Finite Group Databasebeta

← #27173941

Subgroups — #27173941

Order 12928.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692242C211yesmaximal, characteristic
2#20265081C211yesmaximal, characteristic, complemented
3#20265081C211yesmaximal, characteristic, complemented
4#20265071C211yesmaximal, characteristic, complemented
5#20265071C211yesmaximal, characteristic, complemented
6#20265065C211yesmaximal, characteristic, complemented
7#20265065C211yesmaximal, characteristic, complemented
8#20265057C211yesmaximal, characteristic
9#20265050C211yesmaximal, characteristic, complemented
10#20265049C211yesmaximal, characteristic, complemented
11#20265044C211yesmaximal, characteristic, complemented
12#20265028C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic
16C22×Dic202C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18#15692242C418yescomplemented
19C2×#13108053C2211yescharacteristic
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C4×Dic101C2211yescharacteristic
31C2×C4×Dic101C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33C2×C4×Dic101C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35#15692162C2211yescharacteristic
36#15692162C2211yescharacteristic
37#15692162C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45C22×D4×C101C2211yescharacteristic
46C23×C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C2×Dic202—41no—
53C2×Dic202—41no—
54C2×Dic202C2×C414yescomplemented
55C22×Dic101—22no—
56C22×Dic101—41no—
57C22×Dic101—41no—
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66#13108053—22no—
67#13108053—22no—
68#13108053—41no—
69#13108053—41no—
70C22⋊Dic101—22no—
71C22⋊Dic101—41no—
72C22⋊Dic101—41no—
73C22⋊Dic101C2×C414yescomplemented
74C22⋊Dic101C2×C414yescomplemented
75C22⋊Dic101C2×C414yescomplemented
76C22⋊Dic101C2×C414yescomplemented
77C4×Dic101—41no—
78C4×Dic101—41no—
79C4×Dic101—41no—
80C4×Dic101—41no—
81C4×Dic101C2×C414yescomplemented
82C24×C101C2311yescharacteristic
83C24×C101C2311yescharacteristic
84C2×D4×C101—41no—
85C2×D4×C101—41no—
86C2×D4×C101C2×C414yes—
87C22×C4×C101—22no—
88C22×C4×C101—41no—
89C22×C4×C101—41no—
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22⋊C4×C101—22no—
96C22⋊C4×C101—41no—
97C22⋊C4×C101—41no—
98C23×C101—21no—
99C23×C101—21no—
100C23×C101—21no—
101C23×C101—21no—
102C23×C101—21no—
103C23×C101—21no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101—21no—
109C23×C101—21no—
110C23×C101C22×C412yes—
111C23×C101C22×C412yes—
112C23×C101C2411yescharacteristic
113D4×C101—44no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—44no—
128C2×C4×C101C22×C412yes—
129C2×C4×C101#4112yes—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—44no—
150C2×Dic101C22×C412yes—
151C2×Dic101C22×C412yes—
152C2×Dic101C22×C412yes—
153C2×Dic101C22×C412yes—
154C2×Dic101#4112yes—
155Dic202—44no—
156C22×C101—22no—
157C22×C101—22no—
158C22×C101—22no—
159C22×C101—22no—
160C22×C101—22no—
161C22×C101—22no—
162C22×C101—41no—
163C22×C101—41no—
164C22×C101—41no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101C23×C411yescharacteristic
171C22×C101C2×#4111yescharacteristic
172C22×C101C2×#4111yescharacteristic
173C22×C101H(5,𝔽2)11yescharacteristic
174C22×C101H(5,𝔽2)11yescharacteristic
175C22×C101H(5,𝔽2)11yescharacteristic
176C22×C1012−1+411yescharacteristic
177C4×C101—22no—
178C4×C101—22no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101—41no—
182Dic101—22no—
183Dic101—22no—
184Dic101—22no—
185Dic101—22no—
186Dic101—22no—
187Dic101—41no—
188Dic101—41no—
189Dic101—41no—
190C2×C101—42no—
191C2×C101—42no—
192C2×C101C4×#4111yescharacteristic
193C2×C101#51811yescharacteristic
194C2×C101#52011yescharacteristic
195C2×C101#53811yescharacteristic
196C2×C101#53811yescharacteristic
197C2×C101#54211yescharacteristic
198C2×C101#54211yescharacteristic
199#2337—1011nomaximal, Sylow
200C101#233711yescharacteristic, complemented, Sylow
201C2×#124—1011no—
202#391—1011no—
203#391—1011no—
204#388—1011no—
205#388—1011no—
206#388—1011no—
207#388—1011no—
208#386—1011no—
209#386D10111yescharacteristic
210#385—1011no—
211#385—1011no—
212#383—1011no—
213C4×C4⋊C4—1011no—
214C4×C22⋊C4—1011no—
215C4×C22⋊C4—1011no—
216C22×Q8—1011no—
217C22×D4C2×D10111yescharacteristic
218C23×C4—1011no—
219C23×C4C2×D10111yescharacteristic
220#124—1018no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4—1011no—
225C2×C22⋊C4—1011no—
226C2×C22⋊C4—1011no—
227C2×C22⋊C4—1011no—
228C2×C22⋊C4—1011no—
229C2×C22⋊C4—1011no—
230C2×C22⋊C4—1011no—
231C2×C22⋊C4—1011no—
232C2×C22⋊C4C2×D10111yescharacteristic
233C2×C22⋊C4C2×D10111yescharacteristic
234C2×C22⋊C4C2×D10111yescharacteristic
235C2×C42—1011no—
236C2×C42—1011no—
237C2×C42—1011no—
238C2×C42—1011no—
239C2×C42—1011no—
240#95—1011no—
241#95—1011no—
242#95—1011no—
243#95—1011no—
244#95—1011no—
245#95—1011no—
246#95—1011no—
247#95—1011no—
248#95—1011no—
249#95—1011no—
250#95C2×D10111yescharacteristic
251#95C2×D10111yescharacteristic
252C24C22×D10111yescharacteristic
253C24C22×D10111yescharacteristic
254C2×Q8—1014no—
255C2×Q8—4041no—
256C2×Q8—4041no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4C4×D10114yes—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—1011no—
264C22×C4—1011no—
265C22×C4—1011no—
266C22×C4—1011no—
267C22×C4—1011no—
268C22×C4—1011no—
269C22×C4—1011no—
270C22×C4—1011no—
271C22×C4—4041no—
272C22×C4—4041no—
273C22×C4—2022no—
274C22×C4C22×D10111yescharacteristic
275C22×C4C22×D10111yescharacteristic
276C22×C4C22×D10111yescharacteristic
277C22×C4C22×D10111yescharacteristic
278C22×C4C22×D10111yescharacteristic
279C4⋊C4—4041no—
280C4⋊C4—4041no—
281C4⋊C4—2022no—
282C4⋊C4—2022no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—1014no—
287C22⋊C4—1014no—
288C22⋊C4—1014no—
289C22⋊C4—1014no—
290C22⋊C4—4041no—
291C22⋊C4—4041no—
292C22⋊C4—2022no—
293C42—1014no—
294C42—4041no—
295C42—4041no—
296C42—4041no—
297C42—4041no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×C4×D10112yes—
311C23C2×C4×D10112yes—
312C23C23×D10111yescharacteristic
313Q8—4044no—
314D4—44no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—44no—
329C2×C4—1012no—
330C2×C4—1012no—
331C2×C4—1012no—
332C2×C4—1012no—
333C2×C4—1012no—
334C2×C4—4044no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4—2021no—
347C2×C4—2021no—
348C2×C4—2021no—
349C2×C4—2021no—
350C2×C4—2021no—
351C2×C4—2021no—
352C2×C4—2021no—
353C2×C4—2021no—
354C2×C4C2×C4×D10112yes—
355C2×C4#1310807312yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C22×C4×D10111yescharacteristic
371C22#41×D10111yescharacteristic
372C22C2×#1310807311yescharacteristic
373C22#1569230011yescharacteristic
374C22#1569230011yescharacteristic
375C22#1569230011yescharacteristic
376C22#1569230211yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—4041no—
383C4—4041no—
384C4—4041no—
385C4—2022no—
386C4—2022no—
387C4—2022no—
388C4—2022no—
389C4—2022no—
390C2—42no—
391C2—42no—
392C2C4×#1310807311yescharacteristic
393C2#2026565011yescharacteristic
394C2#2026565611yescharacteristic
395C2#2026575911yescharacteristic
396C2#2026576211yescharacteristic
397C2#2026579411yescharacteristic
398C2#2026579511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#386×C101C2
socle, cosocleC23×C101C24
prime radicalC22×D4×C101C22
hypercenter#386D101
hypocenterC101#2337
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20265081C2semidirect
3. #20265081C2semidirect
4. #20265071C2semidirect
5. #20265071C2semidirect
6. #20265065C2semidirect
7. #20265065C2semidirect
9. #20265050C2semidirect
10. #20265049C2semidirect
11. #20265044C2semidirect
12. #20265028C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C4×C22⋊Dic101C2semidirect
18. #15692242C4semidirect
54. C2×Dic202C2×C4semidirect
73. C22⋊Dic101C2×C4semidirect
74. C22⋊Dic101C2×C4semidirect
75. C22⋊Dic101C2×C4semidirect
76. C22⋊Dic101C2×C4semidirect
81. C4×Dic101C2×C4semidirect
200. C101#2337semidirect

Maximal subgroups

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

subgroupindex
C2×#156922422
#202650812
#202650712
#202650652
#202650572
#202650502
#202650492
#202650442
#202650282
C22⋊C4×Dic1012
C4×C22⋊Dic1012
#386×C1012
#2337101