Finite Group Databasebeta

← #27181307

Subgroups — #27181307

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265714C211yesmaximal, characteristic
2C4×Q8×D101C211yesmaximal, characteristic, complemented
3#20265511C211yesmaximal, characteristic, complemented
4#20265497C211yesmaximal, characteristic
5#20265487C211yesmaximal, characteristic, complemented
6#20265486C211yesmaximal, characteristic, complemented
7#20265402C211yesmaximal, characteristic, complemented
8#20265401C211yesmaximal, characteristic, complemented
9#20265331C211yesmaximal, characteristic, complemented
10#20265317C211yesmaximal, characteristic
11#20265282C211yesmaximal, characteristic, complemented
12#20265279C211yesmaximal, characteristic, complemented
13#20265275C211yesmaximal, characteristic, complemented
14#20265232C211yesmaximal, characteristic, complemented
15#441×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108063C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20#15692250—21no—
21#15692246C2211yescharacteristic
22#15692244C2211yescharacteristic
23#15692240—21no—
24#15692231C2211yescharacteristic
25#15692230C2211yescharacteristic
26Q8×Dic101C2211yescharacteristic
27C4×Dic202—21no—
28C4×Dic202C2211yescharacteristic
29#15692220—21no—
30#15692220C2211yescharacteristic
31C4×D404—21no—
32C4×D404C2211yescharacteristic
33#15692212—21no—
34C4⋊C4×D101—21no—
35C4⋊C4×D101C2211yescharacteristic
36C42×D101—21no—
37C42×D101C2211yescharacteristic
38#15692189C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692166C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61Q8×D101—22no—
62Q8×D101—22no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68C2×C4×D101D411yescharacteristic
69C2×C4×D101D411yescharacteristic
70C2×C4×D101C2311yescharacteristic
71#13108064—22no—
72#13108063—22no—
73#13108061—22no—
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101D411yescharacteristic
94C4×Dic101D411yescharacteristic
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98Q16×C101—22no—
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C4⋊C4×C101—21no—
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C42×C101—21no—
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C22×D101—21no—
109C22×D101C2×D411yescharacteristic
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2411yescharacteristic
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127D404—21no—
128D404—21no—
129D404—42no—
130C4×D101—24no—
131C4×D101—22no—
132C4×D101—21no—
133C4×D101—21no—
134C4×D101—41no—
135C4×D101—42no—
136C4×D101—42no—
137Dic202—24no—
138Dic202—21no—
139Dic202#4112yes—
140C8×C101—21no—
141C8×C101—21no—
142#11717579—21no—
143#11717579—21no—
144C22×C101C22×D411yescharacteristic
145C2×D101—21no—
146C2×D101—21no—
147C2×D101—41no—
148C2×D101—41no—
149C2×D101—41no—
150C2×D101#13712yes—
151C4×C101—22no—
152C4×C101—21no—
153C4×C101—21no—
154C4×C101—41no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101C2×#4111yescharacteristic
158C4×C101C2×#4111yescharacteristic
159Dic101—21no—
160Dic101—22no—
161Dic101—21no—
162Dic101—21no—
163Dic101—41no—
164Dic101—41no—
165C2×C101#52511yescharacteristic
166C2×C101C2×#13711yescharacteristic
167C2×C101#57511yescharacteristic
168D101—22no—
169D101—81no—
170#3117—1011nomaximal, Sylow
171C101#311711yescharacteristic, complemented, Sylow
172#529—1011no—
173C2×C4×Q8—1011no—
174#484—1011no—
175#481—1011no—
176#479—1011no—
177#479—1011no—
178#464—1011no—
179#464—1011no—
180#451—1011no—
181#448—1011no—
182#441—1011no—
183#441D10111yescharacteristic, complemented
184#440—1011no—
185#440—1011no—
186#432—1011no—
187C22×Q8—1011no—
188C2×Q16—1011no—
189C2×Q16C2×D10111yescharacteristic
190C2×SD16—1011no—
191C2×SD16—1011no—
192#126—2021no—
193#125—1011no—
194#124—1011no—
195#123—2021no—
196C22⋊Q8—1011no—
197#121—1011no—
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—2021no—
202C4×Q8—2021no—
203C4×Q8C2×D10111yescharacteristic
204C4×Q8C2×D10111yescharacteristic
205C4×D4—1011no—
206C4×D4—2021no—
207#117—2021no—
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—2021no—
210C2×C42—1011no—
211C2×C42—2021no—
212#107—1011no—
213#107—1011no—
214#106—1011no—
215#106C2×D10111yescharacteristic
216C4⋊C8—1011no—
217C4⋊C8—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103—1011no—
222#103C2×D10111yescharacteristic
223#103C2×D10111yescharacteristic
224#102—1011no—
225#102—1011no—
226C22⋊C8—1011no—
227C22⋊C8—1011no—
228#97—1011no—
229#97C2×D10111yescharacteristic
230C2×Q8—1011no—
231C2×Q8—2021no—
232C2×Q8—2022no—
233C2×Q8—2022no—
234C2×Q8C22×D10111yescharacteristic
235C2×Q8C22×D10111yescharacteristic
236C2×D4—1011no—
237C2×D4—2021no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—2021no—
242C22×C4—2021no—
243C22×C4—2021no—
244Q16—22no—
245Q16—2022no—
246SD16—2022no—
247SD16—2022no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8C22×D10111yescharacteristic
251C2×C8C22×D10111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—1011no—
255C4⋊C4—1011no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4C22×D10111yescharacteristic
264C4⋊C4C22×D10111yescharacteristic
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C42—21no—
271C42—21no—
272C42—1011no—
273C42—1011no—
274C42—1011no—
275C42—2021no—
276C42—2021no—
277C42—2021no—
278C42C22×D10111yescharacteristic
279C23—1011no—
280C23—2021no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—1012no—
285Q8—2024no—
286Q8—2021no—
287Q8#1310807412yes—
288D4—4042no—
289D4—2021no—
290D4—2021no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—1011no—
296C2×C4—1011no—
297C2×C4—1011no—
298C2×C4—4042no—
299C2×C4—4041no—
300C2×C4—4042no—
301C2×C4—2022no—
302C2×C4—2024no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4D4×D10111yescharacteristic
309C2×C4D4×D10111yescharacteristic
310C2×C4C23×D10111yescharacteristic
311C8—21no—
312C8—21no—
313C8—2021no—
314C8—2021no—
315C22—1012no—
316C22—4041no—
317C22—4041no—
318C22—4041no—
319C22—2021no—
320C22—2021no—
321C22C2×D4×D10111yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—4041no—
329C4—4041no—
330C4—2021no—
331C4—2022no—
332C4—2021no—
333C4—2021no—
334C4#41×D10111yescharacteristic
335C4C2×#1310807411yescharacteristic
336C2—8081no—
337C2—2022no—
338C2#2026568611yescharacteristic
339C2#137×D10111yescharacteristic
340C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#441×C101C2
socleC22×C101C22×D4
prime radical#15692230C22
hypercenter#441D101
hypocenterC101#3117
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D101C2semidirect
3. #20265511C2semidirect
5. #20265487C2semidirect
6. #20265486C2semidirect
7. #20265402C2semidirect
8. #20265401C2semidirect
9. #20265331C2semidirect
11. #20265282C2semidirect
12. #20265279C2semidirect
13. #20265275C2semidirect
14. #20265232C2semidirect
15. #441×C101C2semidirect
171. C101#3117semidirect
183. #441D101semidirect

Maximal subgroups

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

subgroupindex
#202657142
C4×Q8×D1012
#202655112
#202654972
#202654872
#202654862
#202654022
#202654012
#202653312
#202653172
#202652822
#202652792
#202652752
#202652322
#441×C1012
#3117101