Finite Group Databasebeta

← #27174165

Subgroups — #27174165

Order 12928.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692228C211yesmaximal, characteristic
2C2×#15692215C211yesmaximal, characteristic, complemented
3C2×#15692212C211yesmaximal, characteristic, complemented
4#20265078C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic, complemented
6#20265077C211yesmaximal, characteristic, complemented
7#20265072C211yesmaximal, characteristic, complemented
8#20265066C211yesmaximal, characteristic, complemented
9#20265055C211yesmaximal, characteristic, complemented
10#20265054C211yesmaximal, characteristic, complemented
11#20265028C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
13C4×#13108051C211yesmaximal, characteristic, complemented
14#20265024C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692228C418yescomplemented
23#15692215—24no—
24#15692212—24no—
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41C2×C4×Dic101C2211yescharacteristic
42C2×C4×Dic101C2211yescharacteristic
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
55C23×D101C2311yescharacteristic
56C2×D404—41no—
57C2×D404—41no—
58C2×D404C2×C414yescomplemented
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101—41no—
62C2×C22⋊D101—41no—
63C2×C22⋊D101C2×C414yes—
64C2×C22⋊D101C2×C414yescomplemented
65C22×Dic101—22no—
66C22×Dic101—22no—
67C22×Dic101—41no—
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C2×C4×D101—22no—
75C2×C4×D101—22no—
76C2×C4×D101—22no—
77C2×C4×D101—22no—
78C2×C4×D101—41no—
79C2×C4×D101C2×C414yescomplemented
80#13108053—22no—
81#13108053—41no—
82#13108053C2×C414yescomplemented
83#13108053D414yescomplemented
84C22⋊Dic101—22no—
85#13108051—22no—
86#13108051—22no—
87#13108051—22no—
88#13108051—22no—
89#13108051—22no—
90#13108051—22no—
91#13108051—41no—
92#13108051C2×C414yescomplemented
93C4×Dic101—22no—
94C4×Dic101—22no—
95C4×Dic101—22no—
96C4×Dic101—41no—
97C4×Dic101—41no—
98C24×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C4⋊C4×C101—41no—
106C22⋊C4×C101—22no—
107C22⋊C4×C101—41no—
108C22⋊C4×C101C2×C414yescomplemented
109C42×C101—22no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101C22×C412yes—
117C23×C101C2411yescharacteristic
118C22×D101—21no—
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—21no—
124C22×D101—21no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101C22×C412yes—
131C22×D101C22×C412yes—
132C2×C4×C101—21no—
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×C101C22×C412yes—
148C2×C4×C101C22×C412yes—
149C2×C4×C101C2×D412yes—
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—44no—
164C2×Dic101C22×C412yes—
165C2×Dic101C22×C412yes—
166C2×Dic101C2×D412yes—
167C2×Dic101C2×D412yes—
168C2×Dic101#4112yes—
169C2×Dic101#4112yes—
170C2×Dic101#4112yes—
171D404—44no—
172C4×D101—28no—
173C4×D101—44no—
174C22×C101—22no—
175C22×C101—22no—
176C22×C101—22no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101C23×C411yescharacteristic
182C22×C101C22×D411yescharacteristic
183C22×C101C2×#4111yescharacteristic
184C22×C101C2×#4111yescharacteristic
185C22×C101C2×#4111yescharacteristic
186C22×C101H(5,𝔽2)11yescharacteristic
187C22×C101H(5,𝔽2)11yescharacteristic
188C2×D101—22no—
189C2×D101—22no—
190C2×D101—22no—
191C2×D101—22no—
192C2×D101—22no—
193C2×D101—22no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C4×C101—22no—
203C4×C101—22no—
204C4×C101—22no—
205C4×C101—41no—
206C4×C101—41no—
207C4×C101—41no—
208Dic101—22no—
209Dic101—22no—
210Dic101—22no—
211Dic101—22no—
212Dic101—41no—
213Dic101C4×D414yes—
214C2×C101—42no—
215C2×C101C2×C4×D411yescharacteristic
216C2×C101C4×#4111yescharacteristic
217C2×C101#51811yescharacteristic
218C2×C101#54611yescharacteristic
219C2×C101#55011yescharacteristic
220C2×C101#55311yescharacteristic
221C2×C101#55511yescharacteristic
222D101—42no—
223D101—42no—
224#2363—1011nomaximal, Sylow
225C101#236311yescharacteristic, complemented, Sylow
226C2×#121—1011no—
227C2×C4×D4—1011no—
228C2×#117—1011no—
229#390—1011no—
230#390—1011no—
231#390—1011no—
232#389—1011no—
233#388—1011no—
234#388D10111yescharacteristic, complemented
235#386—1011no—
236#386—1011no—
237C4×C4⋊C4—1011no—
238C4×C22⋊C4—1011no—
239C4×C22⋊C4—1011no—
240C4×C22⋊C4—1011no—
241C22×D4—1011no—
242C22×D4—1011no—
243C22×D4—1011no—
244C23×C4—1011no—
245C23×C4—1011no—
246C23×C4—1011no—
247#121—1018no—
248C4×D4—2024no—
249#117—2024no—
250C2×C4⋊C4—1011no—
251C2×C4⋊C4—1011no—
252C2×C4⋊C4—1011no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4C2×D10111yescharacteristic
255C2×C22⋊C4—1011no—
256C2×C22⋊C4—1011no—
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⋊C4C2×D10111yescharacteristic
265C2×C22⋊C4C2×D10111yescharacteristic
266C2×C22⋊C4C2×D10111yescharacteristic
267C2×C42—1011no—
268C2×C42—1011no—
269C2×C42—1011no—
270C2×C42—1011no—
271C2×C42—1011no—
272C2×C42C2×D10111yescharacteristic
273#95—1011no—
274#95—1011no—
275#95—1011no—
276#95—1011no—
277#95C2×D10111yescharacteristic
278#95C2×D10111yescharacteristic
279C24—1011no—
280C24—1011no—
281C24C22×D10111yescharacteristic
282C2×D4—1014no—
283C2×D4—1014no—
284C2×D4—1014no—
285C2×D4—4041no—
286C2×D4—4041no—
287C2×D4—4041no—
288C2×D4—4041no—
289C2×D4—2022no—
290C2×D4—2022no—
291C22×C4—1011no—
292C22×C4—1011no—
293C22×C4—1011no—
294C22×C4—1011no—
295C22×C4—1011no—
296C22×C4—1014no—
297C22×C4—1011no—
298C22×C4—4041no—
299C22×C4—4041no—
300C22×C4—2022no—
301C22×C4—2022no—
302C22×C4—2022no—
303C22×C4—2022no—
304C22×C4—2022no—
305C22×C4—2022no—
306C22×C4C22×D10111yescharacteristic
307C22×C4C22×D10111yescharacteristic
308C22×C4C22×D10111yescharacteristic
309C22×C4C22×D10111yescharacteristic
310C22×C4C22×D10111yescharacteristic
311C22×C4C22×D10111yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—1014no—
314C4⋊C4—1014no—
315C4⋊C4—4041no—
316C4⋊C4—2022no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—1014no—
320C22⋊C4—4041no—
321C22⋊C4—2022no—
322C22⋊C4—2022no—
323C22⋊C4—2022no—
324C22⋊C4—2022no—
325C22⋊C4—2022no—
326C22⋊C4—2022no—
327C22⋊C4—2022no—
328C22⋊C4C4×D10114yescomplemented
329C42—22no—
330C42—4041no—
331C42—4041no—
332C42—2022no—
333C42—2022no—
334C42—2022no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—1012no—
342C23—1012no—
343C23—2021no—
344C23—2021no—
345C23—2021no—
346C23—2021no—
347C23—2021no—
348C23—2021no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23—2021no—
355C23C2×C4×D10112yes—
356C23C23×D10111yescharacteristic
357D4—4044no—
358D4—4044no—
359D4—2028no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—1012no—
376C2×C4—1012no—
377C2×C4—1012no—
378C2×C4—1012no—
379C2×C4—1012no—
380C2×C4—1012no—
381C2×C4—1012no—
382C2×C4—4044no—
383C2×C4—4044no—
384C2×C4—2028no—
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×C4C2×C4×D10112yes—
397C2×C4C2×C4×D10112yes—
398C2×C4D4×D10112yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—4041no—
407C22—4041no—
408C22—4041no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—2022no—
415C22—2022no—
416C22—2022no—
417C22—2022no—
418C22—2022no—
419C22—2022no—
420C22C22×C4×D10111yescharacteristic
421C22C2×D4×D10111yescharacteristic
422C22#41×D10111yescharacteristic
423C22#41×D10111yescharacteristic
424C22#41×D10111yescharacteristic
425C22#1569230011yescharacteristic
426C22#1569230111yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—1014no—
434C4—4041no—
435C4—2022no—
436C4—2022no—
437C4—2022no—
438C4—2022no—
439C2—42no—
440C2—4042no—
441C2—4042no—
442C2C4×D4×D10111yescharacteristic
443C2#2026564511yescharacteristic
444C2#2026564911yescharacteristic
445C2#2026581911yescharacteristic
446C2#2026584511yescharacteristic
447C2#2026587811yescharacteristic
448C2#2026588911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692228C2
hypercenter#388D101
hypocenterC101#2363
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#15692215C2semidirect
3. C2×#15692212C2semidirect
4. #20265078C2semidirect
5. #20265077C2semidirect
6. #20265077C2semidirect
7. #20265072C2semidirect
8. #20265066C2semidirect
9. #20265055C2semidirect
10. #20265054C2semidirect
11. #20265028C2semidirect
12. C22⋊C4×Dic101C2semidirect
13. C4×#13108051C2semidirect
14. #20265024C2semidirect
15. #388×C101C2semidirect
22. #15692228C4semidirect
58. C2×D404C2×C4semidirect
64. C2×C22⋊D101C2×C4semidirect
79. C2×C4×D101C2×C4semidirect
82. #13108053C2×C4semidirect
83. #13108053D4semidirect
92. #13108051C2×C4semidirect
108. C22⋊C4×C101C2×C4semidirect
225. C101#2363semidirect
234. #388D101semidirect
328. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922282
C2×#156922152
C2×#156922122
#202650782
#202650772
#202650722
#202650662
#202650552
#202650542
#202650282
C22⋊C4×Dic1012
C4×#131080512
#202650242
#388×C1012
#2363101