Finite Group Databasebeta

← #27182990

Subgroups — #27182990

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D101C211yesmaximal, characteristic, complemented
2#20265827C211yesmaximal, characteristic
3#20265526C211yesmaximal, characteristic, complemented
4#20265508C211yesmaximal, characteristic, complemented
5#20265506C211yesmaximal, characteristic, complemented
6#20265460C211yesmaximal, characteristic, complemented
7#20265404C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265380C211yesmaximal, characteristic, complemented
10#20265314C211yesmaximal, characteristic, complemented
11SD16×Dic101C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13C2×#15692179C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#448×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×Q8×D101C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C22×#11717579C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692253—21no—
23#15692253C2211yescharacteristic
24#15692237—21no—
25#15692236—21no—
26#15692234—21no—
27#15692234C2211yescharacteristic
28#15692232—21no—
29#15692231—21no—
30#15692231C2211yescharacteristic
31#15692226C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33D4×Dic101C2211yescharacteristic
34C4×C22⋊D101—21no—
35C4×C22⋊D101C2211yescharacteristic
36C2×C4⋊Dic101C2211yescharacteristic
37C4⋊C4×D101C2211yescharacteristic
38#15692187C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63Q8×D101—22no—
64Q8×D101—41no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101D411yescharacteristic
67C22×Dic101—21no—
68C22×Dic101—21no—
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101C2311yescharacteristic
72#13108062—41no—
73C2×#11717579—41no—
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C8×D101—41no—
77C4⋊Dic101—21no—
78C4⋊Dic101—21no—
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053—21no—
88#13108053D411yescharacteristic
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101D411yescharacteristic
93#13108051—21no—
94#13108051D411yescharacteristic
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97C2×Q8×C101—21no—
98C2×Q8×C101—22no—
99C2×Q8×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C2×D4×C101—21no—
103C2×D4×C101C2311yescharacteristic
104C22×C4×C101—21no—
105C22×C4×C101C2311yescharacteristic
106SD16×C101—22no—
107SD16×C101—41no—
108C2×C8×C101C2311yescharacteristic
109C2×C8×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—42no—
119Q8×C101#4112yescomplemented
120D4×C101—21no—
121D4×C101—21no—
122D4×C101—42no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—42no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—42no—
132C22⋊D101SD1614yescomplemented
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—42no—
138C2×Dic101—42no—
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C4×D101—21no—
143C4×D101—21no—
144C4×D101—42no—
145C4×D101—42no—
146Dic202—21no—
147Dic202—21no—
148Dic202—42no—
149Dic202—42no—
150C8×C101—21no—
151C8×C101—21no—
152#11717579—21no—
153#11717579—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101C2×SD1612yes—
160C22×C101C22×D411yescharacteristic
161C2×D101—21no—
162C2×D101—21no—
163C2×D101C2×SD1612yes—
164C4×C101—22no—
165C4×C101—21no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101C2×#4111yescharacteristic
169C4×C101H(5,𝔽2)11yescharacteristic
170Dic101—21no—
171Dic101—21no—
172Dic101—41no—
173Dic101—41no—
174Dic101—41no—
175Dic101—41no—
176Dic101C2×SD1612yes—
177C2×C101—22no—
178C2×C101—81no—
179C2×C101#54611yescharacteristic
180C2×C101C22×SD1611yescharacteristic
181C2×C101#57811yescharacteristic
182D101—22no—
183#3298—1011nomaximal, Sylow
184C101#329811yescharacteristic, complemented, Sylow
185D4×Q8—1011no—
186#547—1011no—
187#487—1011no—
188#483—1011no—
189#483—1011no—
190#475—1011no—
191#465—1011no—
192#465—1011no—
193#461—1011no—
194#448—1011no—
195#448D10111yescharacteristic, complemented
196C4×SD16—1011no—
197C8×D4—1011no—
198C2×#103—1011no—
199C2×#103—1011no—
200C2×#41—2021no—
201C22×Q8—1011no—
202C22×Q8C2×D10111yescharacteristic
203C2×SD16—1011no—
204C2×SD16C2×D10111yescharacteristic
205C2×SD16C2×D10111yescharacteristic
206C22×C8—1011no—
207C22×C8—1011no—
208C4⋊Q8—1011no—
209C4⋊Q8—1011no—
210C4⋊Q8—2021no—
211#123—2021no—
212#123—2021no—
213C22⋊Q8—1011no—
214C22⋊Q8—1011no—
215C22⋊Q8—2021no—
216C22⋊Q8—2021no—
217C22⋊Q8—2021no—
218#121—1011no—
219#121C2×D10111yescharacteristic
220C4×Q8—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223C4×D4—2021no—
224C2×C4⋊C4—1011no—
225C2×C4⋊C4—1011no—
226#106—1011no—
227#106—1011no—
228#106—1011no—
229C4⋊C8—1011no—
230#103—1011no—
231#103—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#103C2×D10111yescharacteristic
238#103C2×D10111yescharacteristic
239#102—1011no—
240C22⋊C8—1011no—
241C22⋊C8C2×D10111yescharacteristic
242C4×C8—1011no—
243#41—4044no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—1011no—
248C2×Q8—4041no—
249C2×Q8—2021no—
250C2×Q8—2021no—
251C2×Q8—2022no—
252C2×Q8C22×D10111yescharacteristic
253C2×Q8C22×D10111yescharacteristic
254C2×D4—21no—
255C2×D4—1011no—
256C2×D4—2021no—
257C2×D4C22×D10111yescharacteristic
258C22×C4—21no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263C22×C4—2021no—
264C22×C4C22×D10111yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—4041no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8—4041no—
271C2×C8—4041no—
272C2×C8C22×D10111yescharacteristic
273C2×C8C22×D10111yescharacteristic
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—1011no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
282C4⋊C4—2021no—
283C4⋊C4—2021no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4C22×D10111yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—1011no—
289C22⋊C4—1011no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C42—1011no—
295C42—2021no—
296C23—21no—
297C23—1011no—
298C23D4×D10111yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—4042no—
304Q8—4042no—
305Q8—2021no—
306Q8—2021no—
307Q8#1310807312yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—1014no—
312D4—4042no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1011no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—4042no—
323C2×C4—4042no—
324C2×C4—4042no—
325C2×C4—4042no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2021no—
337C8—2021no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1012no—
344C22—2021no—
345C22—2021no—
346C22SD16×D10112yes—
347C22C2×D4×D10111yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—1012no—
353C4—4041no—
354C4—4041no—
355C4—4041no—
356C4—4041no—
357C4—2021no—
358C4—2021no—
359C4C2×#1310807311yescharacteristic
360C4#1569230011yescharacteristic
361C2—22no—
362C2—81no—
363C2—2022no—
364C2#2026581411yescharacteristic
365C2C2×SD16×D10111yescharacteristic
366C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#448×C101C2
socleC22×C101C22×D4
prime radical#20265827C2
hypercenter#448D101
hypocenterC101#3298
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D101C2semidirect
3. #20265526C2semidirect
4. #20265508C2semidirect
5. #20265506C2semidirect
6. #20265460C2semidirect
7. #20265404C2semidirect
8. #20265403C2semidirect
9. #20265380C2semidirect
10. #20265314C2semidirect
11. SD16×Dic101C2semidirect
12. #20265240C2semidirect
13. C2×#15692179C2semidirect
14. #103×D101C2semidirect
15. #448×C101C2semidirect
119. Q8×C101#41semidirect
132. C22⋊D101SD16semidirect
184. C101#3298semidirect
195. #448D101semidirect
307. Q8#13108073semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1012
#202658272
#202655262
#202655082
#202655062
#202654602
#202654042
#202654032
#202653802
#202653142
SD16×Dic1012
#202652402
C2×#156921792
#103×D1012
#448×C1012
#3298101