Finite Group Databasebeta

← #29734439

Subgroups — #29734439

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840375C211yesmaximal, characteristic
2C2×#16840114C211yesmaximal, characteristic, complemented
3#21451641C211yesmaximal, characteristic
4#21447024C212yesmaximal
5#21447017C212yesmaximal
6#21446785C211yesmaximal, characteristic, complemented
7#21446639C211yesmaximal, characteristic, complemented
8#21446084C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#21445588C211yesmaximal, characteristic
11C2×C61×#501C211yesmaximal, characteristic, complemented
12#16840375—24no—
13#16840375—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—21no—
16C2×#14519772—21no—
17C2×#14519768—21no—
18C2×#14519768—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519757C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24#16840114—24no—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839700—21no—
32#16839697—21no—
33#16839681—21no—
34#16839673—21no—
35#16839658C2212yes—
36#16839628C2211yescharacteristic
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#125×C61C2211yescharacteristic
43C61×#501—24no—
44C2×#107×C61C2212yes—
45C2×#106×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×Dic122C2311yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yes—
53#14519772—28no—
54#14519772—28no—
55#14519768—28no—
56#14519768—28no—
57#14519757—22no—
58#14519757—22no—
59#14519757—22no—
60#14519757—22no—
61#14519742—22no—
62#14519742—22no—
63#14519742—22no—
64#14519742—22no—
65#14519734—22no—
66#14519734—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198D411yescharacteristic
81C2×#11758198D411yescharacteristic
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85#14519702—24no—
86#14519702—24no—
87#14519700—24no—
88#14519700—24no—
89#14519689—24no—
90C4×#92418—22no—
91#14519684—21no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61—21no—
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—22no—
98C4⋊Q8×C61—22no—
99#125×C61—22no—
100#125×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#107×C61—24no—
109#106×C61—24no—
110#97×C61—22no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—42no—
113C2×Q8×C61—82no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122C2×D412yes—
137C2×Dic122C2×D412yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61#4114yes—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D412yes—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152#11758198—24no—
153#11758198—24no—
154#11758198—22no—
155#11758198—22no—
156#11758198—82no—
157C4⋊Dic61—24no—
158C4⋊Dic61—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163#11758196—24no—
164#11758196—24no—
165#11758196—22no—
166#11758196—22no—
167#11758196—82no—
168C4×Dic61—24no—
169C4×Dic61—24no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61C22×D411yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—22no—
181Q8×C61—84no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61#12012yes—
192C2×C4×C61#12012yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C612−1+412yes—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C4×D61—42no—
214C4×D61—42no—
215Dic122—24no—
216Dic122—24no—
217Dic122—42no—
218Dic122—42no—
219C8×C61—24no—
220#92418—42no—
221#25686—611nomaximal, Sylow
222C22×C61C2×#12011yescharacteristic
223C22×C61#53211yescharacteristic
224C22×C61#53611yescharacteristic
225C22×C61#57712yes—
226C22×C61#57812yes—
227C2×D61—42no—
228C2×D61—41no—
229C2×D61—42no—
230C2×D61—41no—
231C2×D61—41no—
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—82no—
237C4×C61#54712yes—
238C4×C61#54812yes—
239Dic61—42no—
240Dic61—42no—
241Dic61—41no—
242Dic61—41no—
243Dic61—41no—
244Dic61—82no—
245C2×#541—611no—
246C2×#501D6111yescharacteristic, complemented
247C2×#487—611no—
248#2628—611no—
249#2036—612no—
250#2035—612no—
251#1991—611no—
252#1960—611no—
253#1865—612no—
254#1863—612no—
255#1750—611no—
256C2×C61#243211yescharacteristic
257C2×C61#301011yescharacteristic
258C2×C61#301111yescharacteristic
259C2×C61#317812yes—
260C2×C61#324212yes—
261D61—81no—
262#541—1224no—
263#541—1224no—
264C2×C4⋊Q8—611no—
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×#126—1221no—
267C2×#126—1221no—
268C2×#125—1221no—
269C2×#125—1221no—
270C2×#125C2×D6111yescharacteristic
271C2×C22⋊Q8—611no—
272C2×C22⋊Q8—611no—
273C2×C4×Q8—611no—
274C2×C4×Q8—611no—
275C2×#117—611no—
276#501—24no—
277#487—1224no—
278C2×#107C2×D6112yes—
279C2×#106C2×D6112yes—
280C2×#103—612no—
281C2×#103—612no—
282C2×#103—612no—
283C2×#103—612no—
284C2×C22⋊C8—612no—
285C2×#97C2×D6111yescharacteristic
286C2×C4×C8—611no—
287#400—1221no—
288#399—1221no—
289#395—1221no—
290#393—1221no—
291#389—612no—
292#382—611no—
293#380—612no—
294#340—612no—
295#340—612no—
296#336—612no—
297C61#2568611yescharacteristic, complemented, Sylow
298C22×Q8—21no—
299C22×Q8—611no—
300C22×Q8—611no—
301C23×C4—611no—
302C22×C8—612no—
303C22×C8C22×D6112yes—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—614no—
307C4⋊Q8—1222no—
308#126—1228no—
309#126—1228no—
310#125—22no—
311#125—1228no—
312#125—1228no—
313#125C22⋊D6114yes—
314C22⋊Q8—1222no—
315C22⋊Q8—1222no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C4×Q8—1222no—
319C4×Q8—1222no—
320C4×Q8—1222no—
321C4×Q8—1222no—
322#117—1222no—
323#117—1222no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4C22×D6112yes—
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—611no—
341C2×C22⋊C4—611no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1222no—
345C2×C42—611no—
346C2×C42—1221no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349#107—24no—
350#106—24no—
351#103—1224no—
352#103—1224no—
353#103—1224no—
354#103—1224no—
355C22⋊C8—1224no—
356#97—22no—
357C4×C8—1222no—
358#95—1222no—
359#95—1222no—
360#95—1221no—
361#95—1221no—
362C24—611no—
363C2×Q8—42no—
364C2×Q8—82no—
365C2×Q8—612no—
366C2×Q8—612no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8#1175822614yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—4882no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C22⋊C4—1222no—
418C22⋊C4—1222no—
419C22⋊C4—4882no—
420C42—1224no—
421C42—1224no—
422C42—1221no—
423C42—1221no—
424C42D4×D6112yes—
425C42C2×C22⋊D6112yes—
426C23—1221no—
427C23—1221no—
428C23—1222no—
429C23—1222no—
430C23—1221no—
431C23C2×D4×D6111yescharacteristic
432Q8—84no—
433Q8—1224no—
434Q8—1224no—
435Q8—2442no—
436Q8—2442no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1221no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4#1451974512yes—
465C2×C4#1451974512yes—
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C22×C22⋊D6111yescharacteristic
468C2×C4C2×#1175822612yes—
469C2×C4#1451982512yes—
470C8—24no—
471C8—2442no—
472C22—2441no—
473C22—2441no—
474C22—2442no—
475C22—2442no—
476C22—2441no—
477C22C2×#1451974511yescharacteristic
478C22#1684031311yescharacteristic
479C22#1684033711yescharacteristic
480C22#1684057212yes—
481C22#1684057812yes—
482C4—22no—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—4882no—
488C4—2441no—
489C4—2441no—
490C4—2441no—
491C4—2442no—
492C4—2442no—
493C4#1684041512yes—
494C4#1684042312yes—
495C2—4881no—
496C2#2144944411yescharacteristic
497C2#2145510411yescharacteristic
498C2#2145511311yescharacteristic
499C2#2145649212yes—
500C2#2145708812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#501D61
hypocenterC61#25686
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#16840114C2semidirect
6. #21446785C2semidirect
7. #21446639C2semidirect
8. #21446084C2semidirect
9. #21446070C2semidirect
11. C2×C61×#501C2semidirect
246. C2×#501D61semidirect
297. C61#25686semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168401142
#214516412
#214470242
#214470172
#214467852
#214466392
#214460842
#214460702
#214455882
C2×C61×#5012
#2568661