Finite Group Databasebeta

← #27173726

Subgroups — #27173726

Order 12928.

672 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×D101C211yesmaximal, characteristic
2#20265071C211yesmaximal, characteristic, complemented
3#20265070C211yesmaximal, characteristic, complemented
4#20265069C211yesmaximal, characteristic, complemented
5#20265068C211yesmaximal, characteristic, complemented
6#20265067C211yesmaximal, characteristic, complemented
7#20265066C211yesmaximal, characteristic, complemented
8#20265065C211yesmaximal, characteristic, complemented
9#20265036C211yesmaximal, characteristic, complemented
10#20265034C211yesmaximal, characteristic, complemented
11#20265032C211yesmaximal, characteristic
12#20265032C211yesmaximal, characteristic
13C4×#13108051C211yesmaximal, characteristic, complemented
14#20265024C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19C2×C4⋊Dic101C2211yescharacteristic, complemented
20C2×#13108053C2211yescharacteristic, complemented
21C2×#13108053C2211yescharacteristic, complemented
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2211yescharacteristic
25C2×#13108051C2211yescharacteristic
26C2×#13108051C2211yescharacteristic
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C22⋊C4×D101C418yescomplemented
36C2×C4×Dic101C2211yescharacteristic, complemented
37C2×C4×Dic101C2211yescharacteristic, complemented
38C2×C4×Dic101C2211yescharacteristic, complemented
39#15692162C2211yescharacteristic, complemented
40#15692162C2211yescharacteristic, complemented
41#15692162C2211yescharacteristic, complemented
42#15692162C2211yescharacteristic, complemented
43#15692162C2211yescharacteristic, complemented
44#15692162C2211yescharacteristic, complemented
45C2×C4⋊C4×C101C2211yescharacteristic, complemented
46C2×C22⋊C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic, complemented
50#95×C101C2211yescharacteristic, complemented
51#95×C101C2211yescharacteristic, complemented
52C23×D101—22no—
53C23×D101—22no—
54C23×D101—41no—
55C23×D101—41no—
56C23×D101—41no—
57C23×D101—41no—
58C23×D101C2×C414yes—
59C23×D101C2311yescharacteristic
60C23×D101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C2×C4×D101—41no—
68C2×C4×D101—41no—
69C2×C4×D101—41no—
70C2×C4×D101—41no—
71C2×C4×D101C2×C414yes—
72C2×C4×D101C2×C414yes—
73C4⋊Dic101—41no—
74#13108053—41no—
75#13108053—41no—
76C22⋊Dic101—41no—
77C22⋊Dic101—41no—
78C22⋊Dic101C2×C414yes—
79#13108051—22no—
80#13108051—22no—
81#13108051—22no—
82#13108051—22no—
83#13108051—41no—
84#13108051—41no—
85#13108051—41no—
86#13108051—41no—
87#13108051C2×C414yes—
88#13108051C2×C414yes—
89C4×Dic101—41no—
90C4×Dic101—41no—
91C4×Dic101—41no—
92C24×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96C22×C4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C4⋊C4×C101—41no—
100C22⋊C4×C101—41no—
101C22⋊C4×C101—41no—
102C22⋊C4×C101C2×C414yes—
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×D101—21no—
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—28no—
123C22×D101—44no—
124C22×D101—44no—
125C22×D101—44no—
126C22×D101—44no—
127C22×D101—44no—
128C22×D101—44no—
129C22×D101C22×C412yes—
130C22×D101C22×C412yes—
131C22×D101#4112yes—
132C22×D101#4112yes—
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—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101C22×C412yes—
150C2×C4×C101C22×C412yes—
151C2×Dic101—21no—
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×Dic101C22×C412yes—
168C2×Dic101C22×C412yes—
169C4×D101—44no—
170C4×D101—44no—
171C22×C101—22no—
172C22×C101—22no—
173C22×C101—22no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101C23×C411yescharacteristic
179C22×C101C2×#4111yescharacteristic
180C22×C101C2×#4111yescharacteristic
181C22×C101H(5,𝔽2)11yescharacteristic
182C22×C101H(5,𝔽2)11yescharacteristic
183C22×C101H(5,𝔽2)11yescharacteristic
184C22×C101H(5,𝔽2)11yescharacteristic
185C2×D101—22no—
186C2×D101—22no—
187C2×D101—22no—
188C2×D101—22no—
189C2×D101—22no—
190C2×D101—22no—
191C2×D101—22no—
192C2×D101—22no—
193C2×D101—22no—
194C2×D101—416no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101#11714yes—
200C4×C101—22no—
201C4×C101—22no—
202C4×C101—41no—
203C4×C101—41no—
204C4×C101—41no—
205C4×C101—41no—
206Dic101—22no—
207Dic101—22no—
208Dic101—41no—
209Dic101—41no—
210Dic101—41no—
211Dic101—41no—
212C2×C101—42no—
213C2×C101C2×#11711yescharacteristic
214C2×C101#51811yescharacteristic
215C2×C101#51811yescharacteristic
216C2×C101#53811yescharacteristic
217C2×C101#53811yescharacteristic
218C2×C101#53811yescharacteristic
219C2×C101#53811yescharacteristic
220D101—24no—
221D101—42no—
222#2312—1011nomaximal, Sylow
223C101#231211yescharacteristic, complemented, Sylow
224C22×C22⋊C4—1011no—
225#388—1011no—
226#388—1011no—
227#388—1011no—
228#388—1011no—
229#388—1011no—
230#388—1011no—
231#388—1011no—
232#388D10111yescharacteristic, complemented
233#380—1011no—
234#380—1011no—
235#379—1011no—
236#379—1011no—
237C4×C22⋊C4—1011no—
238C4×C22⋊C4—1011no—
239C25—1011no—
240C23×C4—1011no—
241C23×C4—1011no—
242C2×C4⋊C4—1011no—
243C2×C4⋊C4—1011no—
244C2×C4⋊C4—1011no—
245C2×C4⋊C4C2×D10111yescharacteristic, complemented
246C2×C22⋊C4—1011no—
247C2×C22⋊C4—1011no—
248C2×C22⋊C4—1011no—
249C2×C22⋊C4—1018no—
250C2×C22⋊C4—1011no—
251C2×C22⋊C4—1011no—
252C2×C22⋊C4—1011no—
253C2×C22⋊C4—1011no—
254C2×C22⋊C4—1011no—
255C2×C22⋊C4—1011no—
256C2×C22⋊C4—1011no—
257C2×C22⋊C4—1011no—
258C2×C22⋊C4—1011no—
259C2×C22⋊C4—1011no—
260C2×C22⋊C4C2×D10111yescharacteristic
261C2×C22⋊C4C2×D10111yescharacteristic
262C2×C22⋊C4C2×D10111yescharacteristic
263C2×C42—1011no—
264C2×C42—1011no—
265C2×C42—1011no—
266C2×C42C2×D10111yescharacteristic, complemented
267#95—1011no—
268#95—1011no—
269#95—1011no—
270#95—1011no—
271#95—1011no—
272#95—1011no—
273#95C2×D10111yescharacteristic, complemented
274#95C2×D10111yescharacteristic, complemented
275C24—1011no—
276C24—1014no—
277C24—1011no—
278C24—4041no—
279C24—4041no—
280C24—4041no—
281C24—4041no—
282C24—2022no—
283C24—2022no—
284C24C22×D10111yescharacteristic
285C22×C4—1011no—
286C22×C4—1011no—
287C22×C4—1014no—
288C22×C4—1014no—
289C22×C4—1011no—
290C22×C4—1011no—
291C22×C4—1011no—
292C22×C4—1011no—
293C22×C4—4041no—
294C22×C4—4041no—
295C22×C4—4041no—
296C22×C4—4041no—
297C22×C4C22×D10111yescharacteristic
298C22×C4C22×D10111yescharacteristic
299C22×C4C22×D10111yescharacteristic
300C22×C4C22×D10111yescharacteristic
301C22×C4C22×D10111yescharacteristic
302C22×C4C22×D10111yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—4041no—
305C4⋊C4—4041no—
306C4⋊C4—4041no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—1014no—
310C22⋊C4—1014no—
311C22⋊C4—1014no—
312C22⋊C4—4041no—
313C22⋊C4—4041no—
314C22⋊C4—4041no—
315C22⋊C4—4041no—
316C22⋊C4—4041no—
317C22⋊C4—4041no—
318C22⋊C4—2022no—
319C22⋊C4—2022no—
320C22⋊C4—2022no—
321C22⋊C4—2022no—
322C22⋊C4C4×D10114yes—
323C42—41no—
324C42—4041no—
325C42—4041no—
326C42—4041no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—1012no—
334C23—1012no—
335C23—1012no—
336C23—1012no—
337C23—4044no—
338C23—4044no—
339C23—4044no—
340C23—4044no—
341C23—4044no—
342C23—4044no—
343C23—2028no—
344C23—2021no—
345C23—2021no—
346C23—2021no—
347C23—2021no—
348C23—2021no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23C2×C4×D10112yes—
355C23C23×D10111yescharacteristic
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
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—1012no—
373C2×C4—1012no—
374C2×C4—4044no—
375C2×C4—4044no—
376C2×C4—2021no—
377C2×C4—2021no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4—2021no—
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×C4C2×C4×D10112yes—
393C2×C4C2×C4×D10112yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—1014no—
402C22—40416no—
403C22—4041no—
404C22—4041no—
405C22—4041no—
406C22—4041no—
407C22—2022no—
408C22—2022no—
409C22—2022no—
410C22—2022no—
411C22—2022no—
412C22—2022no—
413C22—2022no—
414C22—2022no—
415C22—2022no—
416C22C22×C4×D10111yescharacteristic
417C22#41×D10111yescharacteristic
418C22#41×D10111yescharacteristic
419C22#1569230011yescharacteristic
420C22#1569230011yescharacteristic
421C22#1569230111yescharacteristic
422C22#1569230111yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—4041no—
430C4—4041no—
431C4—4041no—
432C4—4041no—
433C4—2022no—
434C4—2022no—
435C2—42no—
436C2—4042no—
437C2—2024no—
438C2#117×D10111yescharacteristic
439C2#2026564911yescharacteristic
440C2#2026564911yescharacteristic
441C2#2026575911yescharacteristic
442C2#2026576111yescharacteristic
443C2#2026576311yescharacteristic
444C2#2026576611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC24×D101C22
hypercenter#388D101
hypocenterC101#2312
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20265071C2semidirect
3. #20265070C2semidirect
4. #20265069C2semidirect
5. #20265068C2semidirect
6. #20265067C2semidirect
7. #20265066C2semidirect
8. #20265065C2semidirect
9. #20265036C2semidirect
10. #20265034C2semidirect
13. C4×#13108051C2semidirect
14. #20265024C2semidirect
15. #388×C101C2semidirect
19. C2×C4⋊Dic101C22semidirect
20. C2×#13108053C22semidirect
21. C2×#13108053C22semidirect
35. C22⋊C4×D101C4semidirect
36. C2×C4×Dic101C22semidirect
37. C2×C4×Dic101C22semidirect
38. C2×C4×Dic101C22semidirect
39. #15692162C22semidirect
40. #15692162C22semidirect
41. #15692162C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. C2×C4⋊C4×C101C22semidirect
49. C2×C42×C101C22semidirect
50. #95×C101C22semidirect
51. #95×C101C22semidirect
223. C101#2312semidirect
232. #388D101semidirect
245. C2×C4⋊C4C2×D101semidirect
266. C2×C42C2×D101semidirect
273. #95C2×D101semidirect
274. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1012
#202650712
#202650702
#202650692
#202650682
#202650672
#202650662
#202650652
#202650362
#202650342
#202650322
C4×#131080512
#202650242
#388×C1012
#2312101