Finite Group Databasebeta

← #27183045

Subgroups — #27183045

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265895C211yesmaximal, characteristic, complemented
2#20265826C211yesmaximal, characteristic
3#20265526C211yesmaximal, characteristic, complemented
4#20265518C211yesmaximal, characteristic, complemented
5#20265517C211yesmaximal, characteristic, complemented
6#20265455C211yesmaximal, characteristic, complemented
7#20265418C211yesmaximal, characteristic, complemented
8#20265417C211yesmaximal, characteristic, complemented
9#20265386C211yesmaximal, characteristic, complemented
10#20265340C211yesmaximal, characteristic, complemented
11#20265339C211yesmaximal, characteristic, complemented
12#20265268C211yesmaximal, characteristic, complemented
13C8×C22⋊D101C211yesmaximal, characteristic, complemented
14#20265170C211yesmaximal, characteristic, complemented
15#416×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Dic404C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108062C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692253C2211yescharacteristic
23#15692249—21no—
24#15692248—21no—
25#15692241—21no—
26#15692238—21no—
27#15692236—21no—
28#15692235C2211yescharacteristic
29#15692234C2211yescharacteristic
30#15692234C2211yescharacteristic
31#15692231C2211yescharacteristic
32#15692227—21no—
33C4×Dic202—21no—
34#15692220C2211yescharacteristic
35D4×Dic101C2211yescharacteristic
36C4×C22⋊D101C2211yescharacteristic
37#15692213C2211yescharacteristic
38C2×#13108053—21no—
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54#117×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—22no—
60#13108073—41no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63C2×Dic202C2311yescharacteristic
64C2×C22⋊D101—21no—
65C2×C22⋊D101D411yescharacteristic
66C22×Dic101—21no—
67C22×Dic101—21no—
68C2×C4×D101C2311yescharacteristic
69Dic404—41no—
70#13108064—41no—
71#13108062—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C8×D101—41no—
75C4⋊Dic101—21no—
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101D411yescharacteristic
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051D411yescharacteristic
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104C2×C8×C101—41no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C42×C101—21no—
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101C2×D411yescharacteristic
114Q8×C101—21no—
115Q8×C101—21no—
116D4×C101—21no—
117D4×C101—42no—
118D4×C101#4112yes—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—42no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—22no—
128C22⋊D101—42no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—42no—
134C2×Dic101—42no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C4×D101—21no—
139C4×D101—21no—
140Dic202—21no—
141Dic202—21no—
142Dic202—21no—
143Dic202—21no—
144Dic202—42no—
145C8×C101—21no—
146C8×C101—21no—
147#11717579—21no—
148#11717579—21no—
149C22×C101—22no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101—21no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101C22×D411yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—21no—
159C4×C101—21no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101C2×#4111yescharacteristic
164C4×C101H(5,𝔽2)11yescharacteristic
165Dic101—21no—
166Dic101—21no—
167Dic101—41no—
168Dic101—41no—
169Dic101—41no—
170Dic101—41no—
171Dic101#13512yes—
172C2×C101—41no—
173C2×C101—42no—
174C2×C101#54611yescharacteristic
175C2×C101C2×#13511yescharacteristic
176C2×C101#57811yescharacteristic
177D101—41no—
178#3302—1011nomaximal, Sylow
179C101#330211yescharacteristic, complemented, Sylow
180#556—1011no—
181#547—1011no—
182#487—1011no—
183#485—1011no—
184#485—1011no—
185#474—1011no—
186#467—1011no—
187#467—1011no—
188#462—1011no—
189#452—1011no—
190#452—1011no—
191C4×SD16—1011no—
192C8×D4—1011no—
193#416—1011no—
194#416D10111yescharacteristic, complemented
195C2×#41—1011no—
196C2×#41C2×D10111yescharacteristic
197C2×Q16—1011no—
198C2×Q16—1011no—
199C2×SD16—1011no—
200C22×C8—1011no—
201C22×C8C2×D10111yescharacteristic
202C4⋊Q8—1011no—
203C4⋊Q8—1011no—
204#126—2021no—
205#126—2021no—
206#124—2021no—
207#123—2021no—
208#123—2021no—
209C22⋊Q8—1011no—
210C22⋊Q8—1011no—
211C22⋊Q8—1011no—
212C22⋊Q8—1011no—
213#121—2021no—
214C4×Q8—1011no—
215C4×Q8—2021no—
216C4×D4—1011no—
217C4×D4—1011no—
218#117—1011no—
219#117C2×D10111yescharacteristic
220C2×C4⋊C4—2021no—
221#107—1011no—
222#107—1011no—
223#106—1011no—
224C4⋊C8—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103—1011no—
230#103C2×D10111yescharacteristic
231#103C2×D10111yescharacteristic
232#102—1011no—
233#102C2×D10111yescharacteristic
234#102C2×D10111yescharacteristic
235C22⋊C8—1011no—
236C22⋊C8—1011no—
237C4×C8—1011no—
238#41—22no—
239#41—41no—
240#41—4041no—
241#41—2022no—
242C2×Q8—1011no—
243C2×Q8—1011no—
244C2×Q8—2021no—
245C2×Q8C22×D10111yescharacteristic
246C2×D4—21no—
247C2×D4—1011no—
248C2×D4—2021no—
249C2×D4C22×D10111yescharacteristic
250C22×C4—21no—
251C22×C4—1011no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4C22×D10111yescharacteristic
255Q16—4041no—
256Q16—4041no—
257SD16—2022no—
258C2×C8—41no—
259C2×C8—1011no—
260C2×C8—1011no—
261C2×C8—4041no—
262C2×C8C22×D10111yescharacteristic
263C2×C8C22×D10111yescharacteristic
264C4⋊C4—1011no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1011no—
279C22⋊C4—1011no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—21no—
287C42—1011no—
288C42—2021no—
289C42—2021no—
290C23—21no—
291C23—1011no—
292C23D4×D10111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—4042no—
296Q8—2021no—
297Q8—2021no—
298Q8—2021no—
299Q8—2021no—
300D4—21no—
301D4—42no—
302D4—4042no—
303D4—2022no—
304D4#1310807412yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—1011no—
312C2×C4—1011no—
313C2×C4—1011no—
314C2×C4—4042no—
315C2×C4—4042no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4D4×D10111yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C8—21no—
325C8—21no—
326C8—2021no—
327C8—2021no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—2021no—
335C22—2021no—
336C22—2021no—
337C22C2×D4×D10111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—1012no—
343C4—4041no—
344C4—4041no—
345C4—4041no—
346C4—4041no—
347C4—2021no—
348C4—2021no—
349C4C2×#1310807411yescharacteristic
350C4#1569230011yescharacteristic
351C2—41no—
352C2—42no—
353C2—4041no—
354C2#2026581711yescharacteristic
355C2#135×D10111yescharacteristic
356C2#2026599111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
prime radical#20265826C2
hypercenter#416D101
hypocenterC101#3302
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20265895C2semidirect
3. #20265526C2semidirect
4. #20265518C2semidirect
5. #20265517C2semidirect
6. #20265455C2semidirect
7. #20265418C2semidirect
8. #20265417C2semidirect
9. #20265386C2semidirect
10. #20265340C2semidirect
11. #20265339C2semidirect
12. #20265268C2semidirect
13. C8×C22⋊D101C2semidirect
14. #20265170C2semidirect
15. #416×C101C2semidirect
179. C101#3302semidirect
194. #416D101semidirect

Maximal subgroups

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

subgroupindex
#202658952
#202658262
#202655262
#202655182
#202655172
#202654552
#202654182
#202654172
#202653862
#202653402
#202653392
#202652682
C8×C22⋊D1012
#202651702
#416×C1012
#3302101