Finite Group Databasebeta

← #27174120

Subgroups — #27174120

Order 12928.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692238C211yesmaximal, characteristic
2C2×C4×C22⋊D101C211yesmaximal, characteristic
3C2×#15692214C211yesmaximal, characteristic, complemented
4#20265074C211yesmaximal, characteristic, complemented
5#20265069C211yesmaximal, characteristic, complemented
6#20265066C211yesmaximal, characteristic, complemented
7#20265064C211yesmaximal, characteristic, complemented
8#20265059C211yesmaximal, characteristic, complemented
9#20265056C211yesmaximal, characteristic
10#20265051C211yesmaximal, characteristic, complemented
11#20265050C211yesmaximal, characteristic, complemented
12#20265028C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
14#20265024C211yesmaximal, characteristic, complemented
15C2×#95×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18#15692238C418yescomplemented
19C4×C22⋊D101—24no—
20#15692214—24no—
21C2×C4⋊Dic101C2211yescharacteristic
22C2×C4⋊Dic101C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
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
47C23×C4×C101C2211yescharacteristic
48C23×C4×C101C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C2×C22⋊D101—22no—
56C2×C22⋊D101—22no—
57C2×C22⋊D101C2×C414yes—
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C2×C4×D101—22no—
66C2×C4×D101—22no—
67C2×C4×D101—41no—
68C4⋊Dic101—22no—
69C4⋊Dic101—22no—
70#13108053—22no—
71#13108053—22no—
72#13108053—41no—
73#13108053C2×C414yescomplemented
74#13108053C2×C414yescomplemented
75C22⋊Dic101—22no—
76C22⋊Dic101C2×C414yescomplemented
77C22⋊Dic101D414yes—
78#13108051—22no—
79#13108051—22no—
80#13108051—41no—
81#13108051C2×C414yescomplemented
82#13108051C2×C414yescomplemented
83C4×Dic101—22no—
84C4×Dic101—22no—
85C4×Dic101—22no—
86C4×Dic101—22no—
87C4×Dic101—41no—
88C24×C101C2311yescharacteristic
89C22×C4×C101—22no—
90C22×C4×C101—22no—
91C22×C4×C101—22no—
92C22×C4×C101—22no—
93C22×C4×C101—22no—
94C22×C4×C101—22no—
95C22×C4×C101—41no—
96C22×C4×C101—41no—
97C22×C4×C101C2×C414yescomplemented
98C22×C4×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101C22×C412yes—
109C23×C101C2×D412yes—
110C23×C101#4112yes—
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—28no—
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×C101—44no—
135C2×C4×C101—44no—
136C2×C4×C101C22×C412yes—
137C2×C4×C101C22×C412yes—
138C2×C4×C101#4112yes—
139C2×C4×C101#4112yes—
140C22⋊D101—28no—
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—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101C22×C412yes—
158C2×Dic101C22×C412yes—
159C2×Dic101C22×C412yes—
160C2×Dic101C2×D412yes—
161C2×Dic101C2×D412yes—
162C4×D101—44no—
163C22×C101—22no—
164C22×C101—22no—
165C22×C101—22no—
166C22×C101—22no—
167C22×C101—22no—
168C22×C101—22no—
169C22×C101C4×D414yes—
170C22×C101C23×C411yescharacteristic
171C22×C101C22×D411yescharacteristic
172C22×C101C2×#4111yescharacteristic
173C22×C101C2×#4111yescharacteristic
174C22×C101C2×#4111yescharacteristic
175C22×C101H(5,𝔽2)11yescharacteristic
176C22×C1012−1+411yescharacteristic
177C2×D101—22no—
178C2×D101—22no—
179C2×D101—22no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C4×C101—22no—
185C4×C101—22no—
186C4×C101—22no—
187C4×C101—22no—
188C4×C101—41no—
189C4×C101—41no—
190Dic101—22no—
191Dic101—22no—
192Dic101—22no—
193Dic101—22no—
194Dic101—22no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—24no—
198C2×C101C2×C4×D411yescharacteristic
199C2×C101C4×#4111yescharacteristic
200C2×C101#52011yescharacteristic
201C2×C101#54611yescharacteristic
202C2×C101#54711yescharacteristic
203C2×C101#55111yescharacteristic
204C2×C101#55211yescharacteristic
205D101—42no—
206#2359—1011nomaximal, Sylow
207C101#235911yescharacteristic, complemented, Sylow
208C2×#123—1011no—
209C2×C4×D4—1011no—
210C2×#117—1011no—
211#389—1011no—
212#388—1011no—
213#388—1011no—
214#387—1011no—
215#387—1011no—
216#386—1011no—
217#385—1011no—
218#385—1011no—
219C4×C4⋊C4—1011no—
220C4×C22⋊C4—1011no—
221C4×C22⋊C4—1011no—
222C2×#95D10111yescharacteristic, complemented
223C22×D4—1011no—
224C23×C4—1011no—
225C23×C4C2×D10111yescharacteristic
226C23×C4C2×D10111yescharacteristic
227C23×C4C2×D10111yescharacteristic
228#123—1018no—
229C4×D4—2024no—
230#117—2024no—
231C2×C4⋊C4—1011no—
232C2×C4⋊C4—1011no—
233C2×C4⋊C4—1011no—
234C2×C4⋊C4—1011no—
235C2×C4⋊C4—1011no—
236C2×C4⋊C4—1011no—
237C2×C4⋊C4—1011no—
238C2×C22⋊C4—1011no—
239C2×C22⋊C4—1011no—
240C2×C22⋊C4—1011no—
241C2×C22⋊C4—1011no—
242C2×C22⋊C4—1011no—
243C2×C22⋊C4—1011no—
244C2×C22⋊C4—1011no—
245C2×C22⋊C4—1011no—
246C2×C42—1011no—
247C2×C42—1011no—
248C2×C42—1011no—
249C2×C42—1011no—
250C2×C42—1011no—
251#95—1011no—
252#95—1011no—
253#95—1011no—
254#95—1011no—
255#95—1011no—
256#95—1011no—
257#95C2×D10111yescharacteristic
258#95C2×D10111yescharacteristic
259#95C2×D10111yescharacteristic
260#95C2×D10111yescharacteristic
261C24—1011no—
262C24C22×D10111yescharacteristic
263C2×D4—1014no—
264C2×D4—2022no—
265C2×D4—2022no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—1011no—
275C22×C4—1011no—
276C22×C4—1011no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—1011no—
280C22×C4—1011no—
281C22×C4—4041no—
282C22×C4—2022no—
283C22×C4—2022no—
284C22×C4C4×D10114yescomplemented
285C22×C4C22×D10111yescharacteristic
286C22×C4C22×D10111yescharacteristic
287C22×C4C22×D10111yescharacteristic
288C22×C4C22×D10111yescharacteristic
289C22×C4C22×D10111yescharacteristic
290C22×C4C22×D10111yescharacteristic
291C4⋊C4—1014no—
292C4⋊C4—1014no—
293C4⋊C4—4041no—
294C4⋊C4—2022no—
295C4⋊C4—2022no—
296C4⋊C4—2022no—
297C4⋊C4—2022no—
298C22⋊C4—1014no—
299C22⋊C4—1014no—
300C22⋊C4—1014no—
301C22⋊C4—1014no—
302C22⋊C4—4041no—
303C22⋊C4—2022no—
304C22⋊C4—2022no—
305C22⋊C4—2022no—
306C42—4041no—
307C42—2022no—
308C42—2022no—
309C42—2022no—
310C42—2022no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—1012no—
316C23—2021no—
317C23—2021no—
318C23—2021no—
319C23—2021no—
320C23—2021no—
321C23—2021no—
322C23C2×C4×D10112yes—
323C23D4×D10112yes—
324C23#1310807312yes—
325C23C23×D10111yescharacteristic
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—28no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—1012no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1012no—
348C2×C4—1012no—
349C2×C4—4044no—
350C2×C4—2021no—
351C2×C4—2021no—
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×C4—2021no—
366C2×C4C2×C4×D10112yes—
367C2×C4C2×C4×D10112yes—
368C2×C4#1310807412yes—
369C2×C4#1310807412yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—4041no—
377C22—4041no—
378C22—4041no—
379C22—4041no—
380C22—2022no—
381C22—2022no—
382C22—2022no—
383C22#1569221514yes—
384C22C22×C4×D10111yescharacteristic
385C22C2×D4×D10111yescharacteristic
386C22C2×#1310807311yescharacteristic
387C22C2×#1310807411yescharacteristic
388C22C2×#1310807411yescharacteristic
389C22#1569230011yescharacteristic
390C22#1569230311yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—4041no—
398C4—4041no—
399C4—2022no—
400C4—2022no—
401C4—2022no—
402C4—2022no—
403C4—2022no—
404C2—24no—
405C2—4042no—
406C2C2×#1569221511yescharacteristic
407C2C4×#1310807411yescharacteristic
408C2#2026565811yescharacteristic
409C2#2026581711yescharacteristic
410C2#2026582811yescharacteristic
411C2#2026585311yescharacteristic
412C2#2026586211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC2×#95×C101C2
socle, cosocleC23×C101C24
prime radicalC22×C22⋊D101C22
hypercenterC2×#95D101
hypocenterC101#2359
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
3. C2×#15692214C2semidirect
4. #20265074C2semidirect
5. #20265069C2semidirect
6. #20265066C2semidirect
7. #20265064C2semidirect
8. #20265059C2semidirect
10. #20265051C2semidirect
11. #20265050C2semidirect
12. #20265028C2semidirect
13. C4×C22⋊Dic101C2semidirect
14. #20265024C2semidirect
15. C2×#95×C101C2semidirect
18. #15692238C4semidirect
73. #13108053C2×C4semidirect
74. #13108053C2×C4semidirect
76. C22⋊Dic101C2×C4semidirect
81. #13108051C2×C4semidirect
82. #13108051C2×C4semidirect
97. C22×C4×C101C2×C4semidirect
207. C101#2359semidirect
222. C2×#95D101semidirect
284. C22×C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922382
C2×C4×C22⋊D1012
C2×#156922142
#202650742
#202650692
#202650662
#202650642
#202650592
#202650562
#202650512
#202650502
#202650282
C4×C22⋊Dic1012
#202650242
C2×#95×C1012
#2359101