Finite Group Databasebeta

← #27182696

Subgroups — #27182696

Order 12928.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×D101C211yesmaximal, characteristic, complemented
2Q8×C22⋊D101C211yesmaximal, characteristic, complemented
3#20265834C211yesmaximal, characteristic
4#20265568C211yesmaximal, characteristic, complemented
5#20265435C211yesmaximal, characteristic, complemented
6#20265403C211yesmaximal, characteristic, complemented
7#20265397C211yesmaximal, characteristic, complemented
8#20265385C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265333C211yesmaximal, characteristic, complemented
11#20265331C211yesmaximal, characteristic, complemented
12#20265314C211yesmaximal, characteristic, complemented
13#20265284C211yesmaximal, characteristic, complemented
14#20265246C211yesmaximal, characteristic, complemented
15C2×#137×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×Dic404C2211yescharacteristic
19C2×#13108064C2211yescharacteristic
20C2×#13108064C2211yescharacteristic
21C2×#13108064C2211yescharacteristic
22Q16×D101—24no—
23C2×#13108062C2211yescharacteristic
24C2×C8×D101C2211yescharacteristic
25#15692253—21no—
26#15692253C2211yescharacteristic
27#15692242—21no—
28#15692242C2211yescharacteristic
29#15692234—21no—
30#15692234—21no—
31#15692234C2211yescharacteristic
32#15692234C2211yescharacteristic
33#15692231—21no—
34#15692231C2211yescharacteristic
35#15692226—21no—
36#15692226C2211yescharacteristic
37Q8×Dic101C2211yescharacteristic
38Q8×Dic101C2211yescharacteristic
39C4×C22⋊D101—21no—
40C4×C22⋊D101—21no—
41C4×C22⋊D101C2211yescharacteristic
42#15692187C2211yescharacteristic
43#15692184C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692176C2211yescharacteristic
50C22⋊#11717579C2211yescharacteristic
51#15692167C2211yescharacteristic
52#15692165C2211yescharacteristic
53C2×#41×C101C2211yescharacteristic
54C22×Q8×C101C2211yescharacteristic
55#137×C101—24no—
56C2×Q16×C101C2211yescharacteristic
57C2×Q16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic
59C2×SD16×C101C2211yescharacteristic
60C2×M16×C101C2211yescharacteristic
61C2×Dic202—21no—
62C2×Dic202—21no—
63C2×Dic202C2311yescharacteristic
64C2×Dic202C2311yescharacteristic
65Q8×D101—22no—
66Q8×D101—22no—
67Q8×D101—41no—
68Q8×D101—41no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101D411yescharacteristic
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101C2311yescharacteristic
74Dic404—22no—
75#13108064—22no—
76#13108064—22no—
77#13108064—41no—
78#13108062—41no—
79C2×#11717579C2311yescharacteristic
80C2×#11717579C2311yescharacteristic
81C8×D101—22no—
82C4⋊Dic101—21no—
83C4⋊Dic101—21no—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101D411yescharacteristic
98#13108051—21no—
99#13108051D411yescharacteristic
100C4×Dic101—21no—
101C4×Dic101—21no—
102C4×Dic101C2311yescharacteristic
103#41×C101—22no—
104#41×C101—41no—
105C2×Q8×C101—21no—
106C2×Q8×C101—22no—
107C2×Q8×C101—41no—
108C2×Q8×C101C2311yescharacteristic
109C2×Q8×C101C2311yescharacteristic
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116Q16×C101—22no—
117Q16×C101D414yescomplemented
118SD16×C101—22no—
119SD16×C101—22no—
120M16×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101C2×D411yescharacteristic
126Q8×C101—21no—
127Q8×C101—21no—
128Q8×C101—21no—
129Q8×C101—21no—
130Q8×C101—42no—
131Q8×C101C2×D412yes—
132Q8×C101C2×D412yes—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—42no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—42no—
142C2×C4×C101—42no—
143C2×C4×C101C2×D411yescharacteristic
144C2×C4×C101C2411yescharacteristic
145C22⋊D101—22no—
146C22⋊D101—42no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101C2×D411yescharacteristic
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154C4×D101—21no—
155C4×D101—21no—
156C4×D101—42no—
157C4×D101—42no—
158Dic202—21no—
159Dic202—21no—
160Dic202—21no—
161Dic202—21no—
162Dic202—42no—
163Dic202—42no—
164C8×C101—21no—
165C8×C101C2×D412yes—
166#11717579—21no—
167#11717579—21no—
168C22×C101—21no—
169C22×C101—21no—
170C22×C101—21no—
171C22×C101—41no—
172C22×C101—41no—
173C22×C101—41no—
174C22×C101C22×D411yescharacteristic
175C2×D101—21no—
176C2×D101—21no—
177C2×D101#13712yescomplemented
178C4×C101—22no—
179C4×C101—22no—
180C4×C101—21no—
181C4×C101—41no—
182C4×C101C22×D411yescharacteristic
183C4×C101H(5,𝔽2)11yescharacteristic
184Dic101—21no—
185Dic101—21no—
186Dic101—21no—
187Dic101—41no—
188Dic101—41no—
189Dic101—41no—
190Dic101—41no—
191C2×C101—41no—
192C2×C101—81no—
193C2×C101D4211yescharacteristic
194C2×C101C2×#13711yescharacteristic
195C2×C101#57811yescharacteristic
196D101—22no—
197#3270—1011nomaximal, Sylow
198C101#327011yescharacteristic, complemented, Sylow
199C2×#137D10111yescharacteristic, complemented
200C22×Q16—1011no—
201D4×Q8—1011no—
202#548—1011no—
203#497—1011no—
204#470—1011no—
205#465—1011no—
206#464—1011no—
207#462—1011no—
208#452—1011no—
209#451—1011no—
210#451—1011no—
211#448—1011no—
212#441—1011no—
213#435—1011no—
214C2×#41C2×D10111yescharacteristic
215C22×Q8—1011no—
216C22×Q8—1011no—
217C22×Q8C2×D10111yescharacteristic
218#137—24no—
219C2×Q16—1011no—
220C2×Q16—1011no—
221C2×Q16—1011no—
222C2×Q16—1011no—
223C2×Q16—2024no—
224C2×Q16C2×D10111yescharacteristic
225C2×Q16C2×D10111yescharacteristic
226C2×SD16—1011no—
227C2×SD16C2×D10111yescharacteristic
228C2×SD16C2×D10111yescharacteristic
229C2×M16C2×D10111yescharacteristic
230C22×C8—1011no—
231C4⋊Q8—1011no—
232C4⋊Q8—2021no—
233#124—1011no—
234#124—2021no—
235C22⋊Q8—1011no—
236C22⋊Q8—1011no—
237C22⋊Q8—1011no—
238C22⋊Q8—2021no—
239C22⋊Q8—2021no—
240C22⋊Q8—2021no—
241#121—1011no—
242#121—2021no—
243C4×Q8—1011no—
244C4×Q8—1011no—
245C4×D4—1011no—
246C4×D4—2021no—
247C4×D4—2021no—
248#106—1011no—
249C4⋊C8—1011no—
250#103—1011no—
251#103—1011no—
252#103—1011no—
253#103—1011no—
254#103—1011no—
255#102—1011no—
256C22⋊C8—1011no—
257C22⋊C8—1011no—
258#97—1011no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—1011no—
265C2×Q8—1011no—
266C2×Q8—4041no—
267C2×Q8—4041no—
268C2×Q8—2021no—
269C2×Q8—2021no—
270C2×Q8—2022no—
271C2×Q8—2022no—
272C2×Q8C22×D10111yescharacteristic
273C2×Q8C22×D10111yescharacteristic
274C2×Q8C22×D10111yescharacteristic
275C2×D4—21no—
276C2×D4—1011no—
277C2×D4—2021no—
278C2×D4C22×D10111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1011no—
282C22×C4—2021no—
283C22×C4—2021no—
284C22×C4C22×D10111yescharacteristic
285Q16—22no—
286Q16—4041no—
287Q16—2022no—
288Q16—2022no—
289Q16—2022no—
290Q16C22⋊D10114yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—4041no—
294M16—22no—
295C2×C8—1011no—
296C2×C8—1011no—
297C2×C8—2022no—
298C2×C8C22×D10111yescharacteristic
299C2×C8C22×D10111yescharacteristic
300C4⋊C4—1011no—
301C4⋊C4—1011no—
302C4⋊C4—1011no—
303C4⋊C4—2021no—
304C4⋊C4—2021no—
305C4⋊C4—2021no—
306C4⋊C4—2021no—
307C4⋊C4—2021no—
308C4⋊C4—2021no—
309C4⋊C4—2021no—
310C22⋊C4—1011no—
311C22⋊C4—1011no—
312C22⋊C4—2021no—
313C22⋊C4—2021no—
314C22⋊C4—2021no—
315C22⋊C4—2021no—
316C22⋊C4—2021no—
317C22⋊C4—2021no—
318C42—1011no—
319C42—2021no—
320C42—2021no—
321C23—21no—
322C23—1011no—
323C23D4×D10111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—4042no—
330Q8—4042no—
331Q8—2021no—
332Q8—2021no—
333Q8—2021no—
334Q8—2021no—
335Q8C2×C22⋊D10112yes—
336Q8C2×C22⋊D10112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—4042no—
341D4—2022no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—1011no—
350C2×C4—1011no—
351C2×C4—1011no—
352C2×C4—4042no—
353C2×C4—4042no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4—2021no—
358C2×C4—2021no—
359C2×C4—2021no—
360C2×C4D4×D10111yescharacteristic
361C2×C4C23×D10111yescharacteristic
362C8—21no—
363C8—2021no—
364C8—2021no—
365C8C2×C22⋊D10112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1012no—
373C22—2021no—
374C22—2021no—
375C22C2×D4×D10111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—4041no—
381C4—4041no—
382C4—4041no—
383C4—4041no—
384C4—2021no—
385C4—2021no—
386C4—2021no—
387C4C22×C22⋊D10111yescharacteristic
388C4#1569230011yescharacteristic
389C2—41no—
390C2—81no—
391C2—2022no—
392C2D4×C22⋊D10111yescharacteristic
393C2#137×D10111yescharacteristic
394C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
prime radical#20265834C2
hypercenterC2×#137D101
hypocenterC101#3270
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D101C2semidirect
2. Q8×C22⋊D101C2semidirect
4. #20265568C2semidirect
5. #20265435C2semidirect
6. #20265403C2semidirect
7. #20265397C2semidirect
8. #20265385C2semidirect
9. #20265339C2semidirect
10. #20265333C2semidirect
11. #20265331C2semidirect
12. #20265314C2semidirect
13. #20265284C2semidirect
14. #20265246C2semidirect
15. C2×#137×C101C2semidirect
117. Q16×C101D4semidirect
177. C2×D101#137semidirect
198. C101#3270semidirect
199. C2×#137D101semidirect
290. Q16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1012
Q8×C22⋊D1012
#202658342
#202655682
#202654352
#202654032
#202653972
#202653852
#202653392
#202653332
#202653312
#202653142
#202652842
#202652462
C2×#137×C1012
#3270101