Finite Group Databasebeta

← #27181421

Subgroups — #27181421

Order 12928.

382 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#20265781C211yesmaximal, characteristic
2#20265653C211yesmaximal, characteristic, complemented
3#20265500C211yesmaximal, characteristic, complemented
4#20265499C211yesmaximal, characteristic, complemented
5#20265485C211yesmaximal, characteristic, complemented
6#20265484C211yesmaximal, characteristic, complemented
7#20265374C211yesmaximal, characteristic, complemented
8#20265314C211yesmaximal, characteristic, complemented
9#20265314C211yesmaximal, characteristic, complemented
10#20265278C211yesmaximal, characteristic, complemented
11#20265274C211yesmaximal, characteristic, complemented
12#20265268C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14#20265237C211yesmaximal, characteristic, complemented
15#460×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19#15692247C2211yescharacteristic
20#15692236—21no—
21#15692236—21no—
22#15692227—21no—
23#15692227—21no—
24#15692226C2211yescharacteristic
25#15692226C2211yescharacteristic
26Q8×Dic101C2211yescharacteristic
27C4×Dic202—21no—
28C4×Dic202C2211yescharacteristic
29#15692220—21no—
30#15692220C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33#15692213—21no—
34#15692213C2211yescharacteristic
35#15692212—21no—
36#15692212C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41C4⋊#11717579C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692167C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54#127×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61Q8×D101—22no—
62Q8×D101—22no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C22×Dic101—21no—
66C22×Dic101—21no—
67C2×C4×D101—21no—
68C2×C4×D101C2311yescharacteristic
69#13108062—22no—
70#13108062—22no—
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C4⋊Dic101—21no—
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82#13108053D411yescharacteristic
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87#13108051—21no—
88#13108051D411yescharacteristic
89#13108051D411yescharacteristic
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C2×D4×C101C2311yescharacteristic
100SD16×C101—22no—
101SD16×C101—22no—
102C2×C8×C101C2311yescharacteristic
103C2×C8×C101C2311yescharacteristic
104C4⋊C4×C101—21no—
105C4⋊C4×C101C2311yescharacteristic
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101—21no—
110C22×D101C2×D411yescharacteristic
111Q8×C101—21no—
112Q8×C101#4112yes—
113D4×C101—21no—
114D4×C101—21no—
115D4×C101—21no—
116D4×C101—21no—
117D4×C101—81no—
118D4×C101—81no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101—81no—
125C22⋊D101—81no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—81no—
131C2×Dic101—81no—
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C4×D101—24no—
136C4×D101—21no—
137C4×D101—21no—
138Dic202—24no—
139Dic202—21no—
140Dic202#4112yes—
141C8×C101—21no—
142C8×C101—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101C22×D411yescharacteristic
152C2×D101—21no—
153C2×D101—21no—
154C2×D101—21no—
155C4×C101—22no—
156C4×C101—21no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C101C2×#4111yescharacteristic
161Dic101—21no—
162Dic101—22no—
163Dic101—21no—
164Dic101—21no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168C2×C101—81no—
169C2×C101—81no—
170C2×C101#52511yescharacteristic
171C2×C101#57711yescharacteristic
172C2×C101#57711yescharacteristic
173D101—41no—
174#3126—1011nomaximal, Sylow
175C101#312611yescharacteristic, complemented, Sylow
176#540—1011no—
177#519—1011no—
178#482—1011no—
179#482—1011no—
180#479—1011no—
181#479—1011no—
182#460—1011no—
183#460D10111yescharacteristic, complemented
184#448—1011no—
185#448—1011no—
186#440—1011no—
187#440—1011no—
188C4×SD16—1011no—
189C4×SD16—1011no—
190#433—1011no—
191C22×Q8—1011no—
192C2×SD16—1011no—
193C2×SD16—1011no—
194C2×SD16C2×D10111yescharacteristic
195C2×SD16C2×D10111yescharacteristic
196#127C2×D10111yescharacteristic
197#125—1011no—
198#123—2021no—
199#123—2021no—
200#121—1011no—
201#121—1011no—
202#121—2021no—
203#121—2021no—
204C4×Q8—1011no—
205C4×Q8—1011no—
206C4×Q8—1011no—
207C4×Q8—2021no—
208C4×Q8—2021no—
209C4×Q8C2×D10111yescharacteristic
210C4×D4—1011no—
211C4×D4—1011no—
212#117—1011no—
213#117—1011no—
214#117—2021no—
215#117—2021no—
216#107—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221C4⋊C8C2×D10111yescharacteristic
222#103—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#102—1011no—
227#102—1011no—
228#102C2×D10111yescharacteristic
229#102C2×D10111yescharacteristic
230C22⋊C8—1011no—
231C22⋊C8—1011no—
232#97—1011no—
233C4×C8—1011no—
234C2×Q8—1011no—
235C2×Q8—2021no—
236C2×Q8—2022no—
237C2×Q8—2022no—
238C2×Q8C22×D10111yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—2021no—
242C2×D4—2021no—
243C2×D4C22×D10111yescharacteristic
244C2×D4C22×D10111yescharacteristic
245C22×C4—1011no—
246C22×C4—2021no—
247C22×C4—2021no—
248C22×C4—2021no—
249SD16—22no—
250SD16—22no—
251SD16—2022no—
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—1011no—
262C4⋊C4—1011no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C22⋊C4—1011no—
270C22⋊C4—1011no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C42—21no—
277C42—1011no—
278C42—2021no—
279C42—2021no—
280C42—2021no—
281C42—2021no—
282C42C22×D10111yescharacteristic
283C23—21no—
284C23—21no—
285C23—1011no—
286Q8—21no—
287Q8—1012no—
288Q8—2024no—
289Q8—2021no—
290Q8#1310807312yes—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—81no—
297D4—8081no—
298D4—8081no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—8081no—
302C2×C4—8081no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—1011no—
306C2×C4—2024no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4D4×D10111yescharacteristic
314C2×C4D4×D10111yescharacteristic
315C2×C4C23×D10111yescharacteristic
316C8—21no—
317C8—21no—
318C8—2021no—
319C8—2021no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2021no—
327C22—2021no—
328C22—2021no—
329C22C2×D4×D10111yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—4041no—
335C4—4041no—
336C4—4041no—
337C4—2021no—
338C4—2021no—
339C4—2022no—
340C4—2021no—
341C4#41×D10111yescharacteristic
342C4C2×#1310807311yescharacteristic
343C2—81no—
344C2—81no—
345C2—4041no—
346C2#2026568711yescharacteristic
347C2#2026598311yescharacteristic
348C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#460×C101C2
socleC22×C101C22×D4
prime radical#20265781C2
hypercenter#460D101
hypocenterC101#3126
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #20265653C2semidirect
3. #20265500C2semidirect
4. #20265499C2semidirect
5. #20265485C2semidirect
6. #20265484C2semidirect
7. #20265374C2semidirect
8. #20265314C2semidirect
9. #20265314C2semidirect
10. #20265278C2semidirect
11. #20265274C2semidirect
12. #20265268C2semidirect
13. SD16×Dic101C2semidirect
14. #20265237C2semidirect
15. #460×C101C2semidirect
175. C101#3126semidirect
183. #460D101semidirect

Maximal subgroups

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

subgroupindex
#202657812
#202656532
#202655002
#202654992
#202654852
#202654842
#202653742
#202653142
#202652782
#202652742
#202652682
SD16×Dic1012
#202652372
#460×C1012
#3126101