Finite Group Databasebeta

← #29736727

Subgroups — #29736727

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic, complemented
3C2×#16840108C211yesmaximal, characteristic, complemented
4#21446965C212yesmaximal, complemented
5#21446955C212yesmaximal, complemented
6#21446791C211yesmaximal, characteristic, complemented
7#21446783C211yesmaximal, characteristic, complemented
8#21446142C212yesmaximal, complemented
9#21446137C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C61×#2619C211yesmaximal, characteristic
12#16840390—24no—
13#16840390—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×#14519768—21no—
20C2×#14519754—21no—
21C2×#14519754—21no—
22C2×#14519736C2211yescharacteristic
23C22×C4⋊Dic61C2211yescharacteristic
24#16840126—24no—
25#16840108—24no—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×C22⋊#92418C2212yes—
31C2×#14519688C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839524C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×#127×C61C2211yescharacteristic
39C2×#121×C61C2212yes—
40C2×#117×C61C2211yescharacteristic
41C61×#394—21no—
42C61×#394—21no—
43C61×#392—21no—
44C61×#392—21no—
45C61×#390C2212yes—
46C61×#382C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122—21no—
49C23×Dic61—21no—
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777—22no—
54#14519777—22no—
55#14519777D414yescomplemented
56#14519776—28no—
57#14519769—22no—
58#14519769—22no—
59#14519769—22no—
60#14519769D414yescomplemented
61#14519768—28no—
62#14519754—28no—
63#14519754—28no—
64#14519736—22no—
65#14519736—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81#14519698—24no—
82#14519698—24no—
83#14519698—24no—
84#14519698—24no—
85C22⋊#92418—24no—
86#14519688—22no—
87C4×#92418—22no—
88C22×D4×C61—22no—
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C22×D4×C61C2312yes—
92C23×C4×C61C2311yescharacteristic
93#127×C61—22no—
94#127×C61—22no—
95#121×C61—28no—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#95×C61—21no—
109#95×C61—21no—
110C24×C61—22no—
111C24×C61C2×D411yescharacteristic
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—24no—
115C2×D4×C61—24no—
116C2×D4×C61—44no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—82no—
120C2×D4×C61—82no—
121C2×D4×C61—82no—
122C2×D4×C61—82no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61#4114yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—24no—
139C22×Dic61—82no—
140C2×Dic122—24no—
141C2×Dic122—24no—
142C2×Dic122—82no—
143C2×Dic122—82no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—44no—
150C22⋊C4×C61—82no—
151C22⋊C4×C61—82no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—24no—
157C4⋊Dic61—24no—
158C4⋊Dic61—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61—24no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61#12412yes—
191C23×C61C22×D411yescharacteristic
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—84no—
195D4×C61—84no—
196D4×C61—88no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61#12412yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4112yes—
216C2×C4×C612−1+412yes—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—84no—
226Dic122—84no—
227Dic122—84no—
228#92418—42no—
229#92418—42no—
230#25915—611nomaximal, Sylow
231C22×C61—24no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—82no—
237C22×C61—82no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61C2×#12011yescharacteristic
244C22×C61C2×#12411yescharacteristic
245C22×C61#53611yescharacteristic
246C22×C61#57612yes—
247C22×C61#57712yes—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—82no—
254C4×C61#54812yes—
255C4×C61#54812yes—
256Dic61—42no—
257Dic61—42no—
258Dic61—42no—
259Dic61—42no—
260C2×#544—611no—
261C2×#490—611no—
262C2×#486—611no—
263#2619D6111yescharacteristic
264#2025—612no—
265#2023—612no—
266#1993—611no—
267#1991—611no—
268#1875—612no—
269#1874—612no—
270#1750—611no—
271C2×C61—42no—
272C2×C61—162no—
273C2×C61#243611yescharacteristic
274C2×C61#300811yescharacteristic
275C2×C61#301011yescharacteristic
276C2×C61#313612yes—
277C2×C61#324112yes—
278#544—1224no—
279#544—1224no—
280C2×C4⋊Q8—611no—
281C2×C4⋊Q8—611no—
282C2×C4⋊Q8—1221no—
283C2×#127C2×D6111yescharacteristic
284C2×#125—611no—
285C2×#125—611no—
286C2×#125—1221no—
287C2×C22⋊Q8—1221no—
288C2×C22⋊Q8—1221no—
289C2×#121C2×D6112yes—
290C2×#117—611no—
291C2×#117C2×D6111yescharacteristic
292C22×C4⋊C4—611no—
293#490—1224no—
294#486—1224no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102—612no—
298C2×#102—612no—
299C2×C22⋊C8—612no—
300C2×#97—611no—
301C2×C4×C8—611no—
302#394—21no—
303#394—21no—
304#392—21no—
305#392—21no—
306#390C2×D6112yes—
307#382C2×D6111yescharacteristic
308#340—612no—
309#340—612no—
310#340—612no—
311#340—612no—
312#336—612no—
313C61#2591511yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C22×Q8—1221no—
316C22×D4—22no—
317C22×D4—21no—
318C22×D4—21no—
319C22×D4C22×D6112yes—
320C23×C4—1221no—
321C23×C4C22×D6111yescharacteristic
322C22×C8—612no—
323C22×C8—612no—
324C4⋊Q8—614no—
325C4⋊Q8—1222no—
326C4⋊Q8—1222no—
327C4⋊Q8—1228no—
328C4⋊Q8—1222no—
329#127—22no—
330#127—22no—
331#125—614no—
332#125—1222no—
333#125—1222no—
334#125—1228no—
335#125—1222no—
336C22⋊Q8—1228no—
337C22⋊Q8—1228no—
338#121—28no—
339#117—22no—
340#117—22no—
341#117—1222no—
342#117—1222no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C22⋊D6111yescharacteristic
357C2×C4⋊C4C22⋊D6111yescharacteristic
358C2×C4⋊C4C22×D6112yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4C22⋊D6111yescharacteristic
365C2×C42—1221no—
366C2×C42C22×D6111yescharacteristic
367C2×C42C22×D6111yescharacteristic
368#102—1224no—
369#102—1224no—
370#102—1224no—
371#102—1224no—
372C22⋊C8—1224no—
373#97—1222no—
374C4×C8—1222no—
375#95—21no—
376#95—21no—
377C24—22no—
378C24C2×C22⋊D6111yescharacteristic
379C2×Q8—1224no—
380C2×Q8—1224no—
381C2×Q8—4882no—
382C2×Q8—4882no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—1224no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—4882no—
404C22×C4#1175822414yes—
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—612no—
421C4⋊C4—612no—
422C4⋊C4—612no—
423C4⋊C4—612no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—1224no—
441C42—21no—
442C42—21no—
443C42—1224no—
444C42D4×D6112yes—
445C42D4×D6112yes—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23#1451976412yes—
458C23C22×C22⋊D6111yescharacteristic
459Q8—4884no—
460Q8—4884no—
461D4—44no—
462D4—44no—
463D4—84no—
464D4—84no—
465D4—88no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—4884no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4#1451974512yes—
489C2×C4#1451974512yes—
490C2×C4#1451976412yes—
491C2×C4C2×D4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×#1175822412yes—
494C2×C4#1451982412yes—
495C8—2442no—
496C8—2442no—
497C22—22no—
498C22—24no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22C2×#1451974511yescharacteristic
510C22C2×#1451976411yescharacteristic
511C22#1684033311yescharacteristic
512C22#1684056512yes—
513C22#1684057312yes—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—41no—
518C4—41no—
519C4—82no—
520C4—2442no—
521C4—2442no—
522C4—2442no—
523C4—2442no—
524C4#1684041812yes—
525C4#1684041812yes—
526C2—42no—
527C2—162no—
528C2#2144947811yescharacteristic
529C2#2145508511yescharacteristic
530C2#2145510311yescharacteristic
531C2#2145617312yes—
532C2#2145706612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2619C2
socleC23×C61C22×D4
hypercenter#2619D61
hypocenterC61#25915
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
2. C2×#16840126C2semidirect
3. C2×#16840108C2semidirect
4. #21446965C2semidirect
5. #21446955C2semidirect
6. #21446791C2semidirect
7. #21446783C2semidirect
8. #21446142C2semidirect
9. #21446137C2semidirect
10. #21445585C2semidirect
55. #14519777D4semidirect
60. #14519769D4semidirect
313. C61#25915semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
C2×#168401262
C2×#168401082
#214469652
#214469552
#214467912
#214467832
#214461422
#214461372
#214455852
C61×#26192
#2591561