Finite Group Databasebeta

← #27183124

Subgroups — #27183124

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265834C211yesmaximal, characteristic, complemented
2#20265828C211yesmaximal, characteristic, complemented
3#20265535C211yesmaximal, characteristic, complemented
4#20265512C211yesmaximal, characteristic, complemented
5#20265489C211yesmaximal, characteristic, complemented
6#20265470C211yesmaximal, characteristic, complemented
7#20265423C211yesmaximal, characteristic, complemented
8#20265412C211yesmaximal, characteristic, complemented
9#20265391C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265317C211yesmaximal, characteristic, complemented
12#20265280C211yesmaximal, characteristic, complemented
13#20265246C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#417×C101C211yesmaximal, characteristic, complemented
16C2×#13108074—21no—
17C2×Q8×D101C2211yescharacteristic
18C2×#13108063C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×#13108059C2211yescharacteristic
21C2×C8×D101C2211yescharacteristic
22#15692254C2211yescharacteristic
23#15692242—21no—
24#15692242C2211yescharacteristic
25#15692240—21no—
26#15692238—21no—
27#15692234—21no—
28#15692234C2211yescharacteristic
29#15692233—21no—
30#15692230C2211yescharacteristic
31#15692229C2211yescharacteristic
32#15692226—21no—
33#15692226C2211yescharacteristic
34Q8×Dic101C2211yescharacteristic
35#15692217C2211yescharacteristic
36C4×C22⋊D101—21no—
37C4×C22⋊D101C2211yescharacteristic
38C4⋊C4×D101C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692188C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692165C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54C2×C4⋊C4×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108074—44no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63Q8×D101—22no—
64Q8×D101—41no—
65C2×D404—21no—
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101D411yescharacteristic
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101C2311yescharacteristic
74#13108063—41no—
75#13108061—22no—
76#13108059—41no—
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C8×D101—41no—
80C4⋊Dic101—21no—
81C4⋊Dic101C2311yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91C22⋊Dic101D411yescharacteristic
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051D411yescharacteristic
96C4×Dic101—21no—
97C4×Dic101C2311yescharacteristic
98#41×C101—22no—
99#41×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×D4×C101—21no—
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104C22×C4×C101—21no—
105C22×C4×C101C2311yescharacteristic
106M16×C101—41no—
107C2×C8×C101C2311yescharacteristic
108C2×C8×C101C2311yescharacteristic
109C4⋊C4×C101—21no—
110C4⋊C4×C101C2311yescharacteristic
111C4⋊C4×C101C2311yescharacteristic
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115C22×D101C2×D411yescharacteristic
116Q8×C101—21no—
117Q8×C101#4112yescomplemented
118D4×C101—21no—
119D4×C101—21no—
120D4×C101—42no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—42no—
127C2×C4×C101—42no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—22no—
131C22⋊D101—42no—
132C22⋊D101—42no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139D404—21no—
140D404—21no—
141D404—42no—
142C4×D101—21no—
143C4×D101—21no—
144C4×D101—42no—
145C4×D101—42no—
146C4×D101—42no—
147Dic202—21no—
148Dic202—21no—
149Dic202—42no—
150Dic202—42no—
151C8×C101—21no—
152C8×C101—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—21no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—41no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101AGL(1,ℤ/8)12yes—
168C4×C101—22no—
169C4×C101—21no—
170C4×C101—41no—
171C4×C101—41no—
172C4×C101C2×#4111yescharacteristic
173C4×C101H(5,𝔽2)11yescharacteristic
174Dic101—21no—
175Dic101—21no—
176Dic101—21no—
177Dic101—41no—
178Dic101—41no—
179Dic101—41no—
180C2×C101—41no—
181C2×C101—81no—
182C2×C101#54611yescharacteristic
183C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C101#57811yescharacteristic
185D101—22no—
186D101—81no—
187#3308—1011nomaximal, Sylow
188C101#330811yescharacteristic, complemented, Sylow
189#548—1011no—
190#547—1011no—
191#489—1011no—
192#484—1011no—
193#480—1011no—
194#477—1011no—
195#468—1011no—
196#466—1011no—
197#463—1011no—
198#449—1011no—
199#448—1011no—
200#440—1011no—
201#435—1011no—
202#417D10111yescharacteristic, complemented
203C2×#103—1011no—
204C2×#41—2021no—
205C2×#41C2×D10111yescharacteristic
206C22×Q8—1011no—
207C2×SD16—1011no—
208C2×SD16—1011no—
209C2×D8—1011no—
210C2×M16C2×D10111yescharacteristic
211C22×C8—1011no—
212C4⋊Q8—1011no—
213#124—1011no—
214#124—2021no—
215#123—2021no—
216#123—2021no—
217C22⋊Q8—1011no—
218C22⋊Q8—2021no—
219C22⋊Q8—2021no—
220#121—1011no—
221#121—1011no—
222#121—1011no—
223#121—2021no—
224C4×Q8—1011no—
225C4×D4—1011no—
226C4×D4—1011no—
227C4×D4—2021no—
228C2×C4⋊C4—1011no—
229C2×C4⋊C4C2×D10111yescharacteristic
230#107—1011no—
231#107—1011no—
232#107—1011no—
233C4⋊C8—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#103—1011no—
238#103C2×D10111yescharacteristic
239#103C2×D10111yescharacteristic
240#102—1011no—
241#102—1011no—
242#102C2×D10111yescharacteristic
243#102C2×D10111yescharacteristic
244C22⋊C8—1011no—
245C22⋊C8—1011no—
246#97—1011no—
247#41—22no—
248#41—41no—
249#41—4044no—
250C2×Q8—1011no—
251C2×Q8—4041no—
252C2×Q8—2021no—
253C2×Q8—2021no—
254C2×Q8—2022no—
255C2×Q8C22×D10111yescharacteristic
256C2×D4—21no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4—2021no—
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1011no—
266C22×C4—2021no—
267C22×C4—2021no—
268C22×C4—2021no—
269C22×C4C22×D10111yescharacteristic
270SD16—4041no—
271SD16—2022no—
272D8—4041no—
273M16—41no—
274C2×C8—1011no—
275C2×C8—1011no—
276C2×C8—4041no—
277C2×C8C22×D10111yescharacteristic
278C2×C8C22×D10111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1011no—
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—2021no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4C22×D10111yescharacteristic
288C4⋊C4C22×D10111yescharacteristic
289C22⋊C4—1011no—
290C22⋊C4—1011no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C42—1011no—
299C42—2021no—
300C23—21no—
301C23—1011no—
302C23—2021no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—4042no—
306Q8—4042no—
307Q8—2021no—
308Q8—2021no—
309Q8#1310807412yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—4042no—
314D4—4042no—
315D4—4042no—
316D4—2022no—
317D4—2021no—
318D4—2021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—1011no—
327C2×C4—1011no—
328C2×C4—1011no—
329C2×C4—4042no—
330C2×C4—4042no—
331C2×C4—4042no—
332C2×C4—2021no—
333C2×C4—2021no—
334C2×C4—2021no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4D4×D10111yescharacteristic
338C2×C4C23×D10111yescharacteristic
339C8—21no—
340C8—21no—
341C8—2021no—
342C8—2021no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—1012no—
350C22—4041no—
351C22—4041no—
352C22—4041no—
353C22—2021no—
354C22—2021no—
355C22C2×D4×D10111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—4041no—
361C4—4041no—
362C4—4041no—
363C4—2021no—
364C4—2021no—
365C4—2021no—
366C4C2×#1310807411yescharacteristic
367C4#1569230011yescharacteristic
368C2—41no—
369C2—81no—
370C2—8081no—
371C2—2022no—
372C2#2026581711yescharacteristic
373C2AGL(1,ℤ/8)×D10111yescharacteristic
374C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
hypercenter#417D101
hypocenterC101#3308
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #20265834C2semidirect
2. #20265828C2semidirect
3. #20265535C2semidirect
4. #20265512C2semidirect
5. #20265489C2semidirect
6. #20265470C2semidirect
7. #20265423C2semidirect
8. #20265412C2semidirect
9. #20265391C2semidirect
10. #20265322C2semidirect
11. #20265317C2semidirect
12. #20265280C2semidirect
13. #20265246C2semidirect
14. #103×D101C2semidirect
15. #417×C101C2semidirect
117. Q8×C101#41semidirect
188. C101#3308semidirect
202. #417D101semidirect
309. Q8#13108074semidirect

Maximal subgroups

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

subgroupindex
#202658342
#202658282
#202655352
#202655122
#202654892
#202654702
#202654232
#202654122
#202653912
#202653222
#202653172
#202652802
#202652462
#103×D1012
#417×C1012
#3308101