Finite Group Databasebeta

← #27174118

Subgroups — #27174118

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×#15692236C211yesmaximal, characteristic
2C2×#15692211C211yesmaximal, characteristic, complemented
3#20265072C211yesmaximal, characteristic, complemented
4#20265071C211yesmaximal, characteristic, complemented
5#20265065C211yesmaximal, characteristic, complemented
6#20265061C211yesmaximal, characteristic, complemented
7#20265058C211yesmaximal, characteristic, complemented
8#20265057C211yesmaximal, characteristic
9#20265050C211yesmaximal, characteristic, complemented
10#20265050C211yesmaximal, characteristic, complemented
11C4×#13108053C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
14C2×#15692162C211yesmaximal, characteristic, complemented
15C2×C4×D4×C101C211yesmaximal, characteristic
16C23×Dic101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18#15692236C418yescomplemented
19#15692211—24no—
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33C2×C4×Dic101C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36#15692162C2211yescharacteristic
37#15692162C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46C22×D4×C101C2211yescharacteristic
47C23×C4×C101C2211yescharacteristic
48C23×C4×C101C2211yescharacteristic
49C4×D4×C101—24no—
50C2×C4⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C42×C101C2211yescharacteristic
54C22×Dic101—22no—
55C22×Dic101—22no—
56C22×Dic101—22no—
57C22×Dic101—22no—
58C22×Dic101—41no—
59C22×Dic101C2×C414yescomplemented
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68#13108053—22no—
69#13108053—22no—
70#13108053—22no—
71#13108053—41no—
72#13108053C2×C414yescomplemented
73#13108053C2×C414yescomplemented
74C22⋊Dic101—22no—
75C22⋊Dic101—22no—
76C22⋊Dic101—41no—
77C22⋊Dic101C2×C414yescomplemented
78C22⋊Dic101C2×C414yescomplemented
79C22⋊Dic101C2×C414yescomplemented
80C4×Dic101—22no—
81C4×Dic101—22no—
82C4×Dic101—22no—
83C4×Dic101—41no—
84C24×C101C2311yescharacteristic
85C24×C101C2311yescharacteristic
86C2×D4×C101—22no—
87C2×D4×C101—22no—
88C2×D4×C101C2×C414yes—
89C22×C4×C101—22no—
90C22×C4×C101—22no—
91C22×C4×C101—22no—
92C22×C4×C101—22no—
93C22×C4×C101—41no—
94C22×C4×C101—41no—
95C22×C4×C101C2311yescharacteristic
96C22×C4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C4⋊C4×C101—22no—
101C22⋊C4×C101—22no—
102C22⋊C4×C101D414yes—
103C42×C101—22no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101—21no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101C22×C412yes—
115C23×C101C22×C412yes—
116C23×C101C2×D412yes—
117C23×C101#4112yes—
118C23×C101C2411yescharacteristic
119D4×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—44no—
132C2×C4×C101—44no—
133C2×C4×C101C22×C412yes—
134C2×C4×C101C2×D412yes—
135C2×C4×C101C2×D412yes—
136C2×C4×C101#4112yes—
137C2×Dic101—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—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—44no—
158C2×Dic101C22×C412yes—
159C2×Dic101C22×C412yes—
160C2×Dic101C22×C412yes—
161C2×Dic101C22×C412yes—
162C2×Dic101#4112yes—
163C22×C101—22no—
164C22×C101—22no—
165C22×C101—22no—
166C22×C101—22no—
167C22×C101—22no—
168C22×C101—22no—
169C22×C101—22no—
170C22×C101—22no—
171C22×C101—22no—
172C22×C101—41no—
173C22×C101—41no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101C4×D414yes—
177C22×C101C23×C411yescharacteristic
178C22×C101C22×D411yescharacteristic
179C22×C101C2×#4111yescharacteristic
180C22×C101C2×#4111yescharacteristic
181C22×C101C2×#4111yescharacteristic
182C22×C101H(5,𝔽2)11yescharacteristic
183C22×C1012−1+411yescharacteristic
184C4×C101—22no—
185C4×C101—22no—
186C4×C101—22no—
187C4×C101—22no—
188C4×C101—41no—
189Dic101—22no—
190Dic101—22no—
191Dic101—22no—
192Dic101—22no—
193Dic101—22no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—24no—
198C2×C101—42no—
199C2×C101C2×C4×D411yescharacteristic
200C2×C101C4×#4111yescharacteristic
201C2×C101#52011yescharacteristic
202C2×C101#54611yescharacteristic
203C2×C101#54711yescharacteristic
204C2×C101#55111yescharacteristic
205C2×C101#55211yescharacteristic
206#2359—1011nomaximal, Sylow
207C101#235911yescharacteristic, complemented, Sylow
208C2×#123—1011no—
209C2×C4×D4D10111yescharacteristic
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×#95—1011no—
223C22×D4C2×D10111yescharacteristic
224C23×C4—1011no—
225C23×C4—1011no—
226C23×C4C2×D10111yescharacteristic
227C23×C4C2×D10111yescharacteristic
228#123—1018no—
229C4×D4—24no—
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⋊C4C2×D10111yescharacteristic
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⋊C4C2×D10111yescharacteristic
245C2×C22⋊C4C2×D10111yescharacteristic
246C2×C42—1011no—
247C2×C42—1011no—
248C2×C42—1011no—
249C2×C42—1011no—
250C2×C42C2×D10111yescharacteristic
251#95—1011no—
252#95—1011no—
253#95—1011no—
254#95—1011no—
255#95—1011no—
256#95—1011no—
257#95—1011no—
258#95—1011no—
259#95—1011no—
260#95—1011no—
261C24C22×D10111yescharacteristic
262C24C22×D10111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D10114yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—1014no—
273C22×C4—1011no—
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×C4—2022no—
285C22×C4—2022no—
286C22×C4C22×D10111yescharacteristic
287C22×C4C22×D10111yescharacteristic
288C22×C4C22×D10111yescharacteristic
289C22×C4C22×D10111yescharacteristic
290C22×C4C22×D10111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—1014no—
293C4⋊C4—1014no—
294C4⋊C4—4041no—
295C4⋊C4—2022no—
296C4⋊C4—2022no—
297C4⋊C4—2022no—
298C22⋊C4—22no—
299C22⋊C4—1014no—
300C22⋊C4—1014no—
301C22⋊C4—1014no—
302C22⋊C4—4041no—
303C22⋊C4—2022no—
304C22⋊C4—2022no—
305C22⋊C4C22⋊D10114yes—
306C42—22no—
307C42—4041no—
308C42—2022no—
309C42—2022no—
310C42—2022no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D10112yes—
322C23C2×C4×D10112yes—
323C23C2×C22⋊D10112yes—
324C23#1310807412yes—
325C23C23×D10111yescharacteristic
326D4—28no—
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—44no—
339C2×C4—44no—
340C2×C4—1012no—
341C2×C4—1012no—
342C2×C4—1012no—
343C2×C4—1012no—
344C2×C4—1012no—
345C2×C4—4044no—
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×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—2028no—
366C2×C4C2×C4×D10112yes—
367C2×C4C2×C22⋊D10112yes—
368C2×C4C2×C22⋊D10112yes—
369C2×C4#1310807312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C4×C22⋊D10114yes—
384C22C22×C4×D10111yescharacteristic
385C22C22×C22⋊D10111yescharacteristic
386C22#41×D10111yescharacteristic
387C22C2×#1310807311yescharacteristic
388C22C2×#1310807411yescharacteristic
389C22#1569230011yescharacteristic
390C22#1569230211yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—4041no—
397C4—4041no—
398C4—4041no—
399C4—2022no—
400C4—2022no—
401C4—2022no—
402C4—2022no—
403C4—2022no—
404C2—24no—
405C2—42no—
406C2C2×C4×C22⋊D10111yescharacteristic
407C2C4×#1310807311yescharacteristic
408C2#2026565611yescharacteristic
409C2#2026582111yescharacteristic
410C2#2026582611yescharacteristic
411C2#2026585611yescharacteristic
412C2#2026586711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC2×C4×D4×C101C2
socle, cosocleC23×C101C24
prime radicalC22×D4×C101C22
hypercenterC2×C4×D4D101
hypocenterC101#2359
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15692211C2semidirect
3. #20265072C2semidirect
4. #20265071C2semidirect
5. #20265065C2semidirect
6. #20265061C2semidirect
7. #20265058C2semidirect
9. #20265050C2semidirect
10. #20265050C2semidirect
11. C4×#13108053C2semidirect
12. C22⋊C4×Dic101C2semidirect
13. C4×C22⋊Dic101C2semidirect
14. C2×#15692162C2semidirect
18. #15692236C4semidirect
59. C22×Dic101C2×C4semidirect
72. #13108053C2×C4semidirect
73. #13108053C2×C4semidirect
77. C22⋊Dic101C2×C4semidirect
78. C22⋊Dic101C2×C4semidirect
79. C22⋊Dic101C2×C4semidirect
207. C101#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#156922362
C2×#156922112
#202650722
#202650712
#202650652
#202650612
#202650582
#202650572
#202650502
C4×#131080532
C22⋊C4×Dic1012
C4×C22⋊Dic1012
C2×#156921622
C2×C4×D4×C1012
#2359101