Finite Group Databasebeta

← #27181354

Subgroups — #27181354

Order 12928.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265794C211yesmaximal, characteristic
2#20265655C211yesmaximal, characteristic, complemented
3#20265519C211yesmaximal, characteristic, complemented
4#20265498C211yesmaximal, characteristic
5#20265470C211yesmaximal, characteristic, complemented
6#20265460C211yesmaximal, characteristic, complemented
7#20265397C211yesmaximal, characteristic, complemented
8#20265397C211yesmaximal, characteristic, complemented
9#20265333C211yesmaximal, characteristic, complemented
10#20265281C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12#20265269C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic, complemented
15#448×C101C211yesmaximal, characteristic
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692248—21no—
20#15692242—21no—
21#15692242C2211yescharacteristic
22#15692236—21no—
23#15692234C2211yescharacteristic
24#15692232—21no—
25Q8×Dic101—21no—
26Q8×Dic101—21no—
27Q8×Dic101C2211yescharacteristic
28Q8×Dic101C2211yescharacteristic
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31#15692220C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33#15692214—21no—
34#15692214—21no—
35#15692214C2211yescharacteristic
36#15692214C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C22×Dic101—21no—
62#13108064—22no—
63#13108064—22no—
64C2×#11717579C2311yescharacteristic
65C2×#11717579C2311yescharacteristic
66C4⋊Dic101—21no—
67C4⋊Dic101—21no—
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101D411yescharacteristic
71C4⋊Dic101D411yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78C22⋊Dic101—21no—
79C22⋊Dic101—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101D411yescharacteristic
84C22⋊Dic101D411yescharacteristic
85C4×Dic101—21no—
86C4×Dic101—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93C2×Q8×C101—22no—
94C2×Q8×C101—22no—
95C2×Q8×C101—21no—
96C2×Q8×C101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98C2×D4×C101—21no—
99C2×D4×C101C2311yescharacteristic
100C22×C4×C101—21no—
101C22×C4×C101C2311yescharacteristic
102SD16×C101—22no—
103SD16×C101—22no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110Q8×C101—24no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114Q8×C101#4112yes—
115D4×C101—21no—
116D4×C101—21no—
117D4×C101—81no—
118C2×C4×C101—24no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—81no—
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135Dic202—21no—
136Dic202—21no—
137Dic202—81no—
138C8×C101—21no—
139C8×C101—21no—
140#11717579—21no—
141#11717579—21no—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101C22×D411yescharacteristic
149C4×C101—22no—
150C4×C101—22no—
151C4×C101—21no—
152C4×C101—41no—
153C4×C101C2×#4111yescharacteristic
154C4×C101C2×#4111yescharacteristic
155Dic101—21no—
156Dic101—21no—
157Dic101—21no—
158Dic101—21no—
159Dic101—41no—
160Dic101—41no—
161Dic101—41no—
162Dic101—41no—
163C2×C101—41no—
164C2×C101—81no—
165C2×C101#52511yescharacteristic
166C2×C101#57711yescharacteristic
167C2×C101#57811yescharacteristic
168#3121—1011nomaximal, Sylow
169C101#312111yescharacteristic, complemented, Sylow
170#542—1011no—
171#519—1011no—
172#485—1011no—
173#482—1011no—
174#477—1011no—
175#475—1011no—
176#464—1011no—
177#464—1011no—
178#451—1011no—
179#448D10111yescharacteristic
180#441—1011no—
181#440—1011no—
182C4×Q16—1011no—
183C4×SD16—1011no—
184#433—1011no—
185C22×Q8C2×D10111yescharacteristic
186C2×Q16—1011no—
187C2×Q16—1011no—
188C2×SD16C2×D10111yescharacteristic
189C2×SD16C2×D10111yescharacteristic
190C4⋊Q8—1011no—
191#126—2021no—
192#124—1011no—
193#124—2021no—
194#123—2021no—
195C22⋊Q8—1011no—
196C22⋊Q8—2021no—
197#121C2×D10111yescharacteristic
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8—1011no—
203C4×Q8—2021no—
204C4×Q8—2021no—
205C4×D4—1011no—
206#117—1011no—
207#117—1011no—
208#117—2021no—
209#117—2021no—
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216#103—1011no—
217#103—1011no—
218#103—1011no—
219#103—1011no—
220#103C2×D10111yescharacteristic
221#103C2×D10111yescharacteristic
222#102—1011no—
223#102—1011no—
224C22⋊C8—1011no—
225C22⋊C8C2×D10111yescharacteristic
226#97—1011no—
227C4×C8—1011no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—1011no—
232C2×Q8—2021no—
233C2×Q8C22×D10111yescharacteristic
234C2×Q8C22×D10111yescharacteristic
235C2×D4—21no—
236C2×D4C22×D10111yescharacteristic
237C22×C4—21no—
238C22×C4—2021no—
239C22×C4C22×D10111yescharacteristic
240Q16—2022no—
241Q16—2022no—
242SD16—22no—
243SD16—22no—
244C2×C8—1011no—
245C2×C8—1011no—
246C2×C8C22×D10111yescharacteristic
247C2×C8C22×D10111yescharacteristic
248C4⋊C4—1011no—
249C4⋊C4—1011no—
250C4⋊C4—1011no—
251C4⋊C4—1011no—
252C4⋊C4—1011no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4C22×D10111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—1011no—
263C22⋊C4—1011no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C42—1011no—
270C42—1011no—
271C42—2021no—
272C42—2021no—
273C42—2021no—
274C42—2021no—
275C42—2021no—
276C42—2021no—
277C23—21no—
278C23D4×D10111yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—8081no—
283Q8—2021no—
284Q8—2021no—
285Q8#1310807312yes—
286Q8#1310807312yes—
287D4—21no—
288D4—21no—
289D4—81no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—8081no—
297C2×C4—1011no—
298C2×C4—1011no—
299C2×C4—1011no—
300C2×C4—1011no—
301C2×C4—2021no—
302C2×C4—2021no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4D4×D10111yescharacteristic
306C2×C4C23×D10111yescharacteristic
307C8—21no—
308C8—21no—
309C8—2021no—
310C8—2021no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C2×D4×D10111yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—41no—
322C4—4041no—
323C4—4041no—
324C4—4041no—
325C4—4041no—
326C4—2021no—
327C4—2021no—
328C4—2021no—
329C4—2021no—
330C4C2×#1310807311yescharacteristic
331C4C2×#1310807311yescharacteristic
332C2—41no—
333C2—81no—
334C2#2026568311yescharacteristic
335C2#2026598311yescharacteristic
336C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#448×C101C2
socleC22×C101C22×D4
prime radical#121×C101C22
hypercenter#448D101
hypocenterC101#3121
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
2. #20265655C2semidirect
3. #20265519C2semidirect
5. #20265470C2semidirect
6. #20265460C2semidirect
7. #20265397C2semidirect
8. #20265397C2semidirect
9. #20265333C2semidirect
10. #20265281C2semidirect
11. #20265278C2semidirect
12. #20265269C2semidirect
13. SD16×Dic101C2semidirect
14. #20265235C2semidirect
169. C101#3121semidirect

Maximal subgroups

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

subgroupindex
#202657942
#202656552
#202655192
#202654982
#202654702
#202654602
#202653972
#202653332
#202652812
#202652782
#202652692
SD16×Dic1012
#202652352
#448×C1012
#3121101