Finite Group Databasebeta

← #29736717

Subgroups — #29736717

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840392C211yesmaximal, characteristic
2C2×#16840132C211yesmaximal, characteristic, complemented
3C2×#16840113C211yesmaximal, characteristic, complemented
4#21451588C211yesmaximal, characteristic
5#21446969C212yesmaximal, complemented
6#21446959C212yesmaximal, complemented
7#21446791C211yesmaximal, characteristic, complemented
8#21446130C212yesmaximal
9#21446125C212yesmaximal
10#21445586C211yesmaximal, characteristic
11C61×#1991C211yesmaximal, characteristic, complemented
12#16840392—24no—
13#16840392—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519776C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519767—21no—
20C2×#14519757—21no—
21C2×#14519757C2212yes—
22C2×#14519755—21no—
23C2×#14519735C2211yescharacteristic
24C2×#14519734C2211yescharacteristic
25C2×C4⋊C4×D61C2211yescharacteristic
26#16840132—24no—
27#16840113—24no—
28C2×#14519702C2212yes—
29C2×#14519702C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519689C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839694—21no—
35#16839694—21no—
36#16839689—21no—
37#16839689—21no—
38#16839667C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×#125×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122C2312yes—
50C22×C4×D61—21no—
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519777—22no—
56#14519777D414yes—
57#14519776—28no—
58#14519776—22no—
59#14519776—22no—
60#14519768—22no—
61#14519768—22no—
62#14519767—28no—
63#14519757—28no—
64#14519757—28no—
65#14519755—28no—
66#14519735—22no—
67#14519735—22no—
68#14519734—22no—
69#14519734—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C4⋊C4×D61—22no—
83C4⋊C4×D61—22no—
84C2×#11758198—21no—
85C2×#11758198—22no—
86C2×#11758198D411yescharacteristic
87C2×#11758198D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519702—24no—
91#14519702—24no—
92#14519700—24no—
93#14519700—24no—
94#14519689—24no—
95#14519688—22no—
96#14519684—22no—
97#14519684—22no—
98C22×C8×C61C2312yes—
99#125×C61—22no—
100#125×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C4×C8×C61—22no—
108#95×C61—21no—
109#95×C61—21no—
110C23×D61C2×D411yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×C4×D61—24no—
127C2×C4×D61—82no—
128C2×C4×D61#4114yes—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—24no—
132C2×Dic122—24no—
133C2×Dic122—24no—
134C2×Dic122—24no—
135C2×Dic122—82no—
136C2×Dic122—82no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61C2×D412yes—
147C42×C61C2×D412yes—
148C42×C61C2×D412yes—
149#11758198—24no—
150#11758198—22no—
151#11758198—22no—
152#11758198—44no—
153C4⋊Dic61—24no—
154C4⋊Dic61—24no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—42no—
158C4⋊Dic61—42no—
159C4⋊Dic61—44no—
160C4⋊Dic61C2×D412yes—
161C4⋊Dic61C2×D412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—22no—
166#11758196—22no—
167C4×Dic61—24no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61C22×D411yescharacteristic
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61#12412yes—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61#12012yes—
189C2×C4×C61#12012yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4112yes—
193C2×C4×C612−1+412yes—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61#12412yes—
208C4×D61—24no—
209C4×D61—24no—
210C4×D61—84no—
211Dic122—44no—
212Dic122—44no—
213Dic122—84no—
214Dic122—84no—
215C8×C61—42no—
216#92418—42no—
217#25914—611nomaximal, Sylow
218C22×C61C2×#12011yescharacteristic
219C22×C61C2×#12411yescharacteristic
220C22×C61#53611yescharacteristic
221C22×C61#57712yes—
222C22×C61#57812yes—
223C2×D61—24no—
224C2×D61—22no—
225C2×D61—42no—
226C2×D61—41no—
227C2×D61—41no—
228C4×C61—22no—
229C4×C61—22no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—82no—
233C4×C61#54812yes—
234C4×C61#54812yes—
235Dic61—22no—
236Dic61—41no—
237Dic61—42no—
238Dic61—42no—
239Dic61—41no—
240Dic61—82no—
241C2×#544—611no—
242C2×#491—611no—
243C2×#487—611no—
244#2621—611no—
245#2026—612no—
246#2024—612no—
247#1993—611no—
248#1991D6111yescharacteristic, complemented
249#1873—612no—
250#1872—612no—
251#1750—611no—
252C2×C61#243611yescharacteristic
253C2×C61#300911yescharacteristic
254C2×C61#301111yescharacteristic
255C2×C61#313712yes—
256C2×C61#324212yes—
257D61—42no—
258#544—1224no—
259#544—1224no—
260C2×C4⋊Q8—611no—
261C2×C4⋊Q8—611no—
262C2×C4⋊Q8—611no—
263C2×C4⋊Q8—1221no—
264C2×#125—611no—
265C2×#125—1221no—
266C2×#125C2×D6111yescharacteristic
267C2×C22⋊Q8—612no—
268C2×C22⋊Q8—1221no—
269C2×C22⋊Q8—1221no—
270C2×#117—611no—
271C2×#117—611no—
272C22×C4⋊C4—611no—
273#491—1224no—
274#487—1224no—
275C2×#103—612no—
276C2×#103—612no—
277C2×#103—612no—
278C2×#103—612no—
279C2×C22⋊C8—612no—
280C2×#97—611no—
281C2×C4×C8C2×D6111yescharacteristic
282#398—1221no—
283#398—1221no—
284#397—1221no—
285#397—1221no—
286#391—612no—
287#382C2×D6111yescharacteristic
288#340—612no—
289#340—612no—
290#340C2×D6112yes—
291#340C2×D6112yes—
292#336—612no—
293C61#2591411yescharacteristic, complemented, Sylow
294C22×Q8—612no—
295C22×Q8—1221no—
296C22×Q8—1221no—
297C23×C4—611no—
298C23×C4—1221no—
299C22×C8—612no—
300C22×C8C22×D6112yes—
301C4⋊Q8—614no—
302C4⋊Q8—1222no—
303C4⋊Q8—1228no—
304C4⋊Q8—1222no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307C4⋊Q8—1222no—
308#125—22no—
309#125—1222no—
310#125—1228no—
311#125—1222no—
312#125D24414yes—
313C22⋊Q8—1228no—
314C22⋊Q8—1228no—
315C22⋊Q8—1228no—
316#117—1222no—
317#117—1222no—
318#117—1222no—
319#117—1222no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4C22×D6112yes—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—611no—
340C2×C22⋊C4—611no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1222no—
343C2×C42—611no—
344C2×C42—1221no—
345C2×C42C22×D6111yescharacteristic
346#103—1224no—
347#103—1224no—
348#103—1224no—
349#103—1224no—
350C22⋊C8—1224no—
351#97—1222no—
352C4×C8—22no—
353#95—21no—
354#95—21no—
355#95—1222no—
356#95—1222no—
357C24—611no—
358C2×Q8—1224no—
359C2×Q8—1224no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8—1224no—
363C2×Q8—1224no—
364C2×Q8—4882no—
365C2×Q8—4882no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—611no—
370C22×C4—614no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—1224no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1222no—
380C22×C4—4882no—
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383C22×C4C23×D6111yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—612no—
396C4⋊C4—612no—
397C4⋊C4—1224no—
398C4⋊C4—1224no—
399C4⋊C4—1224no—
400C4⋊C4—1224no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—1224no—
406C4⋊C4—2444no—
407C4⋊C4—2442no—
408C4⋊C4—2442no—
409C4⋊C4C2×D24412yes—
410C4⋊C4C2×D24412yes—
411C22⋊C4—1224no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C22⋊C4—2444no—
415C42—1224no—
416C42—1221no—
417C42—1221no—
418C42C2×D24412yes—
419C42D4×D6112yes—
420C23—612no—
421C23—1221no—
422C23—1222no—
423C23—1222no—
424C23—1221no—
425C23C2×D4×D6111yescharacteristic
426Q8—4884no—
427Q8—4884no—
428Q8—2444no—
429Q8—2444no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—612no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1224no—
443C2×C4—1224no—
444C2×C4—1222no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—4884no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4#1451974712yes—
457C2×C4#1451974712yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×D24411yescharacteristic
460C2×C4#41×D6112yes—
461C2×C4#1451982512yes—
462C8—42no—
463C8—2442no—
464C22—1222no—
465C22—1224no—
466C22—2441no—
467C22—2442no—
468C22—2441no—
469C22C2×#1451974711yescharacteristic
470C22#124×D6111yescharacteristic
471C22#1684033711yescharacteristic
472C22#1684056912yes—
473C22#1684057612yes—
474C4—22no—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—1222no—
480C4—4882no—
481C4—2441no—
482C4—2441no—
483C4—2442no—
484C4—2442no—
485C4#1684042212yes—
486C4#1684042412yes—
487C2—2442no—
488C2#2144947611yescharacteristic
489C2#2145509911yescharacteristic
490C2#2145511711yescharacteristic
491C2#2145618312yes—
492C2#2145709012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1991C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1991D61
hypocenterC61#25914
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#16840132C2semidirect
3. C2×#16840113C2semidirect
5. #21446969C2semidirect
6. #21446959C2semidirect
7. #21446791C2semidirect
11. C61×#1991C2semidirect
248. #1991D61semidirect
293. C61#25914semidirect

Maximal subgroups

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

subgroupindex
C2×#168403922
C2×#168401322
C2×#168401132
#214515882
#214469692
#214469592
#214467912
#214461302
#214461252
#214455862
C61×#19912
#2591461