Finite Group Databasebeta

← #27180346

Subgroups — #27180346

Order 12928.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D101C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265684C211yesmaximal, characteristic, complemented
4#20265338C211yesmaximal, characteristic, complemented
5#20265336C211yesmaximal, characteristic, complemented
6#20265331C211yesmaximal, characteristic, complemented
7#20265326C211yesmaximal, characteristic, complemented
8#20265321C211yesmaximal, characteristic, complemented
9#20265319C211yesmaximal, characteristic, complemented
10#20265314C211yesmaximal, characteristic, complemented
11#20265309C211yesmaximal, characteristic, complemented
12#20265174C211yesmaximal, characteristic, complemented
13#20265173C211yesmaximal, characteristic, complemented
14C2×#15692167C211yesmaximal, characteristic, complemented
15C2×#135×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074—21no—
18C2×#13108074C2211yescharacteristic
19C2×#13108073—21no—
20C2×#13108073C2211yescharacteristic
21#41×D101—24no—
22#41×D101—24no—
23C2×Q8×D101C2211yescharacteristic
24C2×D4×D101—21no—
25C2×D4×D101C2211yescharacteristic
26C22×C4×D101—21no—
27C22×C4×D101C2211yescharacteristic
28#15692275—24no—
29C2×#13108064C2211yescharacteristic, complemented
30C2×#13108062C2211yescharacteristic, complemented
31C2×#13108061C2211yescharacteristic, complemented
32C2×#13108059C2211yescharacteristic, complemented
33C22×#11717579C2211yescharacteristic, complemented
34#15692236—21no—
35#15692231C2211yescharacteristic
36#15692231C2211yescharacteristic, complemented
37#15692226C2211yescharacteristic
38#15692226C2211yescharacteristic, complemented
39#15692223—21no—
40C4×C22⋊D101—21no—
41C4×C22⋊D101—21no—
42#15692214C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50#15692176C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic
52#15692167C2211yescharacteristic
53#15692167C2211yescharacteristic, complemented
54#15692167C2211yescharacteristic, complemented
55C2×#41×C101C2211yescharacteristic
56C2×#41×C101C2211yescharacteristic, complemented
57#135×C101—24no—
58C2×Q16×C101C2211yescharacteristic, complemented
59C2×SD16×C101C2211yescharacteristic, complemented
60C2×SD16×C101C2211yescharacteristic, complemented
61C2×D8×C101C2211yescharacteristic, complemented
62C22×C8×C101C2211yescharacteristic, complemented
63C23×D101—21no—
64C23×D101D411yescharacteristic
65#13108075—22no—
66#13108075—22no—
67#13108074—28no—
68#13108074—22no—
69#13108074D414yescomplemented
70#13108073—28no—
71#13108073—22no—
72#13108073—22no—
73C2×Dic202—21no—
74C2×Dic202C2311yescharacteristic
75Q8×D101—22no—
76Q8×D101—22no—
77C2×D404—21no—
78C2×D404C2311yescharacteristic
79C2×C22⋊D101—21no—
80C2×C22⋊D101—21no—
81C2×C22⋊D101—21no—
82C2×C22⋊D101—21no—
83C2×C22⋊D101—21no—
84D4×D101—28no—
85D4×D101—22no—
86D4×D101—22no—
87C22×Dic101—21no—
88C22×Dic101D411yescharacteristic
89C2×C4×D101—28no—
90C2×C4×D101—21no—
91C2×C4×D101—21no—
92C2×C4×D101—22no—
93C2×C4×D101—22no—
94C2×C4×D101—21no—
95C2×C4×D101D411yescharacteristic
96C2×C4×D101D411yescharacteristic
97C2×C4×D101C2311yescharacteristic
98C2×C4×D101C2311yescharacteristic
99#13108064—22no—
100#13108062—22no—
101#13108061—22no—
102#13108059—22no—
103C2×#11717579—22no—
104C2×#11717579C2311yescharacteristic
105C2×#11717579C2311yescharacteristic
106C4⋊Dic101C2311yescharacteristic
107C4⋊Dic101C2311yescharacteristic
108#13108053—21no—
109#13108053—21no—
110C22⋊Dic101—21no—
111C22⋊Dic101—21no—
112C22⋊Dic101—21no—
113#13108051—21no—
114#13108051—21no—
115C4×Dic101—21no—
116#41×C101—22no—
117#41×C101—22no—
118#41×C101—22no—
119#41×C101D414yescomplemented
120C2×Q8×C101C2311yescharacteristic
121C2×Q8×C101C2311yescharacteristic
122C2×D4×C101—21no—
123C2×D4×C101—21no—
124C2×D4×C101C2311yescharacteristic
125C2×D4×C101C2311yescharacteristic
126C22×C4×C101—21no—
127C22×C4×C101—21no—
128C22×C4×C101C2311yescharacteristic
129Q16×C101—22no—
130SD16×C101—22no—
131SD16×C101—22no—
132D8×C101—22no—
133C2×C8×C101—22no—
134C2×C8×C101C2311yescharacteristic
135C2×C8×C101C2311yescharacteristic
136C23×C101—21no—
137C23×C101—21no—
138C23×C101C2×D411yescharacteristic
139C22×D101—24no—
140C22×D101—22no—
141C22×D101—21no—
142C22×D101—22no—
143C22×D101—41no—
144C22×D101—42no—
145C22×D101—42no—
146C22×D101C2×D411yescharacteristic
147C22×D101C2×D411yescharacteristic
148Q8×C101—21no—
149Q8×C101—21no—
150Q8×C101—21no—
151Q8×C101C2×D412yes—
152D4×C101—24no—
153D4×C101—21no—
154D4×C101—21no—
155D4×C101—21no—
156D4×C101—42no—
157D4×C101C2×D412yes—
158C2×C4×C101—21no—
159C2×C4×C101—24no—
160C2×C4×C101—21no—
161C2×C4×C101—42no—
162C2×C4×C101C2×D412yes—
163C2×C4×C101C2×D411yescharacteristic
164C2×C4×C101C2×D412yes—
165C2×C4×C101C2411yescharacteristic
166C22⋊D101—24no—
167C22⋊D101—24no—
168C22⋊D101—24no—
169C22⋊D101—42no—
170C22⋊D101—42no—
171C2×Dic101—24no—
172C2×Dic101—22no—
173C2×Dic101—21no—
174C2×Dic101—21no—
175C2×Dic101—21no—
176C2×Dic101C2×D411yescharacteristic
177C2×Dic101C2×D411yescharacteristic
178D404—24no—
179D404—21no—
180D404C2×D412yes—
181C4×D101—24no—
182C4×D101—24no—
183C4×D101—22no—
184C4×D101—24no—
185C4×D101—22no—
186C4×D101—21no—
187C4×D101—21no—
188C4×D101—21no—
189C4×D101—21no—
190Dic202—24no—
191Dic202—21no—
192Dic202C2×D412yes—
193C8×C101—21no—
194C8×C101—21no—
195#11717579—21no—
196#11717579—21no—
197C22×C101—21no—
198C22×C101—21no—
199C22×C101—22no—
200C22×C101—41no—
201C22×C101—41no—
202C22×C101—41no—
203C22×C101—41no—
204C22×C101—41no—
205C22×C101#12012yes—
206C22×C101C22×D411yescharacteristic
207C2×D101—21no—
208C2×D101—21no—
209C2×D101—21no—
210C2×D101—22no—
211C2×D101—21no—
212C2×D101—21no—
213C2×D101—44no—
214C2×D101—48no—
215C2×D101—41no—
216C2×D101—41no—
217C2×D101#13512yescomplemented
218C4×C101—22no—
219C4×C101—41no—
220C4×C101#12012yes—
221C4×C101C22×D411yescharacteristic
222C4×C101C22×D411yescharacteristic
223Dic101—21no—
224Dic101—22no—
225Dic101—21no—
226Dic101—41no—
227Dic101—41no—
228C2×C101—22no—
229C2×C101—42no—
230C2×C101—81no—
231C2×C101C2×#12011yescharacteristic
232C2×C101C2×#13511yescharacteristic
233C2×C101#57511yescharacteristic
234D101—22no—
235D101—41no—
236D101—42no—
237#3002—1011nomaximal, Sylow
238C101#300211yescharacteristic, complemented, Sylow
239C22×#41—1011no—
240C2×#135—1011no—
241C2×#135D10111yescharacteristic, complemented
242#525—1011no—
243#452—1011no—
244#452—1011no—
245#451—1011no—
246#450—1011no—
247#449—1011no—
248#449—1011no—
249#448—1011no—
250#447—1011no—
251#416—1011no—
252#416—1011no—
253C2×C22⋊C8—1011no—
254C2×#41—1011no—
255C2×#41—1011no—
256C2×#41—1011no—
257C2×#41—2021no—
258C2×#41—2021no—
259C2×#41—2024no—
260C2×#41—2024no—
261C2×#41C2×D10111yescharacteristic
262C2×#41C2×D10111yescharacteristic, complemented
263C22×Q8—1011no—
264C22×D4—1011no—
265C22×D4—2021no—
266C23×C4—1011no—
267C23×C4—2021no—
268#135—24no—
269#135—2024no—
270C2×Q16—1011no—
271C2×Q16C2×D10111yescharacteristic, complemented
272C2×SD16—1011no—
273C2×SD16—1011no—
274C2×SD16C2×D10111yescharacteristic, complemented
275C2×SD16C2×D10111yescharacteristic, complemented
276C2×D8—1011no—
277C2×D8C2×D10111yescharacteristic, complemented
278C22×C8—1011no—
279C22×C8C2×D10111yescharacteristic, complemented
280#123—2021no—
281C22⋊Q8—1011no—
282C22⋊Q8—1011no—
283#121—1011no—
284#121—1011no—
285#120—2021no—
286C4×D4—2021no—
287C4×D4—2021no—
288#117—1011no—
289#103—1011no—
290#103—1011no—
291#103—1011no—
292#103—1011no—
293#102—1011no—
294#102—1011no—
295#102—1011no—
296#102—1011no—
297C22⋊C8—1011no—
298C22⋊C8—1011no—
299C22⋊C8—1011no—
300C22⋊C8—1011no—
301C24—1011no—
302C24—2021no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—1014no—
307#41—2028no—
308#41—2022no—
309#41—2028no—
310#41—2022no—
311#41—2022no—
312#41—2022no—
313#41—2022no—
314#41C22⋊D10114yescomplemented
315C2×Q8—1011no—
316C2×Q8—2021no—
317C2×Q8—2022no—
318C2×Q8—2022no—
319C2×Q8C22×D10111yescharacteristic
320C2×Q8C22×D10111yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—1011no—
324C2×D4—2021no—
325C2×D4—2021no—
326C2×D4—2022no—
327C2×D4—2028no—
328C2×D4—2021no—
329C2×D4—2021no—
330C2×D4—2021no—
331C2×D4—2021no—
332C2×D4—2022no—
333C2×D4C22×D10111yescharacteristic
334C2×D4C22×D10111yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—1011no—
338C22×C4—1011no—
339C22×C4—1011no—
340C22×C4—1011no—
341C22×C4—1011no—
342C22×C4—2021no—
343C22×C4—2021no—
344C22×C4—2028no—
345C22×C4—2021no—
346C22×C4—2022no—
347C22×C4—2021no—
348C22×C4—2022no—
349C22×C4C22×D10111yescharacteristic
350Q16—22no—
351Q16—2022no—
352SD16—22no—
353SD16—22no—
354SD16—2022no—
355SD16—2022no—
356D8—22no—
357D8—2022no—
358C2×C8—22no—
359C2×C8—1011no—
360C2×C8—1011no—
361C2×C8—2022no—
362C2×C8C22×D10111yescharacteristic
363C2×C8C22×D10111yescharacteristic
364C4⋊C4—1011no—
365C4⋊C4—1011no—
366C4⋊C4—2021no—
367C4⋊C4—2021no—
368C22⋊C4—2021no—
369C22⋊C4—2021no—
370C22⋊C4—2021no—
371C22⋊C4—2021no—
372C22⋊C4—2021no—
373C42—2021no—
374C23—21no—
375C23—21no—
376C23—1011no—
377C23—1011no—
378C23—4041no—
379C23—4042no—
380C23—4042no—
381C23—2022no—
382C23—2024no—
383C23—2021no—
384C23—2022no—
385C23D4×D10111yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—1012no—
390Q8—2024no—
391Q8—2021no—
392Q8C2×C22⋊D10112yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—1012no—
399D4—4042no—
400D4—4042no—
401D4—2024no—
402D4—2024no—
403D4—2024no—
404D4—2024no—
405D4—2021no—
406D4C2×C22⋊D10112yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—1011no—
412C2×C4—1011no—
413C2×C4—2024no—
414C2×C4—2024no—
415C2×C4—2021no—
416C2×C4—2021no—
417C2×C4—2024no—
418C2×C4—2021no—
419C2×C4—2022no—
420C2×C4—2021no—
421C2×C4—2021no—
422C2×C4—2021no—
423C2×C4—2021no—
424C2×C4—2022no—
425C2×C4—2022no—
426C2×C4—2024no—
427C2×C4D4×D10112yes—
428C2×C4D4×D10111yescharacteristic
429C2×C4C2×C22⋊D10112yes—
430C2×C4C23×D10111yescharacteristic
431C8—21no—
432C8—21no—
433C8—2021no—
434C8—2021no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—1012no—
444C22—4044no—
445C22—4048no—
446C22—4041no—
447C22—4041no—
448C22—2022no—
449C22—2021no—
450C22—2021no—
451C22—2021no—
452C22—2021no—
453C22—2021no—
454C22#1569222312yes—
455C22C2×D4×D10111yescharacteristic
456C4—22no—
457C4—41no—
458C4—4041no—
459C4—4041no—
460C4—2021no—
461C4—2021no—
462C4—2022no—
463C4#1569222312yes—
464C4C2×D4×D10111yescharacteristic
465C4C22×C22⋊D10111yescharacteristic
466C2—22no—
467C2—42no—
468C2—81no—
469C2—4042no—
470C2—4041no—
471C2—2022no—
472C2C2×#1569222311yescharacteristic
473C2#135×D10111yescharacteristic
474C2#2026597411yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3002
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#41×D101C2semidirect
2. C2×#15692275C2semidirect
3. #20265684C2semidirect
4. #20265338C2semidirect
5. #20265336C2semidirect
6. #20265331C2semidirect
7. #20265326C2semidirect
8. #20265321C2semidirect
9. #20265319C2semidirect
10. #20265314C2semidirect
11. #20265309C2semidirect
12. #20265174C2semidirect
13. #20265173C2semidirect
14. C2×#15692167C2semidirect
15. C2×#135×C101C2semidirect
29. C2×#13108064C22semidirect
30. C2×#13108062C22semidirect
31. C2×#13108061C22semidirect
32. C2×#13108059C22semidirect
33. C22×#11717579C22semidirect
36. #15692231C22semidirect
38. #15692226C22semidirect
42. #15692214C22semidirect
43. #15692180C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692179C22semidirect
47. #15692177C22semidirect
48. #15692177C22semidirect
49. #15692176C22semidirect
50. #15692176C22semidirect
53. #15692167C22semidirect
54. #15692167C22semidirect
56. C2×#41×C101C22semidirect
58. C2×Q16×C101C22semidirect
59. C2×SD16×C101C22semidirect
60. C2×SD16×C101C22semidirect
61. C2×D8×C101C22semidirect
62. C22×C8×C101C22semidirect
69. #13108074D4semidirect
119. #41×C101D4semidirect
217. C2×D101#135semidirect
238. C101#3002semidirect
241. C2×#135D101semidirect
262. C2×#41C2×D101semidirect
271. C2×Q16C2×D101semidirect
274. C2×SD16C2×D101semidirect
275. C2×SD16C2×D101semidirect
277. C2×D8C2×D101semidirect
279. C22×C8C2×D101semidirect
314. #41C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1012
C2×#156922752
#202656842
#202653382
#202653362
#202653312
#202653262
#202653212
#202653192
#202653142
#202653092
#202651742
#202651732
C2×#156921672
C2×#135×C1012
#3002101