4×C109—41no—
113C2×D4×C109C2311yescharacteristic
114C2×D4×C109C2311yescharacteristic
115C22×C4×C109C2311yescharacteristic
116SD16×C109—22no—
117SD16×C109—41no—
118C2×C8×C109C2311yescharacteristic
119C2×C8×C109C2311yescharacteristic
120C4⋊C4×C109—21no—
121C4⋊C4×C109C2311yescharacteristic
122C22⋊C4×C109—21no—
123C23×C109—21no—
124C23×C109—21no—
125C23×C109—42no—
126C23×C109—42no—
127C23×C109—81no—
128C23×C109C2×D411yescharacteristic
129C22×D109C2×D411yescharacteristic
130Q8×C109—21no—
131Q8×C109—21no—
132D4×C109—21no—
133D4×C109—21no—
134D4×C109—21no—
135D4×C109—42no—
136D4×C109C2×D412yes—
137C2×C4×C109—21no—
138C2×C4×C109—21no—
139C2×C4×C109—21no—
140C2×C4×C109—21no—
141C2×C4×C109C2×D411yescharacteristic
142C2×C4×C109C2411yescharacteristic
143C22⋊D109—22no—
144C22⋊D109—42no—
145C22⋊D109—81no—
146C2×Dic109—21no—
147C2×Dic109—21no—
148C2×Dic109—21no—
149C2×Dic109—21no—
150C2×Dic109—42no—
151C2×Dic109—42no—
152C2×Dic109—81no—
153C2×Dic109C2×D411yescharacteristic
154C2×Dic109C2×D411yescharacteristic
155C2×Dic109C2×D411yescharacteristic
156C4×D109—21no—
157C4×D109—21no—
158C4×D109—42no—
159Dic218—21no—
160Dic218—21no—
161Dic218—21no—
162Dic218—21no—
163Dic218—21no—
164Dic218—42no—
165Dic218—42no—
166Dic218C2×D412yes—
167C8×C109—21no—
168C8×C109—21no—
169#11724784—21no—
170#11724784C2×D412yes—
171C22×C109—21no—
172C22×C109—21no—
173C22×C109—22no—
174C22×C109—41no—
175C22×C109—41no—
176C22×C109—41no—
177C22×C109—41no—
178C22×C109—41no—
179C22×C109—84no—
180C22×C109#13712yescomplemented
181C22×C109C22×D411yescharacteristic
182C2×D109—21no—
183C2×D109—21no—
184C2×D109—21no—
185C4×C109—21no—
186C4×C109—41no—
187C4×C109—41no—
188C4×C109C22×D411yescharacteristic
189C4×C109H(5,𝔽2)11yescharacteristic
190Dic109—22no—
191Dic109—21no—
192Dic109—21no—
193Dic109—21no—
194Dic109—41no—
195Dic109—41no—
196Dic109—41no—
197C2×C109—22no—
198C2×C109—42no—
199C2×C109—81no—
200C2×C109D4211yescharacteristic
201C2×C109C2×#13711yescharacteristic
202C2×C109#57711yescharacteristic
203D109—41no—
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#450D10911yescharacteristic, complemented
218#448—1091no—
219#440—1091no—
220#435—1091no—
221C2×#41—1091no—
222C22×Q8—1091no—
223C22×Q8—1091no—
224C22×D4C2×D10911yescharacteristic
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—2184no—
234C2×SD16C2×D10911yescharacteristic
235C2×SD16C2×D10911yescharacteristic
236C2×M16—1091no—
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
247#121—1091no—
248#121—2181no—
249#120—2181no—
250C4×Q8—1091no—
251C4×D4—1091no—
252C4×D4—1091no—
253C4×D4—2181no—
254C2×C22⋊C4—2181no—
255#107—1091no—
256C4⋊C8—1091no—
257#103—1091no—
258#103—1091no—
259#103—1091no—
260#103—1091no—
261#102C2×D10911yescharacteristic
262#102C2×D10911yescharacteristic
263C22⋊C8—1091no—
264C22⋊C8C2×D10911yescharacteristic
265#97—1091no—
266C24—21no—
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—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1091no—
283C2×D4—2181no—
284C2×D4—2181no—
285C2×D4C22×D10911yescharacteristic
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—41no—
298SD16—1094no—
299SD16—2182no—
300SD16—2182no—
301SD16—2182no—
302M16—2182no—
303C2×C8—1091no—
304C2×C8—1091no—
305C2×C8—2182no—
306C2×C8C22×D10911yescharacteristic
307C2×C8C22×D10911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—1091no—
310C4⋊C4—1091no—
311C4⋊C4—2181no—
312C4⋊C4—2181no—
313C4⋊C4—2181no—
314C4⋊C4—2181no—
315C4⋊C4—2181no—
316C4⋊C4C22×D10911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1091no—
319C22⋊C4—1091no—
320C22⋊C4—2181no—
321C22⋊C4—2181no—
322C22⋊C4—2181no—
323C22⋊C4—2181no—
324C22⋊C4—2181no—
325C22⋊C4—2181no—
326C42—1091no—
327C42—2181no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1091no—
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—21no—
348D4—42no—
349D4—8721no—
350D4—4362no—
351D4—2182no—
352D4D4×D10912yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8721no—
358C2×C4—1091no—
359C2×C4—1091no—
360C2×C4—1091no—
361C2×C4—4362no—
362C2×C4—4362no—
363C2×C4—4362no—
364C2×C4—2181no—
365C2×C4—2181no—
366C2×C4—2181no—
367C2×C4—2181no—
368C2×C4—2181no—
369C2×C4—2181no—
370C2×C4D4×D10911yescharacteristic
371C2×C4C23×D10911yescharacteristic
372C8—21no—
373C8—21no—
374C8—1092no—
375C8—2181no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2181no—
386C22—2181no—
387C22—2181no—
388C22#1681272812yescomplemented
389C22C2×D4×D10911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—4361no—
394C4—4361no—
395C4—4361no—
396C4—2181no—
397C4—2181no—
398C4—2181no—
399C4—2182no—
400C4C2×D4×D10911yescharacteristic
401C4#1681274411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—4361no—
406C2#2029565811yescharacteristic
407C2C2×#1681272811yescharacteristic
408C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#450×C109C2
socleC22×C109C22×D4
prime radical#20295670C2
hypercenter#450D109
hypocenterC109#3266
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16812728C2semidirect
2. C22×#13182017C2semidirect
3. #20295689C2semidirect
5. #20295417C2semidirect
6. #20295271C2semidirect
7. #20295265C2semidirect
8. #20295247C2semidirect
9. #20295229C2semidirect
10. #20295189C2semidirect
11. #20295184C2semidirect
12. #20295163C2semidirect
13. #20295123C2semidirect
14. #20295094C2semidirect
15. #450×C109C2semidirect
18. C2×Q8×D109C22semidirect
20. C2×Dic436C22semidirect
21. C2×Dic436C22semidirect
22. C2×#13182019C22semidirect
28. C2×#13182012C22semidirect
31. #16812697C22semidirect
34. #16812679C22semidirect
41. #16812664C22semidirect
46. #16812632C22semidirect
47. #16812628C22semidirect
48. #16812624C22semidirect
49. #16812624C22semidirect
50. #16812623C22semidirect
51. #16812622C22semidirect
52. #16812611C22semidirect
53. #16812609C22semidirect
86. #13182017D4semidirect
180. C22×C109#137semidirect
205. C109#3266semidirect
217. #450D109semidirect
388. C22#16812728semidirect

Maximal subgroups

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

subgroupindex
C2×#168127282
C22×#131820172
#202956892
#202956702
#202954172
#202952712
#202952652
#202952472
#202952292
#202951892
#202951842
#202951632
#202951232
#202950942
#450×C1092
#3266109