Finite Group Databasebeta

← #27174177

Subgroups — #27174177

Order 12928.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692227C211yesmaximal, characteristic
2C2×#15692215C211yesmaximal, characteristic, complemented
3#20265071C211yesmaximal, characteristic, complemented
4#20265067C211yesmaximal, characteristic, complemented
5#20265061C211yesmaximal, characteristic, complemented
6#20265058C211yesmaximal, characteristic, complemented
7#20265057C211yesmaximal, characteristic, complemented
8#20265056C211yesmaximal, characteristic, complemented
9#20265056C211yesmaximal, characteristic, complemented
10#20265053C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
12C4×#13108051C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
14C2×#15692162C211yesmaximal, characteristic, complemented
15C2×C4×D4×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21#15692227C418yescomplemented
22#15692215—24no—
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
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
54C23×D101C2311yescharacteristic
55C2×C22⋊D101—22no—
56C2×C22⋊D101—22no—
57C2×C22⋊D101—41no—
58C2×C22⋊D101—41no—
59C2×C22⋊D101C2×C414yescomplemented
60C2×C22⋊D101C2×C414yescomplemented
61C22×Dic101—22no—
62C22×Dic101—22no—
63C22×Dic101—41no—
64C22×Dic101—41no—
65C22×Dic101—41no—
66C22×Dic101C2×C414yescomplemented
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C2×C4×D101—22no—
75C2×C4×D101—22no—
76C2×C4×D101—41no—
77#13108053—22no—
78#13108053C2×C414yescomplemented
79C22⋊Dic101—22no—
80C22⋊Dic101—22no—
81C22⋊Dic101—41no—
82C22⋊Dic101C2×C414yescomplemented
83#13108051—22no—
84#13108051—22no—
85#13108051—41no—
86#13108051C2×C414yescomplemented
87C4×Dic101—22no—
88C4×Dic101—41no—
89C4×Dic101—41no—
90C24×C101C2311yescharacteristic
91C24×C101C2311yescharacteristic
92C2×D4×C101—22no—
93C2×D4×C101—22no—
94C2×D4×C101C2×C414yescomplemented
95C22×C4×C101—22no—
96C22×C4×C101—22no—
97C22×C4×C101—22no—
98C22×C4×C101—22no—
99C22×C4×C101—41no—
100C22×C4×C101—41no—
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C4⋊C4×C101—22no—
107C22⋊C4×C101—22no—
108C22⋊C4×C101D414yescomplemented
109C42×C101—22no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101C22×C412yes—
121C23×C101C22×C412yes—
122C23×C101C2×D412yes—
123C23×C101#4112yes—
124C23×C101C2411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—21no—
131C22×D101C22×C412yes—
132D4×C101—28no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—44no—
145C2×C4×C101—44no—
146C2×C4×C101C22×C412yes—
147C2×C4×C101C2×D412yes—
148C2×C4×C101C2×D412yes—
149C2×C4×C101#4112yes—
150C22⋊D101—28no—
151C22⋊D101—44no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—21no—
166C2×Dic101—21no—
167C2×Dic101—21no—
168C2×Dic101—21no—
169C2×Dic101—44no—
170C2×Dic101—44no—
171C2×Dic101C22×C412yes—
172C2×Dic101C22×C412yes—
173C2×Dic101C22×C412yes—
174C2×Dic101#4112yes—
175C4×D101—44no—
176C22×C101—22no—
177C22×C101—22no—
178C22×C101—22no—
179C22×C101—22no—
180C22×C101—22no—
181C22×C101—22no—
182C22×C101—22no—
183C22×C101—22no—
184C22×C101—22no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101C4×D414yes—
190C22×C101C23×C411yescharacteristic
191C22×C101C22×D411yescharacteristic
192C22×C101C2×#4111yescharacteristic
193C22×C101C2×#4111yescharacteristic
194C22×C101C2×#4111yescharacteristic
195C22×C101H(5,𝔽2)11yescharacteristic
196C22×C101H(5,𝔽2)11yescharacteristic
197C2×D101—22no—
198C2×D101—22no—
199C2×D101—22no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C4×C101—22no—
205C4×C101—22no—
206C4×C101—22no—
207C4×C101—22no—
208C4×C101—41no—
209Dic101—22no—
210Dic101—22no—
211Dic101—22no—
212Dic101—22no—
213Dic101—41no—
214Dic101—41no—
215Dic101—41no—
216C2×C101—24no—
217C2×C101—42no—
218C2×C101C2×C4×D411yescharacteristic
219C2×C101C4×#4111yescharacteristic
220C2×C101#51811yescharacteristic
221C2×C101#54611yescharacteristic
222C2×C101#54611yescharacteristic
223C2×C101#55311yescharacteristic
224C2×C101#55311yescharacteristic
225D101—42no—
226#2364—1011nomaximal, Sylow
227C101#236411yescharacteristic, complemented, Sylow
228C2×#121—1011no—
229C2×C4×D4—1011no—
230C2×C4×D4D10111yescharacteristic, complemented
231#388—1011no—
232#388—1011no—
233#387—1011no—
234#387—1011no—
235#386—1011no—
236#386—1011no—
237#386—1011no—
238#386—1011no—
239C4×C22⋊C4—1011no—
240C4×C22⋊C4—1011no—
241C4×C22⋊C4—1011no—
242C2×#95—1011no—
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4C2×D10111yescharacteristic
246C23×C4—1011no—
247C23×C4—1011no—
248C23×C4—1011no—
249C23×C4C2×D10111yescharacteristic
250C23×C4C2×D10111yescharacteristic
251#121—1018no—
252C4×D4—24no—
253C4×D4—2024no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4C2×D10111yescharacteristic
257C2×C22⋊C4—1011no—
258C2×C22⋊C4—1011no—
259C2×C22⋊C4—1011no—
260C2×C22⋊C4—1011no—
261C2×C22⋊C4—1011no—
262C2×C22⋊C4—1011no—
263C2×C22⋊C4—1011no—
264C2×C22⋊C4—1011no—
265C2×C22⋊C4C2×D10111yescharacteristic
266C2×C22⋊C4C2×D10111yescharacteristic
267C2×C42—1011no—
268C2×C42—1011no—
269C2×C42—1011no—
270C2×C42C2×D10111yescharacteristic
271#95—1011no—
272#95—1011no—
273#95—1011no—
274#95—1011no—
275#95—1011no—
276#95—1011no—
277#95—1011no—
278#95—1011no—
279#95—1011no—
280#95—1011no—
281C24—1011no—
282C24C22×D10111yescharacteristic
283C24C22×D10111yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—1014no—
287C2×D4—1014no—
288C2×D4—4041no—
289C2×D4—4041no—
290C2×D4—2022no—
291C2×D4—2022no—
292C2×D4C4×D10114yescomplemented
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—1014no—
300C22×C4—1011no—
301C22×C4—1011no—
302C22×C4—1011no—
303C22×C4—1011no—
304C22×C4—1011no—
305C22×C4—1011no—
306C22×C4—1011no—
307C22×C4—4041no—
308C22×C4—4041no—
309C22×C4—4041no—
310C22×C4—4041no—
311C22×C4—2022no—
312C22×C4—2022no—
313C22×C4—2022no—
314C22×C4—2022no—
315C22×C4C22×D10111yescharacteristic
316C22×C4C22×D10111yescharacteristic
317C22×C4C22×D10111yescharacteristic
318C22×C4C22×D10111yescharacteristic
319C22×C4C22×D10111yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—1014no—
322C4⋊C4—2022no—
323C22⋊C4—22no—
324C22⋊C4—1014no—
325C22⋊C4—1014no—
326C22⋊C4—4041no—
327C22⋊C4—4041no—
328C22⋊C4—2022no—
329C22⋊C4—2022no—
330C22⋊C4—2022no—
331C22⋊C4—2022no—
332C22⋊C4C22⋊D10114yescomplemented
333C42—22no—
334C42—4041no—
335C42—4041no—
336C42—2022no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—1012no—
348C23—2021no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23C2×C4×D10112yes—
355C23C2×C4×D10112yes—
356C23C2×C22⋊D10112yes—
357C23#1310807412yes—
358C23C23×D10111yescharacteristic
359D4—28no—
360D4—4044no—
361D4—2028no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—1012no—
376C2×C4—1012no—
377C2×C4—1012no—
378C2×C4—1012no—
379C2×C4—4044no—
380C2×C4—4044no—
381C2×C4—4044no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4—2021no—
392C2×C4—2021no—
393C2×C4—2021no—
394C2×C4—2021no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4C2×C4×D10112yes—
400C2×C4C2×C22⋊D10112yes—
401C2×C4C2×C22⋊D10112yes—
402C2×C4#1310807312yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—4041no—
417C22—4041no—
418C22—4041no—
419C22—4041no—
420C22—2022no—
421C22—2022no—
422C22—2022no—
423C22C4×C22⋊D10114yes—
424C22C22×C4×D10111yescharacteristic
425C22C22×C22⋊D10111yescharacteristic
426C22#41×D10111yescharacteristic
427C22C2×#1310807311yescharacteristic
428C22C2×#1310807411yescharacteristic
429C22#1569230011yescharacteristic
430C22#1569230111yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—4041no—
437C4—4041no—
438C4—4041no—
439C4—2022no—
440C4—2022no—
441C4—2022no—
442C4—2022no—
443C2—24no—
444C2—42no—
445C2—4042no—
446C2C2×C4×C22⋊D10111yescharacteristic
447C2C4×#1310807311yescharacteristic
448C2#2026564911yescharacteristic
449C2#2026581511yescharacteristic
450C2#2026582111yescharacteristic
451C2#2026587311yescharacteristic
452C2#2026587611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC2×C4×D4×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692227C2
hypercenterC2×C4×D4D101
hypocenterC101#2364
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×#15692215C2semidirect
3. #20265071C2semidirect
4. #20265067C2semidirect
5. #20265061C2semidirect
6. #20265058C2semidirect
7. #20265057C2semidirect
8. #20265056C2semidirect
9. #20265056C2semidirect
10. #20265053C2semidirect
11. C22⋊C4×Dic101C2semidirect
12. C4×#13108051C2semidirect
13. C4×C22⋊Dic101C2semidirect
14. C2×#15692162C2semidirect
15. C2×C4×D4×C101C2semidirect
21. #15692227C4semidirect
59. C2×C22⋊D101C2×C4semidirect
60. C2×C22⋊D101C2×C4semidirect
66. C22×Dic101C2×C4semidirect
78. #13108053C2×C4semidirect
82. C22⋊Dic101C2×C4semidirect
86. #13108051C2×C4semidirect
94. C2×D4×C101C2×C4semidirect
108. C22⋊C4×C101D4semidirect
227. C101#2364semidirect
230. C2×C4×D4D101semidirect
292. C2×D4C4×D101semidirect
332. C22⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922272
C2×#156922152
#202650712
#202650672
#202650612
#202650582
#202650572
#202650562
#202650532
C22⋊C4×Dic1012
C4×#131080512
C4×C22⋊Dic1012
C2×#156921622
C2×C4×D4×C1012
#2364101