Finite Group Databasebeta

← #27174466

Subgroups — #27174466

Order 12928.

864 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
1C22×#13108073C211yesmaximal, characteristic
2C2×#15692239C211yesmaximal, characteristic, complemented
3C2×#15692236C211yesmaximal, characteristic, complemented
4C22×C4⋊Dic101C211yesmaximal, characteristic, complemented
5#20265098C212yesmaximal, complemented
6#20265093C212yesmaximal, complemented
7#20265056C211yesmaximal, characteristic, complemented
8#20265052C212yesmaximal, complemented
9#20265051C212yesmaximal, complemented
10#20265038C211yesmaximal, characteristic, complemented
11#386×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—24no—
13C2×#13108073—24no—
14C2×#13108073—24no—
15C22×Dic202C2211yescharacteristic
16C22×C22⋊D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C23×Dic101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692239—24no—
23#15692236—24no—
24C2×C4⋊Dic101—24no—
25C2×C4⋊Dic101C2212yes—
26C2×C4⋊Dic101C2212yes—
27C2×C4⋊Dic101C2212yes—
28C2×#13108053C2212yes—
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2212yes—
31C2×#13108051C2212yes—
32C2×#13108051C2212yes—
33#15692162C2212yes—
34#15692162C2212yes—
35#15692162C2212yes—
36C22×D4×C101C2211yescharacteristic
37C23×C4×C101C2211yescharacteristic
38C2×C22⋊C4×C101C2211yescharacteristic
39C2×C22⋊C4×C101C2212yes—
40#95×C101C2212yes—
41C23×D101C2311yescharacteristic
42#13108073—232no—
43C2×Dic202—22no—
44C2×Dic202—22no—
45C2×Dic202—22no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51C2×C22⋊D101D414yes—
52C22×Dic101—22no—
53C22×Dic101—22no—
54C22×Dic101—22no—
55C22×Dic101—22no—
56C22×Dic101—22no—
57C22×Dic101—22no—
58C22×Dic101—22no—
59C22×Dic101—22no—
60C22×Dic101D414yes—
61C22×Dic101C2312yes—
62C22×Dic101C2312yes—
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C2×C4×D101—22no—
67C2×C4×D101—22no—
68C2×C4×D101—22no—
69C4⋊Dic101—24no—
70C4⋊Dic101—24no—
71C4⋊Dic101—24no—
72#13108053—24no—
73C22⋊Dic101—22no—
74C22⋊Dic101—24no—
75#13108051—24no—
76#13108051—24no—
77C24×C101C2311yescharacteristic
78C24×C101C2311yescharacteristic
79C2×D4×C101—22no—
80C2×D4×C101—22no—
81C2×D4×C101D414yes—
82C22×C4×C101—22no—
83C22×C4×C101—24no—
84C22×C4×C101C2312yes—
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2312yes—
87C22⋊C4×C101—22no—
88C22⋊C4×C101—24no—
89C23×C101—22no—
90C23×C101—22no—
91C23×C101—21no—
92C23×C101—21no—
93C23×C101—22no—
94C23×C101—22no—
95C23×C101C2×D412yes—
96C23×C101C2×D412yes—
97C23×C101C2×D412yes—
98C23×C101C2×D412yes—
99C23×C101C2411yescharacteristic
100C22×D101—21no—
101C22×D101—21no—
102C22×D101—22no—
103C22×D101—22no—
104C22×D101C2×D412yes—
105D4×C101—28no—
106C2×C4×C101—28no—
107C2×C4×C101—22no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—22no—
111C2×C4×C101—22no—
112C2×C4×C101—22no—
113C2×C4×C101—22no—
114C2×C4×C101C2×D412yes—
115C2×C4×C101#4114yescomplemented
116C22⋊D101—28no—
117C22⋊D101—28no—
118C2×Dic101—28no—
119C2×Dic101—28no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—22no—
127C2×Dic101—22no—
128C2×Dic101—22no—
129C2×Dic101—22no—
130C2×Dic101—22no—
131C2×Dic101—22no—
132C2×Dic101—44no—
133C2×Dic101C2×D412yes—
134C2×Dic101C2×D412yes—
135C2×Dic101C2×D412yes—
136C4×D101—28no—
137Dic202—28no—
138C22×C101—22no—
139C22×C101—22no—
140C22×C101—24no—
141C22×C101—24no—
142C22×C101—22no—
143C22×C101—24no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101—42no—
147C22×C101#12014yes—
148C22×C101C22×D411yescharacteristic
149C22×C101C22×D411yescharacteristic
150C22×C101C22×D411yescharacteristic
151C22×C101C2×#4112yes—
152C22×C1012−1+412yes—
153C2×D101—22no—
154C2×D101—24no—
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—42no—
158C4×C101—22no—
159C4×C101—24no—
160C4×C101—42no—
161Dic101—22no—
162Dic101—22no—
163Dic101—22no—
164Dic101—42no—
165Dic101—42no—
166C2×C101—24no—
167C2×C101—42no—
168C2×C101C2×#12011yescharacteristic
169C2×C101#52511yescharacteristic
170C2×C101#53611yescharacteristic
171C2×C101#54712yes—
172C2×C101#54712yes—
173D101—42no—
174#2406—1011nomaximal, Sylow
175C101#240611yescharacteristic, complemented, Sylow
176C22×#41—1011no—
177C2×#123—1011no—
178C2×#123—1011no—
179C22×C4⋊C4—1011no—
180#395—1012no—
181#394—1012no—
182#386—1011no—
183#386D10111yescharacteristic, complemented
184#385—1012no—
185#385—1012no—
186#381—1011no—
187C2×#41—2024no—
188C2×#41—2024no—
189C2×#41—2024no—
190C22×Q8—1011no—
191C22×D4—1011no—
192C22×D4—1011no—
193C22×D4C2×D10111yescharacteristic
194C23×C4—1011no—
195C23×C4—1011no—
196C23×C4—1011no—
197C23×C4—1011no—
198C23×C4C2×D10111yescharacteristic
199#123—2024no—
200#123—2024no—
201C2×C4⋊C4—1012no—
202C2×C4⋊C4—1012no—
203C2×C4⋊C4—1012no—
204C2×C4⋊C4—1012no—
205C2×C4⋊C4—2024no—
206C2×C22⋊C4—1011no—
207C2×C22⋊C4—1012no—
208C2×C22⋊C4—1012no—
209C2×C22⋊C4—1012no—
210C2×C22⋊C4C2×D10111yescharacteristic
211C2×C22⋊C4C2×D10112yes—
212#95—1012no—
213#95—1012no—
214#95—1012no—
215#95C2×D10112yes—
216C24—1011no—
217C24C22×D10111yescharacteristic
218C24C22×D10111yescharacteristic
219#41—20232no—
220C2×Q8—2022no—
221C2×Q8—2022no—
222C2×Q8—2022no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—1014no—
226C2×D4—2022no—
227C2×D4—2022no—
228C2×D4—2022no—
229C2×D4—2022no—
230C2×D4—2022no—
231C2×D4D40414yes—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—1014no—
235C22×C4—1011no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1012no—
239C22×C4—1012no—
240C22×C4—2022no—
241C22×C4—2022no—
242C22×C4—2022no—
243C22×C4—2022no—
244C22×C4—2022no—
245C22×C4—2022no—
246C22×C4—2022no—
247C22×C4—2022no—
248C22×C4—2022no—
249C22×C4—2022no—
250C22×C4—2022no—
251C22×C4C22×D10112yes—
252C22×C4C22×D10111yescharacteristic
253C22×C4C22×D10112yes—
254C4⋊C4—2024no—
255C4⋊C4—2024no—
256C4⋊C4—2024no—
257C4⋊C4—2024no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—2024no—
261C22⋊C4—2022no—
262C22⋊C4—2024no—
263C22⋊C4—2024no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—1012no—
271C23—2021no—
272C23—2021no—
273C23—2022no—
274C23—2022no—
275C23D4×D10112yes—
276C23D4×D10112yes—
277C23C2×D40412yes—
278C23C2×D40412yes—
279C23C23×D10111yescharacteristic
280Q8—2028no—
281D4—28no—
282D4—2028no—
283D4—2028no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—1012no—
293C2×C4—1012no—
294C2×C4—1012no—
295C2×C4—4044no—
296C2×C4—2028no—
297C2×C4—2022no—
298C2×C4—2022no—
299C2×C4—2022no—
300C2×C4—2022no—
301C2×C4—2022no—
302C2×C4—2022no—
303C2×C4—2028no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4—2028no—
311C2×C4C2×D40412yes—
312C2×C4#1310807314yescomplemented
313C22—22no—
314C22—24no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—22no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—4041no—
323C22—4042no—
324C22—4041no—
325C22—2024no—
326C22—2022no—
327C22#1569222514yes—
328C22C2×D4×D10111yescharacteristic
329C22C2×D4×D10111yescharacteristic
330C22C22×D40411yescharacteristic
331C22C2×#1310807312yes—
332C22#1569230212yes—
333C4—22no—
334C4—24no—
335C4—42no—
336C4—4042no—
337C4—4042no—
338C4—2022no—
339C4—2022no—
340C4—2022no—
341C2—24no—
342C2—42no—
343C2—4042no—
344C2C2×#1569222511yescharacteristic
345C2#2026568311yescharacteristic
346C2#2026574911yescharacteristic
347C2#2026582712yes—
348C2#2026583112yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#386×C101C2
prime radicalC22×#13108073C2
hypercenter#386D101
hypocenterC101#2406
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#15692239C2semidirect
3. C2×#15692236C2semidirect
4. C22×C4⋊Dic101C2semidirect
5. #20265098C2semidirect
6. #20265093C2semidirect
7. #20265056C2semidirect
8. #20265052C2semidirect
9. #20265051C2semidirect
10. #20265038C2semidirect
11. #386×C101C2semidirect
115. C2×C4×C101#41semidirect
175. C101#2406semidirect
183. #386D101semidirect
312. C2×C4#13108073semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×#156922392
C2×#156922362
C22×C4⋊Dic1012
#202650982
#202650932
#202650562
#202650522
#202650512
#202650382
#386×C1012
#2406101