Finite Group Databasebeta

← #27181394

Subgroups — #27181394

Order 12928.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265798C211yesmaximal, characteristic, complemented
2#20265512C211yesmaximal, characteristic, complemented
3#20265490C211yesmaximal, characteristic, complemented
4#20265472C211yesmaximal, characteristic, complemented
5#20265457C211yesmaximal, characteristic, complemented
6#20265399C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265339C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic
10#20265277C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12C4×#13108064C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#520×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20#15692255C2211yescharacteristic
21#15692250—21no—
22#15692243C2211yescharacteristic
23#15692241—21no—
24#15692237—21no—
25#15692234C2211yescharacteristic
26#15692233—21no—
27#15692229C2211yescharacteristic
28C4×Dic202C2211yescharacteristic
29C4×D404C2211yescharacteristic
30#15692188C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692188C2211yescharacteristic
33#15692188C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35C4⋊#11717579C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692178C2211yescharacteristic
39#15692178C2211yescharacteristic
40#15692176C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C2×#41×C101C2211yescharacteristic
49C4×Q8×C101C2211yescharacteristic
50C4×Q8×C101C2211yescharacteristic
51C4×D4×C101—21no—
52C4×D4×C101—21no—
53C4×D4×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#117×C101—21no—
56#117×C101C2211yescharacteristic
57C2×C4⋊C4×C101—21no—
58C2×C4⋊C4×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C4×D101—21no—
64#13108064—22no—
65#13108062—22no—
66#13108061—22no—
67#13108059—22no—
68C2×#11717579C2311yescharacteristic
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C4⋊Dic101—21no—
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77C22⋊Dic101—21no—
78#13108051—21no—
79#13108051—21no—
80#13108051—21no—
81#13108051—21no—
82C4×Dic101—21no—
83#41×C101—22no—
84#41×C101—22no—
85C2×Q8×C101C2311yescharacteristic
86C2×D4×C101—21no—
87C2×D4×C101C2311yescharacteristic
88C22×C4×C101—21no—
89C22×C4×C101—21no—
90C22×C4×C101—21no—
91C22×C4×C101—21no—
92C22×C4×C101C2311yescharacteristic
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101D411yescharacteristic
97C4⋊C4×C101D411yescharacteristic
98C4⋊C4×C101C2311yescharacteristic
99C4⋊C4×C101C2311yescharacteristic
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101D411yescharacteristic
103C22⋊C4×C101D411yescharacteristic
104C42×C101—21no—
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—21no—
111Q8×C101—21no—
112Q8×C101#4112yes—
113D4×C101—24no—
114D4×C101—21no—
115D4×C101#4112yes—
116C2×C4×C101—24no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—81no—
123C2×C4×C101—81no—
124C2×C4×C101—81no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—81no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135D404—21no—
136D404—21no—
137C4×D101—81no—
138Dic202—21no—
139Dic202—21no—
140Dic202—81no—
141#11717579—21no—
142#11717579—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—21no—
146C22×C101—22no—
147C22×C101—21no—
148C22×C101—21no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101C22×D411yescharacteristic
152C2×D101—41no—
153C2×D101—41no—
154C2×D101—41no—
155C4×C101—21no—
156C4×C101—22no—
157C4×C101—21no—
158C4×C101—21no—
159C4×C101—21no—
160C4×C101—21no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101C2×#4111yescharacteristic
164C4×C101C2×#4111yescharacteristic
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168C2×C101—41no—
169C2×C101—42no—
170C2×C101#52511yescharacteristic
171C2×C101#57611yescharacteristic
172C2×C101#57811yescharacteristic
173D101—81no—
174#3124—1011nomaximal, Sylow
175C101#312411yescharacteristic, complemented, Sylow
176#542—1011no—
177#520D10111yescharacteristic, complemented
178#484—1011no—
179#480—1011no—
180#477—1011no—
181#474—1011no—
182#464—1011no—
183#463—1011no—
184#452—1011no—
185#449—1011no—
186#440—1011no—
187#440—1011no—
188C4×Q16—1011no—
189C4×D8—1011no—
190#433—1011no—
191C2×#41C2×D10111yescharacteristic
192C2×Q16—1011no—
193C2×SD16—1011no—
194C2×SD16—1011no—
195C2×D8—1011no—
196C4⋊Q8—1011no—
197#126—2021no—
198#124—1011no—
199#124—2021no—
200#123—2021no—
201C22⋊Q8—1011no—
202C22⋊Q8—2021no—
203#121—1011no—
204C4×Q8—1011no—
205C4×Q8C2×D10111yescharacteristic
206C4×Q8C2×D10111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1011no—
210C4×D4C2×D10111yescharacteristic
211C4×D4C2×D10111yescharacteristic
212#117—21no—
213#117C2×D10111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D10111yescharacteristic
216#107—1011no—
217#107—1011no—
218#107—1011no—
219#107—1011no—
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222#103—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230C22⋊C8—1011no—
231C22⋊C8—1011no—
232#97—1011no—
233C4×C8—1011no—
234#41—22no—
235#41—22no—
236C2×Q8—1011no—
237C2×Q8—2021no—
238C2×Q8C22×D10111yescharacteristic
239C2×D4—21no—
240C2×D4—1011no—
241C2×D4—2021no—
242C2×D4C22×D10111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2021no—
248C22×C4C22×D10111yescharacteristic
249Q16—2022no—
250SD16—2022no—
251SD16—2022no—
252D8—2022no—
253C2×C8—1011no—
254C2×C8—1011no—
255C2×C8—1011no—
256C2×C8—1011no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4C22⋊D10111yescharacteristic
266C4⋊C4C22⋊D10111yescharacteristic
267C4⋊C4C22×D10111yescharacteristic
268C4⋊C4C22×D10111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4C22⋊D10111yescharacteristic
277C22⋊C4C22⋊D10111yescharacteristic
278C42—21no—
279C42—21no—
280C42—2021no—
281C42C22×D10111yescharacteristic
282C42C22×D10111yescharacteristic
283C23—21no—
284C23—2021no—
285C23C2×C22⋊D10111yescharacteristic
286Q8—21no—
287Q8—8081no—
288Q8—2021no—
289Q8—2021no—
290Q8#1310807412yes—
291D4—24no—
292D4—21no—
293D4—8081no—
294D4—2021no—
295D4—2021no—
296D4#1310807412yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—8081no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4C2×C22⋊D10111yescharacteristic
311C2×C4C2×C22⋊D10111yescharacteristic
312C2×C4C2×C22⋊D10111yescharacteristic
313C2×C4C2×C22⋊D10111yescharacteristic
314C2×C4C2×C22⋊D10111yescharacteristic
315C2×C4C23×D10111yescharacteristic
316C8—2021no—
317C8—2021no—
318C8—2021no—
319C8—2021no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—4041no—
327C22—4041no—
328C22—4041no—
329C22C22×C22⋊D10111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—4041no—
339C4—4041no—
340C4—4041no—
341C4C2×#1310807411yescharacteristic
342C4C2×#1310807411yescharacteristic
343C2—41no—
344C2—42no—
345C2—8081no—
346C2#2026568811yescharacteristic
347C2#2026598011yescharacteristic
348C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#520×C101C2
socleC22×C101C22×D4
prime radical#20265322C2
hypercenter#520D101
hypocenterC101#3124
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20265798C2semidirect
2. #20265512C2semidirect
3. #20265490C2semidirect
4. #20265472C2semidirect
5. #20265457C2semidirect
6. #20265399C2semidirect
7. #20265392C2semidirect
8. #20265339C2semidirect
10. #20265277C2semidirect
11. #20265275C2semidirect
12. C4×#13108064C2semidirect
13. C4×#13108059C2semidirect
14. #20265238C2semidirect
15. #520×C101C2semidirect
175. C101#3124semidirect
177. #520D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202655122
#202654902
#202654722
#202654572
#202653992
#202653922
#202653392
#202653222
#202652772
#202652752
C4×#131080642
C4×#131080592
#202652382
#520×C1012
#3124101