Finite Group Databasebeta

← #27174198

Subgroups — #27174198

Order 12928.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692241C211yesmaximal, characteristic
2C2×#15692220C211yesmaximal, characteristic, complemented
3C2×#15692215C211yesmaximal, characteristic
4#20265081C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic
6#20265066C211yesmaximal, characteristic, complemented
7#20265066C211yesmaximal, characteristic, complemented
8#20265065C211yesmaximal, characteristic, complemented
9#20265049C211yesmaximal, characteristic, complemented
10#20265048C211yesmaximal, characteristic, complemented
11C4×#13108053C211yesmaximal, characteristic, complemented
12#20265028C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C4×#13108051C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692241C418yescomplemented
21#15692220—24no—
22#15692215—24no—
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C2×C4⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C42×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C2×Dic202—22no—
56C2×Dic202—22no—
57C2×Dic202C2×C414yescomplemented
58C2×C22⋊D101—22no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101C2×C414yes—
61C22×Dic101—22no—
62C22×Dic101—22no—
63C22×Dic101—41no—
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C2×C4×D101—22no—
72C2×C4×D101—22no—
73C2×C4×D101—41no—
74#13108053—22no—
75#13108053—22no—
76#13108053—22no—
77#13108053—22no—
78#13108053D414yes—
79C22⋊Dic101—22no—
80C22⋊Dic101C2×C414yescomplemented
81#13108051—22no—
82#13108051—22no—
83#13108051—41no—
84#13108051C2×C414yescomplemented
85#13108051C2×C414yescomplemented
86C4×Dic101—22no—
87C4×Dic101—22no—
88C4×Dic101—22no—
89C4×Dic101—22no—
90C4×Dic101—41no—
91C4×Dic101C2×C414yescomplemented
92C24×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96C22×C4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C4⋊C4×C101—22no—
100C22⋊C4×C101—22no—
101C22⋊C4×C101—41no—
102C22⋊C4×C101C2×C414yescomplemented
103C42×C101—41no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101—21no—
109C23×C101—21no—
110C23×C101C22×C412yes—
111C23×C101C2411yescharacteristic
112C22×D101—21no—
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—21no—
117C22×D101—21no—
118C22×D101C22×C412yes—
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—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101C22×C412yes—
135C2×C4×C101C22×C412yes—
136C2×C4×C101C2×D412yes—
137C22⋊D101—28no—
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—21no—
150C2×Dic101—21no—
151C2×Dic101—44no—
152C2×Dic101C22×C412yes—
153C2×Dic101C22×C412yes—
154C2×Dic101C22×C412yes—
155C2×Dic101C2×D412yes—
156C2×Dic101C2×D412yes—
157C2×Dic101#4112yes—
158C2×Dic101#4112yes—
159C2×Dic101#4112yes—
160C4×D101—44no—
161Dic202—28no—
162C22×C101—22no—
163C22×C101—22no—
164C22×C101—22no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101C23×C411yescharacteristic
170C22×C101C22×D411yescharacteristic
171C22×C101C2×#4111yescharacteristic
172C22×C101C2×#4111yescharacteristic
173C22×C101C2×#4111yescharacteristic
174C22×C101H(5,𝔽2)11yescharacteristic
175C22×C1012−1+411yescharacteristic
176C2×D101—22no—
177C2×D101—22no—
178C2×D101—22no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C4×C101—22no—
184C4×C101—22no—
185C4×C101—22no—
186C4×C101—41no—
187C4×C101—41no—
188C4×C101—41no—
189Dic101—22no—
190Dic101—22no—
191Dic101—22no—
192Dic101—22no—
193Dic101—22no—
194Dic101—41no—
195Dic101C4×D414yes—
196C2×C101—42no—
197C2×C101C2×C4×D411yescharacteristic
198C2×C101C4×#4111yescharacteristic
199C2×C101#52011yescharacteristic
200C2×C101#54611yescharacteristic
201C2×C101#54811yescharacteristic
202C2×C101#55611yescharacteristic
203C2×C101#55911yescharacteristic
204D101—42no—
205#2367—1011nomaximal, Sylow
206C101#236711yescharacteristic, complemented, Sylow
207C2×#124—1011no—
208C2×C4×Q8—1011no—
209C2×C4×D4—1011no—
210#391—1011no—
211#390—1011no—
212#388—1011no—
213#388—1011no—
214#388—1011no—
215#388D10111yescharacteristic, complemented
216#385—1011no—
217#385—1011no—
218C4×C4⋊C4—1011no—
219C4×C4⋊C4—1011no—
220C4×C22⋊C4—1011no—
221C4×C22⋊C4—1011no—
222C22×Q8—1011no—
223C22×D4—1011no—
224C23×C4—1011no—
225C23×C4—1011no—
226#124—1018no—
227C4×Q8—2024no—
228C4×D4—2024no—
229C2×C4⋊C4—1011no—
230C2×C4⋊C4—1011no—
231C2×C4⋊C4—1011no—
232C2×C4⋊C4—1011no—
233C2×C4⋊C4—1011no—
234C2×C4⋊C4C2×D10111yescharacteristic
235C2×C22⋊C4—1011no—
236C2×C22⋊C4—1011no—
237C2×C22⋊C4—1011no—
238C2×C22⋊C4—1011no—
239C2×C22⋊C4—1011no—
240C2×C22⋊C4—1011no—
241C2×C22⋊C4—1011no—
242C2×C22⋊C4C2×D10111yescharacteristic
243C2×C22⋊C4C2×D10111yescharacteristic
244C2×C22⋊C4C2×D10111yescharacteristic
245C2×C42—1011no—
246C2×C42—1011no—
247C2×C42—1011no—
248C2×C42—1011no—
249C2×C42—1011no—
250C2×C42—1011no—
251C2×C42C2×D10111yescharacteristic
252#95—1011no—
253#95—1011no—
254#95—1011no—
255#95—1011no—
256#95—1011no—
257#95—1011no—
258#95C2×D10111yescharacteristic
259#95C2×D10111yescharacteristic
260C24—1011no—
261C24C22×D10111yescharacteristic
262C2×Q8—1014no—
263C2×Q8—2022no—
264C2×Q8—2022no—
265C2×D4—1014no—
266C2×D4—2022no—
267C2×D4—2022no—
268C22×C4—1011no—
269C22×C4—1011no—
270C22×C4—1011no—
271C22×C4—1011no—
272C22×C4—1011no—
273C22×C4—1011no—
274C22×C4—1011no—
275C22×C4—4041no—
276C22×C4—4041no—
277C22×C4—2022no—
278C22×C4—2022no—
279C22×C4—2022no—
280C22×C4—2022no—
281C22×C4C22×D10111yescharacteristic
282C22×C4C22×D10111yescharacteristic
283C22×C4C22×D10111yescharacteristic
284C22×C4C22×D10111yescharacteristic
285C22×C4C22×D10111yescharacteristic
286C22×C4C22×D10111yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—1014no—
289C4⋊C4—2022no—
290C4⋊C4—2022no—
291C4⋊C4—2022no—
292C4⋊C4—2022no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—1014no—
296C22⋊C4—1014no—
297C22⋊C4—1014no—
298C22⋊C4—4041no—
299C22⋊C4—2022no—
300C22⋊C4—2022no—
301C22⋊C4—2022no—
302C22⋊C4C4×D10114yescomplemented
303C42—41no—
304C42—1014no—
305C42—4041no—
306C42—2022no—
307C42—2022no—
308C42—2022no—
309C42—2022no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—1012no—
317C23—2021no—
318C23—2021no—
319C23—2021no—
320C23—2021no—
321C23—2021no—
322C23—2021no—
323C23C2×C4×D10112yes—
324C23C23×D10111yescharacteristic
325Q8—2028no—
326D4—2028no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—1012no—
343C2×C4—1012no—
344C2×C4—1012no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1012no—
348C2×C4—1012no—
349C2×C4—1012no—
350C2×C4—4044no—
351C2×C4—4044no—
352C2×C4—2021no—
353C2×C4—2021no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4—2021no—
358C2×C4—2021no—
359C2×C4—2021no—
360C2×C4—2021no—
361C2×C4—2021no—
362C2×C4—2021no—
363C2×C4—2021no—
364C2×C4—2021no—
365C2×C4C2×C4×D10112yes—
366C2×C4C2×C4×D10112yes—
367C2×C4D4×D10112yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—4041no—
376C22—4041no—
377C22—4041no—
378C22—4041no—
379C22—2022no—
380C22—2022no—
381C22—2022no—
382C22C22×C4×D10111yescharacteristic
383C22C2×D4×D10111yescharacteristic
384C22#41×D10111yescharacteristic
385C22#41×D10111yescharacteristic
386C22#41×D10111yescharacteristic
387C22#1569230011yescharacteristic
388C22#1569230211yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—1014no—
396C4—4041no—
397C4—2022no—
398C4—2022no—
399C4—2022no—
400C4—2022no—
401C4—2022no—
402C2—42no—
403C2—4042no—
404C2C4×D4×D10111yescharacteristic
405C2#2026564511yescharacteristic
406C2#2026565611yescharacteristic
407C2#2026581611yescharacteristic
408C2#2026583511yescharacteristic
409C2#2026589411yescharacteristic
410C2#2026591111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC22×C22⋊D101C22
hypercenter#388D101
hypocenterC101#2367
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#15692220C2semidirect
4. #20265081C2semidirect
6. #20265066C2semidirect
7. #20265066C2semidirect
8. #20265065C2semidirect
9. #20265049C2semidirect
10. #20265048C2semidirect
11. C4×#13108053C2semidirect
12. #20265028C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C4×#13108051C2semidirect
15. #388×C101C2semidirect
20. #15692241C4semidirect
57. C2×Dic202C2×C4semidirect
80. C22⋊Dic101C2×C4semidirect
84. #13108051C2×C4semidirect
85. #13108051C2×C4semidirect
91. C4×Dic101C2×C4semidirect
102. C22⋊C4×C101C2×C4semidirect
206. C101#2367semidirect
215. #388D101semidirect
302. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922412
C2×#156922202
C2×#156922152
#202650812
#202650772
#202650662
#202650652
#202650492
#202650482
C4×#131080532
#202650282
C22⋊C4×Dic1012
C4×#131080512
#388×C1012
#2367101