Finite Group Databasebeta

← #27184186

Subgroups — #27184186

Order 12928.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265910C211yesmaximal, characteristic
2#20265905C211yesmaximal, characteristic, complemented
3#20265571C211yesmaximal, characteristic, complemented
4#20265568C211yesmaximal, characteristic, complemented
5#20265548C211yesmaximal, characteristic
6#20265399C211yesmaximal, characteristic, complemented
7#20265398C211yesmaximal, characteristic, complemented
8#20265386C211yesmaximal, characteristic, complemented
9#20265382C211yesmaximal, characteristic, complemented
10#20265381C211yesmaximal, characteristic, complemented
11#20265376C211yesmaximal, characteristic
12#20265305C211yesmaximal, characteristic, complemented
13#20265276C211yesmaximal, characteristic, complemented
14#20265271C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×Dic404C2211yescharacteristic
17C2×Dic404C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108063C2211yescharacteristic
20C2×#13108063C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108062C2211yescharacteristic
23C2×#13108061C2211yescharacteristic
24#15692255C2211yescharacteristic
25#15692253C2211yescharacteristic
26#15692253C2211yescharacteristic
27#15692251C2211yescharacteristic
28#15692247—21no—
29#15692245—21no—
30#15692245—21no—
31#15692243C2211yescharacteristic
32#15692242C2211yescharacteristic
33#15692241—21no—
34#15692238—21no—
35#15692237—21no—
36C4×Dic202—21no—
37C4×Dic202C2211yescharacteristic
38#15692220C2211yescharacteristic
39#15692220C2211yescharacteristic
40#15692220C2211yescharacteristic
41#15692217C2211yescharacteristic
42C4×C22⋊D101—21no—
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic
59C2×Dic202D411yescharacteristic
60C2×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62C2×Dic202C2311yescharacteristic
63C2×D404C2311yescharacteristic
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C4×D101—21no—
67Dic404—22no—
68Dic404—41no—
69#13108064—41no—
70#13108063—22no—
71#13108063—41no—
72#13108062—41no—
73#13108062—41no—
74#13108061—41no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C4⋊Dic101—21no—
78C4⋊Dic101—21no—
79C4⋊Dic101D411yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053D411yescharacteristic
88#13108053D411yescharacteristic
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101C2311yescharacteristic
98C4×Dic101C2311yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101C2311yescharacteristic
101C22×C4×C101—21no—
102SD16×C101—22no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C22⋊C4×C101—21no—
109C42×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C22×D101—21no—
113Q8×C101—21no—
114Q8×C101—21no—
115D4×C101—21no—
116D4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—81no—
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101—42no—
125C22⋊D101—81no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133D404—21no—
134D404—21no—
135C4×D101—81no—
136Dic202—22no—
137Dic202—21no—
138Dic202—21no—
139Dic202—21no—
140Dic202—21no—
141Dic202—21no—
142Dic202—21no—
143C8×C101—21no—
144C8×C101#4112yes—
145#11717579—21no—
146#11717579—21no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101C22×D411yescharacteristic
151C2×D101—41no—
152C2×D101—41no—
153C2×D101—41no—
154C4×C101—21no—
155C4×C101—21no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C101H(5,𝔽2)11yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—21no—
164Dic101—41no—
165Dic101—41no—
166Dic101—41no—
167Dic101#13712yes—
168C2×C101—81no—
169C2×C101#55011yescharacteristic
170C2×C101C2×#13711yescharacteristic
171C2×C101#57711yescharacteristic
172D101—81no—
173#3400—1011nomaximal, Sylow
174C101#340011yescharacteristic, complemented, Sylow
175#559—1011no—
176#557—1011no—
177#497—1011no—
178#497—1011no—
179#492—1011no—
180#464—1011no—
181#464—1011no—
182#462—1011no—
183#461—1011no—
184#461—1011no—
185#460—1011no—
186#446—1011no—
187#440—1011no—
188#440D10111yescharacteristic, complemented
189C4×Q16—1011no—
190C2×Q16—1011no—
191C2×Q16—1011no—
192C2×Q16—1011no—
193C2×SD16—1011no—
194C2×SD16—1011no—
195C2×SD16—1011no—
196C2×SD16—1011no—
197C2×SD16—1011no—
198C2×SD16C2×D10111yescharacteristic
199C4⋊Q8—1011no—
200C4⋊Q8—1011no—
201C4⋊Q8—1011no—
202#127—1011no—
203#125—2021no—
204#125—2021no—
205#125—2021no—
206#124—1011no—
207#124—1011no—
208#124—2021no—
209#123—2021no—
210#123—2021no—
211C4×Q8—1011no—
212C4×Q8—1011no—
213C4×Q8—1011no—
214C4×Q8—1011no—
215C4×Q8—2021no—
216C4×Q8C2×D10111yescharacteristic
217C4×D4—1011no—
218C4×D4—2021no—
219C4×D4C2×D10111yescharacteristic
220#107C2×D10111yescharacteristic
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223C4⋊C8—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103C2×D10111yescharacteristic
228#102—1011no—
229#102C2×D10111yescharacteristic
230#97—1011no—
231#97C2×D10111yescharacteristic
232C4×C8—1011no—
233C2×Q8—1011no—
234C2×Q8—1011no—
235C2×Q8—1011no—
236C2×Q8—1011no—
237C2×Q8C22×D10111yescharacteristic
238C2×D4—1011no—
239C2×D4—2021no—
240C2×D4—2021no—
241C2×D4C22×D10111yescharacteristic
242C22×C4—21no—
243C22×C4—2021no—
244Q16—4041no—
245Q16—4041no—
246Q16—2022no—
247SD16—22no—
248SD16—4041no—
249SD16—4041no—
250SD16—4041no—
251SD16—4041no—
252SD16—2022no—
253C2×C8—1011no—
254C2×C8—1011no—
255C2×C8C22×D10111yescharacteristic
256C2×C8C22×D10111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—1011no—
259C4⋊C4—1011no—
260C4⋊C4—1011no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4C22×D10111yescharacteristic
271C4⋊C4C22×D10111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C22⋊C4—2021no—
278C42—21no—
279C42—1011no—
280C42—1011no—
281C42—2021no—
282C42—2021no—
283C42—2021no—
284C42C22×D10111yescharacteristic
285C23—21no—
286C23—2021no—
287Q8—21no—
288Q8—21no—
289Q8—2022no—
290Q8—2021no—
291Q8—2021no—
292Q8—2021no—
293Q8—2021no—
294Q8—2021no—
295Q8—2021no—
296D4—21no—
297D4—21no—
298D4—8081no—
299D4—4042no—
300D4—2021no—
301D4—2021no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—81no—
306C2×C4—8081no—
307C2×C4—1011no—
308C2×C4—1011no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4D4×D10111yescharacteristic
315C2×C4D4×D10111yescharacteristic
316C2×C4C23×D10111yescharacteristic
317C8—21no—
318C8—2021no—
319C8—2021no—
320C8#1310807412yes—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—4041no—
325C22—4041no—
326C22—4041no—
327C22C2×D4×D10111yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—1012no—
334C4—4041no—
335C4—4041no—
336C4—4041no—
337C4—2021no—
338C4—2021no—
339C4—2021no—
340C4C2×#1310807411yescharacteristic
341C4#1569230111yescharacteristic
342C2—81no—
343C2—8081no—
344C2#2026584811yescharacteristic
345C2#137×D10111yescharacteristic
346C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radical#15692251C22
hypercenter#440D101
hypocenterC101#3400
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20265905C2semidirect
3. #20265571C2semidirect
4. #20265568C2semidirect
6. #20265399C2semidirect
7. #20265398C2semidirect
8. #20265386C2semidirect
9. #20265382C2semidirect
10. #20265381C2semidirect
12. #20265305C2semidirect
13. #20265276C2semidirect
14. #20265271C2semidirect
15. #440×C101C2semidirect
174. C101#3400semidirect
188. #440D101semidirect

Maximal subgroups

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

subgroupindex
#202659102
#202659052
#202655712
#202655682
#202655482
#202653992
#202653982
#202653862
#202653822
#202653812
#202653762
#202653052
#202652762
#202652712
#440×C1012
#3400101