Finite Group Databasebeta

← #27173642

Subgroups — #27173642

Order 12928.

824 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692227C211yesmaximal, characteristic
2C2×D4×Dic101C211yesmaximal, characteristic, complemented
3C2×#15692215C211yesmaximal, characteristic, complemented
4C2×#15692215C211yesmaximal, characteristic, complemented
5C22×C4×Dic101C211yesmaximal, characteristic, complemented
6#20265077C211yesmaximal, characteristic, complemented
7#20265074C211yesmaximal, characteristic, complemented
8#20265069C211yesmaximal, characteristic, complemented
9#20265065C211yesmaximal, characteristic, complemented
10#20265056C211yesmaximal, characteristic, complemented
11#20265035C211yesmaximal, characteristic, complemented
12#20265028C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14#20265024C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C23×Dic101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692227C418yescomplemented
23D4×Dic101—24no—
24#15692215—24no—
25#15692215—24no—
26C2×C4⋊Dic101C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×#13108053C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×C22⋊Dic101C2211yescharacteristic
33C2×C22⋊Dic101C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×C4×Dic101—24no—
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41C2×C4×Dic101C2211yescharacteristic
42C2×C4×Dic101C2211yescharacteristic
43C2×C4×Dic101C2211yescharacteristic
44C2×C4×Dic101C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48#15692162C2211yescharacteristic
49C22×D4×C101C2211yescharacteristic
50C23×C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54#95×C101C2211yescharacteristic
55#95×C101C2211yescharacteristic
56C23×D101C2311yescharacteristic
57C2×C22⋊D101—22no—
58C2×C22⋊D101—22no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101C2×C414yescomplemented
62C2×C22⋊D101C2×C414yescomplemented
63C22×Dic101—22no—
64C22×Dic101—22no—
65C22×Dic101—22no—
66C22×Dic101—22no—
67C22×Dic101—22no—
68C22×Dic101—22no—
69C22×Dic101—22no—
70C22×Dic101C2×C414yescomplemented
71C22×Dic101D414yescomplemented
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C2×C4×D101—22no—
80C2×C4×D101—22no—
81C2×C4×D101—22no—
82C4⋊Dic101—22no—
83#13108053—22no—
84#13108053—22no—
85#13108053—22no—
86#13108053C2×C414yescomplemented
87C22⋊Dic101—22no—
88C22⋊Dic101—22no—
89C22⋊Dic101C2×C414yescomplemented
90#13108051—22no—
91#13108051—22no—
92#13108051—22no—
93#13108051C2×C414yescomplemented
94C4×Dic101—22no—
95C4×Dic101—22no—
96C4×Dic101—22no—
97C4×Dic101—22no—
98C4×Dic101—22no—
99C4×Dic101D414yescomplemented
100C24×C101C2311yescharacteristic
101C24×C101C2311yescharacteristic
102C2×D4×C101—22no—
103C2×D4×C101—22no—
104C2×D4×C101C2×C414yescomplemented
105C22×C4×C101—22no—
106C22×C4×C101—22no—
107C22×C4×C101—22no—
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22⋊C4×C101—22no—
114C22⋊C4×C101—22no—
115C22⋊C4×C101—22no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101C22×C412yes—
127C23×C101C22×C412yes—
128C23×C101C2×D412yes—
129C23×C101#4112yes—
130C23×C101C2411yescharacteristic
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—21no—
134C22×D101—21no—
135C22×D101—21no—
136C22×D101—21no—
137C22×D101C22×C412yes—
138D4×C101—28no—
139C2×C4×C101—28no—
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×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101C22×C412yes—
153C2×C4×C101C2×D412yes—
154C2×C4×C101#4112yes—
155C22⋊D101—28no—
156C22⋊D101—28no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—28no—
164C2×Dic101—28no—
165C2×Dic101—21no—
166C2×Dic101—21no—
167C2×Dic101—21no—
168C2×Dic101—21no—
169C2×Dic101—21no—
170C2×Dic101—21no—
171C2×Dic101—44no—
172C2×Dic101C22×C412yes—
173C2×Dic101C22×C412yes—
174C2×Dic101C22×C412yes—
175C2×Dic101C2×D412yes—
176C2×Dic101C2×D412yes—
177C2×Dic101C2×D412yes—
178C2×Dic101C2×D412yes—
179C2×Dic101#4112yes—
180C2×Dic101#4112yes—
181C4×D101—44no—
182C22×C101—22no—
183C22×C101—22no—
184C22×C101—22no—
185C22×C101—22no—
186C22×C101—22no—
187C22×C101—22no—
188C22×C101—22no—
189C22×C101—22no—
190C22×C101—22no—
191C22×C101—41no—
192C22×C101—41no—
193C22×C101—41no—
194C22×C101—41no—
195C22×C101C4×D414yescomplemented
196C22×C101C23×C411yescharacteristic
197C22×C101C22×D411yescharacteristic
198C22×C101C22×D411yescharacteristic
199C22×C101C2×#4111yescharacteristic
200C22×C101C2×#4111yescharacteristic
201C22×C101C2×#4111yescharacteristic
202C22×C101C2×#4111yescharacteristic
203C2×D101—22no—
204C2×D101—22no—
205C2×D101—22no—
206C2×D101—41no—
207C2×D101—41no—
208C2×D101—41no—
209C2×D101—41no—
210C4×C101—22no—
211C4×C101—22no—
212C4×C101—22no—
213C4×C101—41no—
214C4×C101—41no—
215Dic101—22no—
216Dic101—22no—
217Dic101—22no—
218Dic101—41no—
219Dic101—41no—
220Dic101C4×D414yescomplemented
221Dic101#12114yescomplemented
222C2×C101—24no—
223C2×C101—42no—
224C2×C101C2×C4×D411yescharacteristic
225C2×C101C2×C4×D411yescharacteristic
226C2×C101C4×#4111yescharacteristic
227C2×C101C2×#12111yescharacteristic
228C2×C101#52511yescharacteristic
229C2×C101#52911yescharacteristic
230C2×C101#53211yescharacteristic
231D101—42no—
232C4×#121—1011nomaximal, Sylow
233C101C4×#12111yescharacteristic, complemented, Sylow
234C2×#121—1011no—
235C2×C4×D4—1011no—
236C2×C4×D4—1011no—
237C2×C4×D4—1011no—
238C22×C42—1011no—
239#390—1011no—
240#389—1011no—
241#388—1011no—
242#388—1011no—
243#386—1011no—
244#386D10111yescharacteristic, complemented
245#380—1011no—
246C4×C4⋊C4—1011no—
247C4×C22⋊C4—1011no—
248C4×C22⋊C4—1011no—
249C22×D4—1011no—
250C22×D4—1011no—
251C22×D4C2×D10111yescharacteristic
252C23×C4—1011no—
253C23×C4—1011no—
254C23×C4—1011no—
255C23×C4—1011no—
256C23×C4C2×D10111yescharacteristic
257#121—1018no—
258C4×D4—2024no—
259C4×D4—2024no—
260C4×D4—2024no—
261C2×C4⋊C4—1011no—
262C2×C4⋊C4—1011no—
263C2×C4⋊C4—1011no—
264C2×C4⋊C4—1011no—
265C2×C4⋊C4—1011no—
266C2×C22⋊C4—1011no—
267C2×C22⋊C4—1011no—
268C2×C22⋊C4—1011no—
269C2×C22⋊C4—1011no—
270C2×C22⋊C4—1011no—
271C2×C22⋊C4—1011no—
272C2×C22⋊C4—1011no—
273C2×C22⋊C4C2×D10111yescharacteristic
274C2×C22⋊C4C2×D10111yescharacteristic
275C2×C22⋊C4C2×D10111yescharacteristic
276C2×C42—1011no—
277C2×C42—1011no—
278C2×C42—1011no—
279C2×C42—1011no—
280C2×C42—1011no—
281C2×C42—1011no—
282C2×C42—2024no—
283#95—1011no—
284#95—1011no—
285#95—1011no—
286#95—1011no—
287#95C2×D10111yescharacteristic
288#95C2×D10111yescharacteristic
289C24—1011no—
290C24C22×D10111yescharacteristic
291C24C22×D10111yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—1014no—
295C2×D4—1014no—
296C2×D4—2022no—
297C2×D4—2022no—
298C2×D4—2022no—
299C2×D4—2022no—
300C2×D4C4×D10114yescomplemented
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—1011no—
305C22×C4—1011no—
306C22×C4—1014no—
307C22×C4—1014no—
308C22×C4—1011no—
309C22×C4—1011no—
310C22×C4—1011no—
311C22×C4—1011no—
312C22×C4—1011no—
313C22×C4—2022no—
314C22×C4—2022no—
315C22×C4—2022no—
316C22×C4—2022no—
317C22×C4—2022no—
318C22×C4—2022no—
319C22×C4—2022no—
320C22×C4—2022no—
321C22×C4—2022no—
322C22×C4—2022no—
323C22×C4C22×D10111yescharacteristic
324C22×C4C22×D10111yescharacteristic
325C22×C4C22×D10111yescharacteristic
326C22×C4C22×D10111yescharacteristic
327C22×C4C22×D10111yescharacteristic
328C4⋊C4—1014no—
329C4⋊C4—2022no—
330C4⋊C4—2022no—
331C4⋊C4—2022no—
332C4⋊C4—2022no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—1014no—
337C22⋊C4—1014no—
338C22⋊C4—2022no—
339C22⋊C4—2022no—
340C22⋊C4—2022no—
341C22⋊C4—2022no—
342C22⋊C4—2022no—
343C42—1014no—
344C42—2022no—
345C42—2022no—
346C42—2022no—
347C42—2022no—
348C42—2022no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—1012no—
360C23—2021no—
361C23—2021no—
362C23—2021no—
363C23—2021no—
364C23—2021no—
365C23—2021no—
366C23C2×C4×D10112yes—
367C23C2×C4×D10112yes—
368C23D4×D10112yes—
369C23#1310807312yes—
370C23C23×D10111yescharacteristic
371D4—28no—
372D4—2028no—
373D4—2028no—
374C2×C4—28no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—1012no—
388C2×C4—1012no—
389C2×C4—1012no—
390C2×C4—1012no—
391C2×C4—1012no—
392C2×C4—1012no—
393C2×C4—1012no—
394C2×C4—1012no—
395C2×C4—1012no—
396C2×C4—4044no—
397C2×C4—4044no—
398C2×C4—2028no—
399C2×C4—2028no—
400C2×C4—2021no—
401C2×C4—2021no—
402C2×C4—2021no—
403C2×C4—2021no—
404C2×C4—2021no—
405C2×C4—2021no—
406C2×C4—2021no—
407C2×C4—2021no—
408C2×C4—2021no—
409C2×C4—2021no—
410C2×C4—2021no—
411C2×C4—2021no—
412C2×C4C2×C4×D10112yes—
413C2×C4D4×D10112yes—
414C2×C4#1310807312yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—4041no—
429C22—4041no—
430C22—4041no—
431C22—4041no—
432C22—2022no—
433C22—2022no—
434C22—2022no—
435C22#1569221514yescomplemented
436C22C22×C4×D10111yescharacteristic
437C22C2×D4×D10111yescharacteristic
438C22C2×D4×D10111yescharacteristic
439C22#41×D10111yescharacteristic
440C22#41×D10111yescharacteristic
441C22C2×#1310807311yescharacteristic
442C22C2×#1310807311yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—1014no—
449C4—1014no—
450C4—4041no—
451C4—4041no—
452C4—2022no—
453C4—2022no—
454C4—2022no—
455C2—24no—
456C2—42no—
457C2—4042no—
458C2C4×D4×D10111yescharacteristic
459C2C2×#1569221511yescharacteristic
460C2C4×#1310807311yescharacteristic
461C2#121×D10111yescharacteristic
462C2#2026568311yescharacteristic
463C2#2026570411yescharacteristic
464C2#2026572211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#386×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692227C2
hypercenter#386D101
hypocenterC101C4×#121
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic101C2semidirect
3. C2×#15692215C2semidirect
4. C2×#15692215C2semidirect
5. C22×C4×Dic101C2semidirect
6. #20265077C2semidirect
7. #20265074C2semidirect
8. #20265069C2semidirect
9. #20265065C2semidirect
10. #20265056C2semidirect
11. #20265035C2semidirect
12. #20265028C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. #20265024C2semidirect
15. #386×C101C2semidirect
22. #15692227C4semidirect
61. C2×C22⋊D101C2×C4semidirect
62. C2×C22⋊D101C2×C4semidirect
70. C22×Dic101C2×C4semidirect
71. C22×Dic101D4semidirect
86. #13108053C2×C4semidirect
89. C22⋊Dic101C2×C4semidirect
93. #13108051C2×C4semidirect
99. C4×Dic101D4semidirect
104. C2×D4×C101C2×C4semidirect
195. C22×C101C4×D4semidirect
220. Dic101C4×D4semidirect
221. Dic101#121semidirect
233. C101C4×#121semidirect
244. #386D101semidirect
300. C2×D4C4×D101semidirect
435. C22#15692215semidirect

Maximal subgroups

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

subgroupindex
C2×#156922272
C2×D4×Dic1012
C2×#156922152
C22×C4×Dic1012
#202650772
#202650742
#202650692
#202650652
#202650562
#202650352
#202650282
C22⋊C4×Dic1012
#202650242
#386×C1012
C4×#121101