Finite Group Databasebeta

← #27181417

Subgroups — #27181417

Order 12928.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265780C211yesmaximal, characteristic
2#20265655C211yesmaximal, characteristic, complemented
3#20265502C211yesmaximal, characteristic, complemented
4#20265498C211yesmaximal, characteristic, complemented
5#20265486C211yesmaximal, characteristic, complemented
6#20265482C211yesmaximal, characteristic, complemented
7#20265375C211yesmaximal, characteristic, complemented
8#20265375C211yesmaximal, characteristic, complemented
9#20265316C211yesmaximal, characteristic, complemented
10#20265280C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12SD16×Dic101C211yesmaximal, characteristic, complemented
13#20265262C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic, complemented
15#448×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18#15692251C2211yescharacteristic
19#15692246C2211yescharacteristic
20#15692239—21no—
21#15692237—21no—
22#15692229C2211yescharacteristic
23#15692227—21no—
24#15692226—21no—
25Q8×Dic101—21no—
26Q8×Dic101—21no—
27Q8×Dic101C2211yescharacteristic
28Q8×Dic101C2211yescharacteristic
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32#15692217C2211yescharacteristic
33#15692214—21no—
34#15692214—21no—
35#15692214C2211yescharacteristic
36#15692214C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic
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×D404C2311yescharacteristic
60C2×C22⋊D101—21no—
61C2×C22⋊D101—21no—
62C2×C22⋊D101—21no—
63C22×Dic101—21no—
64C2×C4×D101—21no—
65#13108061—22no—
66#13108061—22no—
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2311yescharacteristic
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101D411yescharacteristic
75C4⋊Dic101D411yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79#13108053—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101D411yescharacteristic
83C22⋊Dic101D411yescharacteristic
84#13108051—21no—
85#13108051—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
110C22×D101—21no—
111Q8×C101—24no—
112Q8×C101—21no—
113Q8×C101—21no—
114Q8×C101#4112yes—
115Q8×C101#4112yes—
116D4×C101—21no—
117D4×C101—21no—
118D4×C101—81no—
119C2×C4×C101—24no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—81no—
128C22⋊D101—81no—
129C22⋊D101—81no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—81no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138D404—21no—
139D404—21no—
140C4×D101—81no—
141C8×C101—21no—
142C8×C101—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—21no—
146C22×C101—21no—
147C22×C101—21no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101C22×D411yescharacteristic
152C2×D101—41no—
153C2×D101—41no—
154C2×D101—41no—
155C4×C101—22no—
156C4×C101—22no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C101C2×#4111yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—21no—
164Dic101—21no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168C2×C101—41no—
169C2×C101—81no—
170C2×C101#52511yescharacteristic
171C2×C101#57711yescharacteristic
172C2×C101#57711yescharacteristic
173D101—81no—
174#3126—1011nomaximal, Sylow
175C101#312611yescharacteristic, complemented, Sylow
176#540—1011no—
177#519—1011no—
178#482—1011no—
179#482—1011no—
180#479—1011no—
181#479—1011no—
182#460—1011no—
183#460—1011no—
184#448—1011no—
185#448D10111yescharacteristic, complemented
186#440—1011no—
187#440—1011no—
188C4×SD16—1011no—
189C4×SD16—1011no—
190#433—1011no—
191C22×Q8C2×D10111yescharacteristic
192C2×SD16—1011no—
193C2×SD16—1011no—
194C2×SD16C2×D10111yescharacteristic
195C2×SD16C2×D10111yescharacteristic
196#127—1011no—
197#125—1011no—
198#123—2021no—
199#123—2021no—
200#121—1011no—
201#121—2021no—
202#121—2021no—
203#121C2×D10111yescharacteristic
204C4×Q8—1011no—
205C4×Q8—1011no—
206C4×Q8—1011no—
207C4×Q8—1011no—
208C4×Q8—2021no—
209C4×Q8—2021no—
210C4×D4—1011no—
211C4×D4—1011no—
212#117—1011no—
213#117—1011no—
214#117—2021no—
215#117—2021no—
216#107—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222#103—1011no—
223#103—1011no—
224#103C2×D10111yescharacteristic
225#103C2×D10111yescharacteristic
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230C22⋊C8—1011no—
231C22⋊C8C2×D10111yescharacteristic
232#97—1011no—
233C4×C8—1011no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D10111yescharacteristic
238C2×Q8C22×D10111yescharacteristic
239C2×D4—21no—
240C2×D4—1011no—
241C2×D4—2021no—
242C2×D4—2021no—
243C2×D4—2021no—
244C2×D4C22×D10111yescharacteristic
245C22×C4—21no—
246C22×C4—2021no—
247C22×C4—2021no—
248C22×C4C22×D10111yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—2022no—
252SD16—2022no—
253C2×C8—1011no—
254C2×C8—1011no—
255C2×C8C22×D10111yescharacteristic
256C2×C8C22×D10111yescharacteristic
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—1011no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1011no—
271C22⋊C4—1011no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C42—1011no—
277C42—1011no—
278C42—2021no—
279C42—2021no—
280C42—2021no—
281C42—2021no—
282C42—2021no—
283C23—21no—
284C23—2021no—
285C23D4×D10111yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1310807312yes—
290Q8#1310807312yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—8081no—
295D4—8081no—
296D4—8081no—
297D4—2021no—
298D4—2021no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—8081no—
306C2×C4—8081no—
307C2×C4—1011no—
308C2×C4—1011no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4D4×D10111yescharacteristic
315C2×C4C23×D10111yescharacteristic
316C8—21no—
317C8—21no—
318C8—2021no—
319C8—2021no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—4041no—
327C22—4041no—
328C22—4041no—
329C22C2×D4×D10111yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—4041no—
335C4—4041no—
336C4—4041no—
337C4—2021no—
338C4—2021no—
339C4—2021no—
340C4—2021no—
341C4C2×#1310807311yescharacteristic
342C4C2×#1310807311yescharacteristic
343C2—41no—
344C2—81no—
345C2—8081no—
346C2#2026568311yescharacteristic
347C2#2026598311yescharacteristic
348C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#448×C101C2
socleC22×C101C22×D4
prime radical#20265780C2
hypercenter#448D101
hypocenterC101#3126
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. #20265502C2semidirect
4. #20265498C2semidirect
5. #20265486C2semidirect
6. #20265482C2semidirect
7. #20265375C2semidirect
8. #20265375C2semidirect
9. #20265316C2semidirect
10. #20265280C2semidirect
11. #20265278C2semidirect
12. SD16×Dic101C2semidirect
13. #20265262C2semidirect
14. #20265235C2semidirect
15. #448×C101C2semidirect
175. C101#3126semidirect
185. #448D101semidirect

Maximal subgroups

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

subgroupindex
#202657802
#202656552
#202655022
#202654982
#202654862
#202654822
#202653752
#202653162
#202652802
#202652782
SD16×Dic1012
#202652622
#202652352
#448×C1012
#3126101