Finite Group Databasebeta

← #29689264

Subgroups — #29689264

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C2×#16839820C212yesmaximal, complemented
3#21446615C212yesmaximal, complemented
4#21446615C212yesmaximal, complemented
5#21446611C212yesmaximal, complemented
6#21446611C212yesmaximal, complemented
7#21445731C211yesmaximal, characteristic, complemented
8#21445728C212yesmaximal, complemented
9C61×#1780C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11#16840269—28no—
12C2×#14519758C411yescharacteristic
13C2×#14519758C411yescharacteristic
14C2×#14519753C412yes—
15C2×#14519748C412yes—
16C2×#14519745C411yescharacteristic
17C2×#14519745C411yescharacteristic
18C2×C4×C22⋊D61C2212yes—
19C2×C4×C22⋊D61C2212yes—
20C2×#14519736C2211yescharacteristic
21#16839820—28no—
22C2×#14519706C2212yescomplemented
23C2×#14519706C2212yescomplemented
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2211yescharacteristic, complemented
26C2×C22⋊#92418C2212yescomplemented
27C2×C22⋊#92418C2211yescharacteristic, complemented
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32C2×#14519687C2212yescomplemented
33C2×C8×Dic61C2212yescomplemented
34C22×#41×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2212yes—
36C2×C22⋊C8×C61C2211yescharacteristic, complemented
37C2×C22⋊C8×C61C2212yescomplemented
38C2×C22⋊C8×C61C2211yescharacteristic, complemented
39C24×D61C2×C411yescharacteristic
40C22×C22⋊D61C2×C412yes—
41C22×C22⋊D61C2×C412yes—
42C23×Dic61C2×C411yescharacteristic
43C22×C4×D61—24no—
44C22×C4×D61—41no—
45C22×C4×D61—41no—
46C22×C4×D61—42no—
47C22×C4×D61C2×C412yes—
48C22×C4×D61C2312yes—
49C22×#92418C2312yes—
50C22×#92418C2312yes—
51#14519758—24no—
52#14519758—24no—
53#14519753—28no—
54#14519748—28no—
55#14519745—24no—
56#14519745—24no—
57C4×C22⋊D61—24no—
58C4×C22⋊D61—24no—
59C4×C22⋊D61—24no—
60C4×C22⋊D61—24no—
61#14519736D418yes—
62C2×C4⋊Dic61C2×C412yes—
63C2×#11758196C2×C412yes—
64C2×#11758196C2×C412yes—
65C2×C22⋊Dic61C2×C412yes—
66C2×C22⋊Dic61C2×C412yes—
67C2×C22⋊Dic61C2×C411yescharacteristic
68C2×C22⋊Dic61C2×C411yescharacteristic
69C2×#11758198C2×C412yes—
70C2×#11758198C2×C412yes—
71C2×C4×Dic61C2312yes—
72#14519706—24no—
73#14519706—24no—
74C22⋊#92418—24no—
75C22⋊#92418—41no—
76C22⋊#92418—42no—
77C22⋊#92418—41no—
78#14519689—42no—
79#14519689—42no—
80#14519689—42no—
81#14519689—42no—
82#14519687—24no—
83C8×Dic61—24no—
84C2×#41×C61—24no—
85C2×#41×C61—42no—
86C2×#41×C61—41no—
87C2×#41×C61—41no—
88C22×Q8×C61C2×C411yescharacteristic
89C22×D4×C61C2×C411yescharacteristic
90C22×D4×C61C2×C411yescharacteristic
91C22×D4×C61C2×C411yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C22×C8×C61C2312yes—
97C22⋊C8×C61—24no—
98C22⋊C8×C61—41no—
99C22⋊C8×C61—42no—
100C22⋊C8×C61—41no—
101C24×C61C22×C411yescharacteristic
102C24×C61C22×C411yescharacteristic
103C24×C61C22×C411yescharacteristic
104C23×D61—22no—
105C23×D61—24no—
106C23×D61—41no—
107C23×D61—41no—
108C23×D61—42no—
109C23×D61C22×C412yes—
110#41×C61—48no—
111#41×C61—48no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61—41no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C2×D414yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—22no—
147C22×Dic61—41no—
148C22×Dic61—41no—
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C412yes—
151C22×Dic61C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—48no—
162C2×C4×D61—48no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61C22⋊C414yes—
172C22⋊Dic61C22⋊C414yes—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61C22⋊C418yes—
176#11758196—24no—
177#11758196—24no—
178C4×Dic61C2×D414yes—
179C4×Dic61C2×D414yes—
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—22no—
189C23×C61—22no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61C2×C22⋊C412yes—
201C23×C61C23×C411yescharacteristic
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—44no—
209C22×D61—48no—
210C22×D61—48no—
211C22×D61—44no—
212C22×D61—44no—
213C22×D61#13114yes—
214Q8×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217D4×C61—44no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C22×D412yes—
234C2×C4×C61H(5,𝔽2)12yes—
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C22⋊D61—48no—
238C22⋊D61—48no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—44no—
247C2×Dic61C2×C22⋊C414yes—
248C2×Dic61C2×C22⋊C414yes—
249C4×D61—48no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C8×C61—42no—
255C8×C61—42no—
256#92418—42no—
257#92418—42no—
258#21792—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61#51811yescharacteristic
276C22×C61#51811yescharacteristic
277C22×C61C2×#13112yes—
278C22×C61#56812yes—
279C2×D61—24no—
280C2×D61—24no—
281C2×D61—24no—
282C2×D61—416no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61#53512yes—
291C4×C61#53512yes—
292Dic61—24no—
293Dic61—24no—
294Dic61—42no—
295C2×#525—611no—
296C2×#433—612no—
297#1955—612no—
298#1955—612no—
299#1954—612no—
300#1954—612no—
301#1780—611no—
302#1780D6111yescharacteristic, complemented
303#1779—612no—
304C2×C61—42no—
305C2×C61—81no—
306C2×C61—81no—
307C2×C61#230911yescharacteristic
308C2×C61#288012yes—
309C2×C61#297612yes—
310C2×C61#297711yescharacteristic
311C2×C61#297811yescharacteristic
312D61—44no—
313C22×#41C2×D6111yescharacteristic
314C24×C4—611no—
315#525—1228no—
316C2×#123—611no—
317C2×#123—611no—
318C2×C22⋊Q8—612no—
319C2×#121—612no—
320C2×#120—611no—
321C2×#120—611no—
322C2×C4×D4—612no—
323C2×C4×D4—612no—
324C2×#117—611no—
325#433—1228no—
326C2×C4⋊C8—612no—
327C2×C4⋊C8—612no—
328C2×C22⋊C8—612no—
329C2×C22⋊C8—611no—
330C2×C22⋊C8—612no—
331C2×C22⋊C8—611no—
332C2×C22⋊C8—612no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—612no—
335C2×C22⋊C8—612no—
336C2×C22⋊C8C2×D6112yes—
337C2×C22⋊C8C2×D6111yescharacteristic, complemented
338C2×C22⋊C8C2×D6112yescomplemented
339C2×C22⋊C8C2×D6111yescharacteristic, complemented
340C2×#97—612no—
341C2×C4×C8—612no—
342C61#2179211yescharacteristic, complemented, Sylow
343C25—611no—
344C2×#41—24no—
345C2×#41—42no—
346C2×#41—41no—
347C2×#41—41no—
348C22×Q8C4×D6111yescharacteristic
349C22×D4—612no—
350C22×D4—612no—
351C22×D4C4×D6111yescharacteristic
352C22×D4C4×D6111yescharacteristic
353C22×D4C4×D6111yescharacteristic
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4—1224no—
358C23×C4—2441no—
359C23×C4—2441no—
360C23×C4—2442no—
361C23×C4C22×D6111yescharacteristic
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C22×C8—612no—
365C22×C8—612no—
366C22×C8C22×D6112yes—
367C22×C8C22×D6112yes—
368#123—1224no—
369#123—1224no—
370C22⋊Q8—1228no—
371#121—1228no—
372#120—1224no—
373#120—1224no—
374C4×D4—1224no—
375C4×D4—1224no—
376C4×D4—1224no—
377C4×D4—1224no—
378#117—618no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—612no—
388C2×C42—612no—
389C4⋊C8—1224no—
390C4⋊C8—1224no—
391C22⋊C8—24no—
392C22⋊C8—41no—
393C22⋊C8—42no—
394C22⋊C8—41no—
395C22⋊C8—1224no—
396C22⋊C8—2442no—
397C22⋊C8—2442no—
398C22⋊C8—2442no—
399C22⋊C8—2442no—
400C22⋊C8—2441no—
401C22⋊C8—2442no—
402C22⋊C8—2441no—
403#97—1224no—
404C4×C8—1224no—
405C24—612no—
406C24—1224no—
407C24—1222no—
408C24—2441no—
409C24—2441no—
410C24—2442no—
411C24C2×C4×D6111yescharacteristic
412C24C2×C4×D6111yescharacteristic
413C24C2×C4×D6111yescharacteristic
414#41—48no—
415#41—48no—
416C2×Q8—22no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—2442no—
433C2×D4—2442no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1224no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2448no—
459C22×C4—2448no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4C2×C4×D6111yescharacteristic
463C22×C4C2×C4×D6111yescharacteristic
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4D4×D6114yes—
466C22×C4C23×D6111yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C4⋊C4—618no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C22⋊C4—614no—
483C22⋊C4—614no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C42—614no—
489C42—614no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—614no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23—2448no—
512C23—2444no—
513C23—2444no—
514C23—2448no—
515C23—2444no—
516C23C22⋊C4×D6112yes—
517C23C22×C4×D6111yescharacteristic
518Q8—44no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—2448no—
523D4—2448no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—614no—
539C2×C4—614no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—2448no—
548C2×C4—2444no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4C22⋊C4×D6112yes—
554C2×C4C2×D4×D6112yes—
555C2×C4#1451982212yes—
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—42no—
559C8—42no—
560C8—2442no—
561C8—2442no—
562C22—24no—
563C22—22no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—1224no—
578C22—1224no—
579C22—1224no—
580C22—24416no—
581C22—2442no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22C2×C22⋊C4×D6111yescharacteristic
586C22#1684023511yescharacteristic
587C22#1684023511yescharacteristic
588C22#131×D6112yes—
589C22#1684052712yes—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—1224no—
594C4—1224no—
595C4—2442no—
596C4#1684032712yes—
597C4#1684032712yes—
598C2—42no—
599C2—81no—
600C2—81no—
601C2—2444no—
602C2#2144840011yescharacteristic
603C2#2145426712yes—
604C2#2145486212yes—
605C2#2145487111yescharacteristic
606C2#2145487711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1780C2
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenter#1780D61
hypocenterC61#21792
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. C2×#16839820C2semidirect
3. #21446615C2semidirect
4. #21446615C2semidirect
5. #21446611C2semidirect
6. #21446611C2semidirect
7. #21445731C2semidirect
8. #21445728C2semidirect
9. C61×#1780C2semidirect
22. C2×#14519706C22semidirect
23. C2×#14519706C22semidirect
25. C2×C22⋊#92418C22semidirect
26. C2×C22⋊#92418C22semidirect
27. C2×C22⋊#92418C22semidirect
32. C2×#14519687C22semidirect
33. C2×C8×Dic61C22semidirect
36. C2×C22⋊C8×C61C22semidirect
37. C2×C22⋊C8×C61C22semidirect
38. C2×C22⋊C8×C61C22semidirect
302. #1780D61semidirect
337. C2×C22⋊C8C2×D61semidirect
338. C2×C22⋊C8C2×D61semidirect
339. C2×C22⋊C8C2×D61semidirect
342. C61#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C2×#168398202
#214466152
#214466112
#214457312
#214457282
C61×#17802
#2179261