Finite Group Databasebeta

← #30616763

Subgroups — #30616763

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic
2C2×C4×#11758217C211yesmaximal, characteristic
3#21462012C212yesmaximal, complemented
4#21461962C212yesmaximal, complemented
5#21461846C214yesmaximal
6C4×#14519838C214yesmaximal, complemented
7C2×#16840315C211yesmaximal, characteristic
8C22×#11758217C2212yes—
9#16840783—21no—
10#16840782—21no—
11#16840779—21no—
12#16840778—21no—
13#16840765C2212yes—
14C2×#14519838C2218yes—
15#16840732C2212yes—
16C4×#11758217—21no—
17C4×#11758217—21no—
18C4×#11758217C2212yes—
19C4×#11758217C2212yes—
20C4×#11758217C2212yes—
21C2×#14519832—21no—
22C2×C4×#92420—21no—
23#16840642C2218yes—
24#16840315—21no—
25#16840315—21no—
26#16840315—21no—
27#16840315—21no—
28#16840315—21no—
29#16840315—21no—
30#16840315—21no—
31#16840315—21no—
32C2×#14519776C2211yescharacteristic
33C2×#14519776C2211yescharacteristic
34C2×#14519768C411yescharacteristic
35C2×#14519768C411yescharacteristic
36C2×#14519754C412yes—
37C2×#14519754C2212yes—
38C2×C4×Dic122C412yes—
39C2×C4×Dic122C2212yes—
40C2×#14519736C411yescharacteristic
41C22×C4×Dic61C2211yescharacteristic
42C2×#117×C61C411yescharacteristic
43C2×#11758217—22no—
44C2×#11758217—28no—
45C2×#11758217C2314yes—
46C2×#11758217C2314yes—
47C22×#92420—22no—
48#14519844—22no—
49C4⋊#92420—22no—
50#14519838—216no—
51C22⋊#92420—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54#14519832—22no—
55#14519832—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58C4×#92420—22no—
59C22×Dic122C2×C412yes—
60C22×Dic122C2312yes—
61C23×Dic61C2×C411yescharacteristic
62C23×Dic61C2311yescharacteristic
63#14519776—21no—
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519776—21no—
68#14519776—21no—
69#14519776—21no—
70#14519776—21no—
71#14519768—21no—
72#14519768—21no—
73#14519768—21no—
74#14519768—21no—
75#14519768—21no—
76#14519768—21no—
77#14519768—21no—
78#14519768—21no—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82#14519754—22no—
83#14519754—22no—
84#14519754—22no—
85#14519754—22no—
86#14519754—22no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122—22no—
94C4×Dic122—22no—
95#14519736—21no—
96#14519736—21no—
97#14519736—21no—
98#14519736—21no—
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×C4⋊Dic61C2×C412yes—
102C2×C4⋊Dic61C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758196C2×C412yes—
106C2×#11758196C2312yes—
107C2×C22⋊Dic61C2×C412yes—
108C2×C4×Dic61C2×C412yes—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D411yescharacteristic
117C2×C4×Dic61D411yescharacteristic
118C2×C4×Dic61D411yescharacteristic
119C2×C4×Dic61C2311yescharacteristic
120C2×C4×Dic61C2311yescharacteristic
121C23×C4×C61C2×C411yescharacteristic
122#117×C61—21no—
123#117×C61—21no—
124#117×C61—21no—
125#117×C61—21no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61C2×C412yes—
129C2×C42×C61C2×C412yes—
130C24×C61C22×C411yescharacteristic
131C22×C4×C61—24no—
132C22×C4×C61C22⋊C411yescharacteristic
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C412yes—
140C22×Dic61—24no—
141C22×Dic61—42no—
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22×C412yes—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2411yescharacteristic
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—28no—
161C2×Dic122—44no—
162#11758217—28no—
163#11758217—24no—
164#11758217—28no—
165#11758217#4118yes—
166C2×#92420—28no—
167C2×#92420—24no—
168C2×#92420—24no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61—22no—
176C42×C61—22no—
177C22⋊Dic61—22no—
178C22⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61C2×C22⋊C411yescharacteristic
217C23×C61C2×C22⋊C411yescharacteristic
218C23×C61C23×C411yescharacteristic
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—42no—
228C2×C4×C61#10114yescomplemented
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255C2×Dic61C22×D411yescharacteristic
256C2×Dic61C2×#4114yes—
257Dic122—44no—
258Dic122—44no—
259Dic122—44no—
260Dic122—44no—
261#92420—28no—
262#92420—44no—
263#21425—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—44no—
272C22×C61C2×#10112yes—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61C4×#4112yes—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61C2×#10112yes—
280C4×C61C2×#10112yes—
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—42no—
287Dic61—42no—
288Dic61#41312yes—
289Dic61#52512yes—
290Dic61#53212yes—
291C2×#533—611no—
292#2920—611no—
293C2×C4×M16—611no—
294#1997—612no—
295#1966—612no—
296#1901—614no—
297C4×#101—614no—
298C2×C61—22no—
299C2×C61—22no—
300C2×C61—41no—
301C2×C61#229711yescharacteristic
302C2×C61C22×#10111yescharacteristic
303C2×C61C2×#41311yescharacteristic
304C22×M16—612no—
305#533—1221no—
306#533—1221no—
307#533—1221no—
308#533—1221no—
309#533—1221no—
310#533—1221no—
311#533—1221no—
312#533—1221no—
313C2×C4⋊Q8—611no—
314C2×C4⋊Q8—611no—
315C2×#125—611no—
316C2×#125—611no—
317C2×C22⋊Q8—612no—
318C2×C22⋊Q8—612no—
319C2×C4×Q8—612no—
320C2×C4×Q8—612no—
321C2×#117—611no—
322C2×#117#692311yescharacteristic
323C22×C42—611no—
324#432—1221no—
325#432—1221no—
326#431—1221no—
327#431—1221no—
328#423—612no—
329C2×#101—618no—
330#407—612no—
331C4×M16—612no—
332C4×M16—612no—
333C4×M16—612no—
334C4×M16—1221no—
335C4×M16—1221no—
336C2×#97—1221no—
337C2×C4×C8—1221no—
338#343—618no—
339C61#2142511yescharacteristic, complemented, Sylow
340C22×Q8—612no—
341C22×Q8—612no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4C2×#692311yescharacteristic
345C2×M16—614no—
346C2×M16—614no—
347C2×M16—1228no—
348C2×M16—1222no—
349C22×C8—1222no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358#125—1221no—
359#125—1221no—
360#125—1221no—
361#125—1221no—
362#125—1221no—
363#125—1221no—
364#125—1221no—
365#125—1221no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C4×Q8—1222no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—1221no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4C2×#692311yescharacteristic
399C2×C4⋊C4C2×#692311yescharacteristic
400C2×C22⋊C4—612no—
401C2×C22⋊C4C2×#692312yes—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—612no—
415C2×C42C2×#692312yes—
416C4⋊C8—1222no—
417C4⋊C8—1222no—
418#101—12216no—
419C22⋊C8—1222no—
420C22⋊C8—1222no—
421#97—1222no—
422#97—1222no—
423#97—1222no—
424C4×C8—1222no—
425C4×C8—1222no—
426C4×C8—1222no—
427C24C22×#692311yescharacteristic
428C2×Q8—1228no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C22×C4—24no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—1224no—
450C22×C4—2442no—
451C22×C4C22⋊#692311yescharacteristic
452C22×C4C22⋊#692311yescharacteristic
453C22×C4C22⋊#692311yescharacteristic
454C22×C4C22⋊#692311yescharacteristic
455C22×C4C22×#692312yes—
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692312yes—
459M16—618no—
460M16—1224no—
461M16—1228no—
462M16—1228no—
463C2×C8—1224no—
464C2×C8—1228no—
465C2×C8—1224no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—22no—
491C42—22no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—1222no—
509C42—1222no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23C2×C22⋊#692311yescharacteristic
514C23C2×C22⋊#692311yescharacteristic
515C23C23×#692311yescharacteristic
516Q8—2444no—
517Q8—2444no—
518Q8—2444no—
519Q8—2444no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—614no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4#1451983814yescomplemented
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C2×C22⋊#692311yescharacteristic
558C8—1228no—
559C8—2444no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22C2×#1451983812yes—
569C22C22×C22⋊#692311yescharacteristic
570C22#41×#692312yes—
571C4—21no—
572C4—21no—
573C4—42no—
574C4—42no—
575C4—612no—
576C4—612no—
577C4—612no—
578C4—1221no—
579C4—1221no—
580C4—1221no—
581C4—1221no—
582C4—1221no—
583C4—2442no—
584C4—2442no—
585C4C2×#1451983812yes—
586C4C2×#1451983812yes—
587C2—22no—
588C2—22no—
589C2—41no—
590C2#2146228011yescharacteristic
591C2C22×#1451983811yescharacteristic
592C2D61⋊#41311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenterC2×#117#6923
hypocenterC61#21425
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21462012C2semidirect
4. #21461962C2semidirect
6. C4×#14519838C2semidirect
228. C2×C4×C61#101semidirect
339. C61#21425semidirect
553. C2×C4#14519838semidirect

Maximal subgroups

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

subgroupindex
#214624952
C2×C4×#117582172
#214620122
#214619622
#214618462
C4×#145198382
C2×#168403152
#2142561