Finite Group Databasebeta

← #30603271

Subgroups — #30603271

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519841C212yesmaximal
2C2×C4×C4⋊#6923C211yesmaximal, characteristic, complemented
3C2×#16840632C212yesmaximal, complemented
4#21461483C211yesmaximal, characteristic, complemented
5#21448306C211yesmaximal, characteristic
6C22×#11758213C2212yes—
7C22×C4⋊#6923—21no—
8C22×C4×#6923C2212yes—
9C2×#14519841—24no—
10C2×#14519841—24no—
11C2×#14519841—24no—
12C4×C4⋊#6923—24no—
13C2×(C2×#6923)⋊C4—21no—
14#16840632—24no—
15#16840628—21no—
16#16840628—21no—
17C22×#11758224C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×C42×D61C2211yescharacteristic
20#16839668C411yescharacteristic, complemented
21#16839667C411yescharacteristic, complemented
22#16839665C411yescharacteristic, complemented
23#16839638—21no—
24#16839638—21no—
25C22⋊C4×Dic61—21no—
26C22⋊C4×Dic61—21no—
27C61×#390C411yescharacteristic, complemented
28C23×#6923—22no—
29C2×#11758213—22no—
30C2×#11758213—22no—
31C2×#11758213—22no—
32C2×#11758213—22no—
33C2×#11758213—22no—
34C2×#11758213D414yes—
35C22×#92420—22no—
36C2×C4⋊#6923—24no—
37C2×C4⋊#6923—24no—
38C2×C4⋊#6923—24no—
39C2×C4×#6923—22no—
40C2×C4×#6923—22no—
41C2×C4×#6923—22no—
42C2×C4×#6923—22no—
43C2×C4×#6923—22no—
44C2×C4×#6923D414yes—
45#14519841—216no—
46(C2×#6923)⋊C4—24no—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—21no—
52C2×#11758224—21no—
53C2×#11758224D412yes—
54C2×#11758224Q812yes—
55C22×Dic122C2×C411yescharacteristic
56C22×C22⋊D61—21no—
57C23×Dic61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61#14519735—22no—
62#14519735—22no—
63C2×C4⋊Dic61C2×C411yescharacteristic
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2×C411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C42×D61—22no—
72C42×D61—22no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77C22×D4×C61C2×C411yescharacteristic
78C2×C4⋊C4×C61C2×C411yescharacteristic
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C42×C61C2×C411yescharacteristic
82C24×C61—21no—
83C23×D61C2×D411yescharacteristic
84C22×#6923—24no—
85C22×#6923—24no—
86C22×#6923—24no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—42no—
90C22×#6923—42no—
91C2×D4×C61—21no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61C4⋊C412yescomplemented
95C22×C4×C61—21no—
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22×C411yescharacteristic
99C2×C22⋊D61—22no—
100C2×C22⋊D61—22no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C2×C22⋊D61—41no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—21no—
109C22×Dic61—42no—
110C22×Dic61—42no—
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114#11758224—24no—
115#11758224—24no—
116#11758224—24no—
117#11758224—24no—
118#11758224—42no—
119#11758224—42no—
120#11758224—42no—
121#11758224—42no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61C2×D412yes—
132C2×C4×D61C2×D412yes—
133C2×C4×D61C2×Q811yescharacteristic
134C2×C4×D61#4111yescharacteristic
135C2×C4×D61#4112yes—
136C2×C4×D61#4112yes—
137C2×C4×D61#4111yescharacteristic
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122C4⋊C412yescomplemented
142C2×#92420—24no—
143C2×#92420—42no—
144C2×#92420—42no—
145C4⋊#6923—216no—
146C4×#6923—24no—
147C4×#6923—24no—
148C4×#6923—42no—
149C4×#6923—42no—
150#11758213—24no—
151#11758213—24no—
152#11758213—42no—
153#11758213—42no—
154C4⋊C4×C61—22no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—22no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—42no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61C2×C22⋊C411yescharacteristic
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61#12011yescharacteristic
180C22×D61#12111yescharacteristic
181C22×D61C22⋊Q811yescharacteristic
182C22×D61#12311yescharacteristic
183C22×D61#12411yescharacteristic
184C22×D61#12711yescharacteristic
185C2×#6923—28no—
186C2×#6923—44no—
187C2×#6923—44no—
188C2×#6923—44no—
189C2×#6923—44no—
190C2×#6923—44no—
191C2×#6923—44no—
192D4×C61—24no—
193D4×C61—42no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#10412yescomplemented
203C2×C4×C61C2×C4⋊C411yescharacteristic
204C2×C4×C61#11711yescharacteristic
205C2×C4×C61C4×D412yes—
206C2×C4×C61C4×D412yes—
207C22⋊D61—42no—
208C22⋊D61—42no—
209C22⋊D61—42no—
210C22⋊D61—42no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—44no—
220C2×Dic61—44no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61#10412yescomplemented
225C2×Dic61#12111yescharacteristic
226C2×Dic61#12112yes—
227C2×Dic61#12112yes—
228C2×Dic61C22⋊Q811yescharacteristic
229C4×D61—24no—
230C4×D61—24no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61#12012yes—
234C4×D61#12112yes—
235C4×D61C22⋊Q812yes—
236C4×D61C22⋊Q812yes—
237C4×D61#12312yes—
238C4×D61#12312yes—
239C4×D61#12412yes—
240C4×D61#12612yes—
241Dic122—24no—
242Dic122—42no—
243#92420—44no—
244#11075—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61#38011yescharacteristic
253C22×C61#38611yescharacteristic
254C22×C61#39011yescharacteristic
255C22×C61C2×#10411yescharacteristic
256C22×C61C2×#10411yescharacteristic
257C22×C61#42111yescharacteristic
258C22×C61#42111yescharacteristic
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61#39211yescharacteristic
272C2×D61#39311yescharacteristic
273C2×D61#39711yescharacteristic
274C2×D61#39911yescharacteristic
275#6923—48no—
276C4×C61—22no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61#38512yes—
280C4×C61#38812yes—
281Dic61—22no—
282Dic61—41no—
283Dic61—42no—
284Dic61—81no—
285Dic61#39412yes—
286Dic61#39612yes—
287C22×#104—612no—
288#2297—611no—
289C2×C4×C4⋊C4—611no—
290C2×#339—612no—
291#1728—611no—
292C2×C61—82no—
293C2×C61#143911yescharacteristic
294C2×C61#180611yescharacteristic
295C2×C61#180611yescharacteristic
296C2×C61#189611yescharacteristic
297C2×C61#189711yescharacteristic
298C2×C61#198011yescharacteristic
299C2×C61#198011yescharacteristic
300D61—21no—
301D61—21no—
302D61—21no—
303D61—21no—
304C22×#41—611no—
305C22×M16—612no—
306C2×#117—611no—
307C22×C4⋊C4—1221no—
308C22×C42—611no—
309C22×C42—612no—
310C2×#104—1224no—
311C2×#104—1224no—
312C2×#104—1224no—
313#391—611no—
314#391—611no—
315#390—611no—
316#390#692311yescharacteristic, complemented
317#386—1221no—
318#386—1221no—
319C4×C4⋊C4—1224no—
320C4×C22⋊C4—1221no—
321C4×C22⋊C4—1221no—
322C2×#95—1221no—
323#339—1224no—
324#336—1221no—
325#336—1221no—
326C61#1107511yescharacteristic, complemented, Sylow
327C2×#41—612no—
328C2×#41—612no—
329C2×#41—1221no—
330C2×#41—1221no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C22×Q8—611no—
336C22×D4—1221no—
337C22×D4C2×#692311yescharacteristic
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—1221no—
342C23×C4—1222no—
343C2×M16—614no—
344C2×M16—1222no—
345C2×M16—1222no—
346C2×M16—1222no—
347C2×M16—1222no—
348C2×M16—1222no—
349C22×C8—1222no—
350#117—1222no—
351#117—1222no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4C2×#692311yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C42—611no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—614no—
366C2×C42—1222no—
367C2×C42—1222no—
368C2×C42—1222no—
369C2×C42—1222no—
370C2×C42—1222no—
371C2×C42—1222no—
372C2×C42—1221no—
373C2×C42—1222no—
374C2×C42C2×#692311yescharacteristic
375#104—12216no—
376#95—1221no—
377#95—1221no—
378#95—1224no—
379#95—1221no—
380#95—1221no—
381C24—21no—
382C24—611no—
383#41—1224no—
384#41—1224no—
385#41—1224no—
386#41—1224no—
387#41—2442no—
388#41—2442no—
389#41—2442no—
390#41—2442no—
391C2×Q8—612no—
392C2×Q8—1221no—
393C2×Q8—1222no—
394C2×Q8—1222no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—2441no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4C4⋊#692312yescomplemented
405C22×C4—21no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—1222no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4C22⋊#692311yescharacteristic
439C22×C4C22⋊#692311yescharacteristic
440C22×C4C22×#692311yescharacteristic
441M16—1224no—
442M16—1224no—
443M16—2442no—
444M16—2442no—
445C2×C8—1224no—
446C2×C8—2442no—
447C2×C8—2442no—
448C4⋊C4—22no—
449C4⋊C4—12216no—
450C4⋊C4—1222no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—2441no—
454C22⋊C4—2441no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C42—22no—
458C42—1222no—
459C42—1222no—
460C42—1224no—
461C42—1224no—
462C42—1222no—
463C42—2442no—
464C42—2442no—
465C42—2442no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—611no—
474C23—611no—
475C23—611no—
476C23—611no—
477C23—611no—
478C23—611no—
479C23—1221no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23C2×C22⋊#692311yescharacteristic
484Q8—1224no—
485Q8—2442no—
486D4—24no—
487D4—42no—
488D4—2442no—
489D4—2442no—
490D4—2442no—
491D4—2442no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—611no—
501C2×C4—611no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1228no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2444no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4#1451984112yescomplemented
538C2×C4C2×C4⋊#692311yescharacteristic
539C2×C4#1451985611yescharacteristic
540C2×C4D4×#692312yes—
541C2×C4D4×#692312yes—
542C8—2444no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—611no—
551C22—611no—
552C22—611no—
553C22—611no—
554C22—1221no—
555C22—1221no—
556C22—1221no—
557C22—1221no—
558C22—1221no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22#1684070411yescharacteristic
567C22#1684071011yescharacteristic
568C22#1684071411yescharacteristic
569C22C2×#1451984111yescharacteristic
570C22D61⋊#10411yescharacteristic
571C22#1684075911yescharacteristic
572C22#1684076011yescharacteristic
573C4—22no—
574C4—41no—
575C4—81no—
576C4—612no—
577C4—612no—
578C4—1222no—
579C4—4881no—
580C4—2448no—
581C4—2442no—
582C4—2441no—
583C4#1684070912yes—
584C4#1684071212yes—
585C2—82no—
586C2—1221no—
587C2—1221no—
588C2—1221no—
589C2—1221no—
590C2#2146100211yescharacteristic
591C2#2146169111yescharacteristic
592C2#2146169311yescharacteristic
593C2#2146183711yescharacteristic
594C2#2146183911yescharacteristic
595C2#2146198311yescharacteristic
596C2#2146198411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758224C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#11075
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊#6923C2semidirect
3. C2×#16840632C2semidirect
4. #21461483C2semidirect
20. #16839668C4semidirect
21. #16839667C4semidirect
22. #16839665C4semidirect
27. C61×#390C4semidirect
94. C2×D4×C61C4⋊C4semidirect
141. C2×Dic122C4⋊C4semidirect
202. C2×C4×C61#104semidirect
224. C2×Dic61#104semidirect
316. #390#6923semidirect
326. C61#11075semidirect
404. C2×D4C4⋊#6923semidirect
537. C2×C4#14519841semidirect

Maximal subgroups

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

subgroupindex
C22×#145198412
C2×C4×C4⋊#69232
C2×#168406322
#214614832
#214483062
#1107561