Finite Group Databasebeta

← #27182636

Subgroups — #27182636

Order 12928.

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×SD16×D101C211yesmaximal, characteristic, complemented
2Q8×C22⋊D101C211yesmaximal, characteristic, complemented
3#20265815C211yesmaximal, characteristic
4#20265570C211yesmaximal, characteristic, complemented
5#20265423C211yesmaximal, characteristic, complemented
6#20265415C211yesmaximal, characteristic, complemented
7#20265401C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265331C211yesmaximal, characteristic, complemented
11#20265326C211yesmaximal, characteristic, complemented
12#20265316C211yesmaximal, characteristic, complemented
13#20265278C211yesmaximal, characteristic, complemented
14#20265246C211yesmaximal, characteristic, complemented
15C2×#137×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C2×#13108064C2211yescharacteristic, complemented
19C2×#13108063C2211yescharacteristic
20C2×#13108062C2211yescharacteristic
21C2×#13108061C2211yescharacteristic
22C2×#13108061C2211yescharacteristic
23SD16×D101—24no—
24C2×C8×D101C2211yescharacteristic
25#15692253—21no—
26#15692253C2211yescharacteristic, complemented
27#15692244C2211yescharacteristic
28#15692238—21no—
29#15692234—21no—
30#15692234C2211yescharacteristic, complemented
31#15692231—21no—
32#15692231C2211yescharacteristic
33#15692231C2211yescharacteristic
34#15692229C2211yescharacteristic
35#15692227—21no—
36#15692223—21no—
37Q8×Dic101C2211yescharacteristic, complemented
38D4×Dic101C2211yescharacteristic
39C4×C22⋊D101—21no—
40C4×C22⋊D101C2211yescharacteristic
41C2×#13108051—21no—
42#15692189C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692179C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic
49#15692176C2211yescharacteristic
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic
52#15692165C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic
54C22×Q8×C101C2211yescharacteristic, complemented
55#137×C101—24no—
56C2×Q16×C101C2211yescharacteristic, complemented
57C2×Q16×C101C2211yescharacteristic, complemented
58C2×SD16×C101C2211yescharacteristic
59C2×SD16×C101C2211yescharacteristic
60C2×M16×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C2×Dic202—21no—
63C2×Dic202C2311yescharacteristic
64Q8×D101—22no—
65Q8×D101—41no—
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101D411yescharacteristic
70D4×D101—22no—
71D4×D101—41no—
72C22×Dic101—21no—
73C2×C4×D101—21no—
74C2×C4×D101C2311yescharacteristic
75#13108064—41no—
76#13108063—22no—
77#13108062—22no—
78#13108061—22no—
79#13108061—41no—
80C2×#11717579C2311yescharacteristic
81C2×#11717579C2311yescharacteristic
82C8×D101—22no—
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⋊Dic101D411yescharacteristic
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051D411yescharacteristic
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×C101—22no—
118SD16×C101—22no—
119SD16×C101D414yescomplemented
120M16×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—42no—
127C22×D101—42no—
128C22×D101—81no—
129C22×D101C2×D411yescharacteristic
130Q8×C101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—21no—
135Q8×C101—42no—
136Q8×C101C2×D412yes—
137D4×C101—21no—
138D4×C101—42no—
139D4×C101C2×D412yes—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—42no—
146C2×C4×C101—42no—
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2411yescharacteristic
149C22⋊D101—22no—
150C22⋊D101—42no—
151C22⋊D101—81no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—81no—
156C2×Dic101C2×D411yescharacteristic
157C2×Dic101C2×D411yescharacteristic
158C2×Dic101C2×D411yescharacteristic
159D404—21no—
160D404—21no—
161C4×D101—21no—
162C4×D101—21no—
163C4×D101—42no—
164Dic202—21no—
165Dic202—21no—
166Dic202—42no—
167C8×C101—21no—
168C8×C101C2×D412yes—
169#11717579—21no—
170#11717579—21no—
171C22×C101—22no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101C22×D411yescharacteristic
178C2×D101—21no—
179C2×D101—21no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—84no—
184C2×D101#13712yescomplemented
185C4×C101—22no—
186C4×C101—21no—
187C4×C101—41no—
188C4×C101—41no—
189C4×C101C22×D411yescharacteristic
190C4×C101H(5,𝔽2)11yescharacteristic
191Dic101—21no—
192Dic101—21no—
193Dic101—21no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—42no—
198C2×C101—41no—
199C2×C101D4211yescharacteristic
200C2×C101C2×#13711yescharacteristic
201C2×C101#57711yescharacteristic
202D101—22no—
203D101—81no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137D10111yescharacteristic, complemented
207C22×SD16—1011no—
208D4×Q8—1011no—
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467—1011no—
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450—1011no—
218#448—1011no—
219#440—1011no—
220#435—1011no—
221C2×#41C2×D10111yescharacteristic
222C22×Q8—1011no—
223C22×Q8C2×D10111yescharacteristic, complemented
224C22×D4—1011no—
225#137—24no—
226C2×Q16—1011no—
227C2×Q16C2×D10111yescharacteristic, complemented
228C2×Q16C2×D10111yescharacteristic, complemented
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16—1011no—
233C2×SD16—2024no—
234C2×SD16C2×D10111yescharacteristic
235C2×SD16C2×D10111yescharacteristic
236C2×M16C2×D10111yescharacteristic, complemented
237C22×C8—1011no—
238C4⋊Q8—1011no—
239C4⋊Q8—2021no—
240#124—1011no—
241#123—2021no—
242C22⋊Q8—1011no—
243C22⋊Q8—1011no—
244C22⋊Q8—1011no—
245C22⋊Q8—2021no—
246C22⋊Q8—2021no—
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8—1011no—
251C4×D4—1011no—
252C4×D4—1011no—
253C4×D4—2021no—
254C2×C22⋊C4—2021no—
255#107—1011no—
256C4⋊C8—1011no—
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103—1011no—
261#102—1011no—
262#102—1011no—
263C22⋊C8—1011no—
264C22⋊C8—1011no—
265#97—1011no—
266C24—2021no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1011no—
273C2×Q8—4041no—
274C2×Q8—2021no—
275C2×Q8—2022no—
276C2×Q8C22×D10111yescharacteristic
277C2×Q8C22×D10111yescharacteristic
278C2×Q8C22×D10111yescharacteristic
279C2×D4—21no—
280C2×D4—1011no—
281C2×D4—1011no—
282C2×D4—4041no—
283C2×D4—2021no—
284C2×D4—2021no—
285C2×D4—2022no—
286C2×D4C22×D10111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1011no—
290C22×C4—2021no—
291C22×C4—2021no—
292C22×C4C22×D10111yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—4041no—
296SD16—22no—
297SD16—4041no—
298SD16—2022no—
299SD16—2022no—
300SD16—2022no—
301SD16C22⋊D10114yescomplemented
302M16—22no—
303C2×C8—1011no—
304C2×C8—1011no—
305C2×C8—2022no—
306C2×C8C22×D10111yescharacteristic
307C2×C8C22×D10111yescharacteristic
308C4⋊C4—1011no—
309C4⋊C4—1011no—
310C4⋊C4—1011no—
311C4⋊C4—2021no—
312C4⋊C4—2021no—
313C4⋊C4—2021no—
314C4⋊C4—2021no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C22⋊C4—1011no—
318C22⋊C4—1011no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C42—1011no—
327C42—2021no—
328C23—21no—
329C23—8081no—
330C23—1011no—
331C23—4042no—
332C23—4042no—
333C23—2021no—
334C23D4×D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—4042no—
342Q8—2021no—
343Q8—2021no—
344Q8C2×C22⋊D10112yes—
345D4—21no—
346D4—42no—
347D4—8081no—
348D4—4042no—
349D4—2022no—
350D4—2021no—
351D4—2021no—
352D4C2×C22⋊D10112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—8081no—
361C2×C4—1011no—
362C2×C4—1011no—
363C2×C4—1011no—
364C2×C4—4042no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4—2021no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4D4×D10111yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—21no—
373C8—2021no—
374C8—2021no—
375C8C2×C22⋊D10112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—8084no—
383C22—1012no—
384C22—4041no—
385C22—4041no—
386C22—4041no—
387C22—2021no—
388C22—2021no—
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—4041no—
395C4—4041no—
396C4—4041no—
397C4—2021no—
398C4—2021no—
399C4—2021no—
400C4C22×C22⋊D10111yescharacteristic
401C4#1569230011yescharacteristic
402C2—41no—
403C2—42no—
404C2—8081no—
405C2—2022no—
406C2D4×C22⋊D10111yescharacteristic
407C2#137×D10111yescharacteristic
408C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
prime radical#20265815C2
hypercenterC2×#137D101
hypocenterC101#3266
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×SD16×D101C2semidirect
2. Q8×C22⋊D101C2semidirect
4. #20265570C2semidirect
5. #20265423C2semidirect
6. #20265415C2semidirect
7. #20265401C2semidirect
8. #20265380C2semidirect
9. #20265339C2semidirect
10. #20265331C2semidirect
11. #20265326C2semidirect
12. #20265316C2semidirect
13. #20265278C2semidirect
14. #20265246C2semidirect
15. C2×#137×C101C2semidirect
18. C2×#13108064C22semidirect
26. #15692253C22semidirect
30. #15692234C22semidirect
37. Q8×Dic101C22semidirect
42. #15692189C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692179C22semidirect
47. #15692179C22semidirect
50. C22⋊#11717579C22semidirect
52. #15692165C22semidirect
54. C22×Q8×C101C22semidirect
56. C2×Q16×C101C22semidirect
57. C2×Q16×C101C22semidirect
60. C2×M16×C101C22semidirect
119. SD16×C101D4semidirect
184. C2×D101#137semidirect
205. C101#3266semidirect
206. C2×#137D101semidirect
223. C22×Q8C2×D101semidirect
227. C2×Q16C2×D101semidirect
228. C2×Q16C2×D101semidirect
236. C2×M16C2×D101semidirect
301. SD16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1012
Q8×C22⋊D1012
#202658152
#202655702
#202654232
#202654152
#202654012
#202653802
#202653392
#202653312
#202653262
#202653162
#202652782
#202652462
C2×#137×C1012
#3266101