Finite Group Databasebeta

← #29714375

Subgroups — #29714375

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3C2×#16839972C211yesmaximal, characteristic, complemented
4#21447022C212yesmaximal
5#21446827C212yesmaximal
6#21446767C211yesmaximal, characteristic, complemented
7#21446742C211yesmaximal, characteristic, complemented
8#21446163C212yesmaximal, complemented
9#21446072C212yesmaximal, complemented
10#21445799C211yesmaximal, characteristic
11C4×C22⋊Q8×C61C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13#16840319—24no—
14#16840319—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769—21no—
18C2×#14519756—21no—
19C2×#14519756C2212yes—
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2212yes—
22C2×#14519736—21no—
23#16840057—24no—
24#16839972—24no—
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×C22⋊#92418C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C22⋊Q8×C61C2211yescharacteristic
37C2×C4×Q8×C61C2211yescharacteristic
38C22×C42×C61C2211yescharacteristic
39C61×#391C2211yescharacteristic
40C61×#389C2211yescharacteristic
41C61×#387—21no—
42C61×#385—21no—
43C61×#380C2211yescharacteristic
44C4×C4⋊C4×C61—21no—
45C4×C4⋊C4×C61C2211yescharacteristic
46C4×C22⋊C4×C61—21no—
47C22×Dic122—21no—
48C22×Dic122C2312yes—
49C2×#11758202—24no—
50C2×#11758202—24no—
51C2×#11758202—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519777—22no—
55#14519777—22no—
56#14519777D414yes—
57#14519777D414yes—
58#14519769—28no—
59#14519756—28no—
60#14519756—24no—
61#14519756—24no—
62C4×Dic122—28no—
63C4×Dic122—24no—
64C4×Dic122—24no—
65#14519736—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—22no—
74#14519710—24no—
75C4⋊#92418—22no—
76C4⋊#92418—22no—
77#14519701—24no—
78#14519700—24no—
79#14519700—24no—
80#14519700—42no—
81C22⋊#92418—42no—
82C22×Q8×C61C2311yescharacteristic
83C23×C4×C61D411yescharacteristic
84C23×C4×C61D411yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—22no—
99C2×C42×C61—22no—
100C2×C42×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61D411yescharacteristic
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107#95×C61—21no—
108#95×C61—21no—
109#95×C61—21no—
110#95×C61—21no—
111C24×C61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61#4112yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—41no—
132C22×C4×C61—41no—
133C22×C4×C61Q1614yescomplemented
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—22no—
141C22×Dic61—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—24no—
147C2×Dic122—22no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122C2×D414yes—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61#4112yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—42no—
162C42×C61—22no—
163C42×C61—22no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—42no—
167C42×C61—42no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—28no—
171#11758202—48no—
172C4⋊Dic61—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61—24no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D414yes—
177#11758196—28no—
178#11758196—28no—
179C4×Dic61—28no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61C2×Q1612yes—
193C23×C61C22×D411yescharacteristic
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—41no—
197Q8×C61—41no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—28no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—44no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61#12012yes—
227C2×C4×C61#12012yes—
228C2×C4×C61C2×Q1612yes—
229C2×C4×C61C2×Q1612yes—
230C2×C4×C61#13712yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243Dic122—24no—
244Dic122—24no—
245Dic122—48no—
246Dic122—42no—
247Dic122—42no—
248#92418—42no—
249#92418—42no—
250#24155—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61C2×#12011yescharacteristic
259C22×C61#52511yescharacteristic
260C22×C61#53711yescharacteristic
261C22×C61C22×Q1611yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C22×C61#57511yescharacteristic
264C22×C61#57511yescharacteristic
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61#45114yes—
274C4×C61#54611yescharacteristic
275C4×C61#54711yescharacteristic
276C4×C61#54811yescharacteristic
277C4×C61#55011yescharacteristic
278Dic61—44no—
279Dic61—44no—
280C2×#533—611no—
281C2×#477—611no—
282C2×#462—611no—
283C4×C22⋊Q8D6111yescharacteristic
284#2036—612no—
285#2001—612no—
286#1987—611no—
287#1981—611no—
288#1878—612no—
289#1863—612no—
290#1797—611no—
291C2×C61—42no—
292C2×C61#240711yescharacteristic
293C2×C61C2×#45111yescharacteristic
294C2×C61#300511yescharacteristic
295C2×C61#310411yescharacteristic
296C2×C61#311611yescharacteristic
297C2×C61#324911yescharacteristic
298C2×C61#325211yescharacteristic
299C22×Q16—612no—
300#533—1224no—
301#533—1224no—
302C2×C4⋊Q8—611no—
303C2×C4⋊Q8—611no—
304C2×#125—1221no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8—1221no—
307C2×C22⋊Q8C2×D6111yescharacteristic
308C2×C4×Q8—612no—
309C2×C4×Q8—1221no—
310C2×C4×Q8C2×D6111yescharacteristic
311C2×#117—1221no—
312C22×C42C2×D6111yescharacteristic
313#477—1224no—
314#462—1224no—
315C2×#107—612no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8—611no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#103—612no—
322C2×C22⋊C8—612no—
323#391C2×D6111yescharacteristic
324#389C2×D6111yescharacteristic
325#387—21no—
326#385—21no—
327#380C2×D6111yescharacteristic
328C4×C4⋊C4—21no—
329C4×C4⋊C4C2×D6111yescharacteristic
330C4×C22⋊C4—21no—
331#340—612no—
332#340—612no—
333#336—612no—
334C61#2415511yescharacteristic, complemented, Sylow
335C22×Q8—612no—
336C22×Q8—1221no—
337C22×Q8C22×D6111yescharacteristic
338C23×C4C22⋊D6111yescharacteristic
339C23×C4C22⋊D6111yescharacteristic
340C23×C4C22×D6111yescharacteristic
341C2×Q16—1224no—
342C2×Q16—1224no—
343C2×Q16—2442no—
344C22×C8—612no—
345C22×C8—612no—
346C4⋊Q8—614no—
347C4⋊Q8—614no—
348C4⋊Q8—1222no—
349C4⋊Q8—1222no—
350#125—1228no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1228no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—1224no—
359C4×Q8—1224no—
360C4×Q8—1228no—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1222no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—1222no—
382C2×C42C22⋊D6111yescharacteristic
383C2×C42C22⋊D6111yescharacteristic
384C2×C42C22×D6111yescharacteristic
385C2×C42C22×D6111yescharacteristic
386#107—1224no—
387C4⋊C8—1222no—
388C4⋊C8—1222no—
389#103—1224no—
390#103—1224no—
391#103—1224no—
392#103—2442no—
393C22⋊C8—2442no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399C24C2×C22⋊D6111yescharacteristic
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—614no—
406C2×Q8—1222no—
407C2×Q8—1224no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8#1175822212yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4#1175820214yescomplemented
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—614no—
455C4⋊C4—1228no—
456C4⋊C4—1228no—
457C4⋊C4—1222no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4#1175822212yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—1228no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42—42no—
471C42—1228no—
472C42D4×D6112yes—
473C42D4×D6112yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23C2×#1175820212yes—
481C23C22×C22⋊D6111yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—1224no—
487Q8—1224no—
488Q8—2448no—
489Q8—2442no—
490Q8—2442no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4#1451974512yes—
526C2×C4#1451974512yes—
527C2×C4C2×#1175820212yes—
528C2×C4C2×#1175820212yes—
529C2×C4#1451980712yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4C2×#1175822211yescharacteristic
533C2×C4C2×#1175822211yescharacteristic
534C2×C4#1451982311yescharacteristic
535C2×C4#1451982411yescharacteristic
536C8—2442no—
537C8—2442no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#1451974511yescharacteristic
546C22#1684027311yescharacteristic
547C22#1684034011yescharacteristic
548C22C22×#1175820211yescharacteristic
549C22C2×#1451980711yescharacteristic
550C22#1684056011yescharacteristic
551C22#1684056011yescharacteristic
552C4—22no—
553C4—22no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—2444no—
561C4—2444no—
562C4#1683991714yes—
563C4#1684040811yescharacteristic
564C4#1684041711yescharacteristic
565C4#1684042111yescharacteristic
566C4#1684043311yescharacteristic
567C2—42no—
568C2#2144916411yescharacteristic
569C2C2×#1683991711yescharacteristic
570C2#2145505911yescharacteristic
571C2#2145585811yescharacteristic
572C2#2145598111yescharacteristic
573C2#2145716111yescharacteristic
574C2#2145719211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC4×C22⋊Q8D61
hypocenterC61#24155
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
3. C2×#16839972C2semidirect
6. #21446767C2semidirect
7. #21446742C2semidirect
8. #21446163C2semidirect
9. #21446072C2semidirect
133. C22×C4×C61Q16semidirect
334. C61#24155semidirect
432. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×#168400572
C2×#168399722
#214470222
#214468272
#214467672
#214467422
#214461632
#214460722
#214457992
C4×C22⋊Q8×C612
#2415561