Finite Group Databasebeta

← #29779846

Subgroups — #29779846

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454180C211yesmaximal, characteristic, complemented
2C4×#14519766C211yesmaximal, characteristic
3#21446968C212yesmaximal, complemented
4#21446964C212yesmaximal
5#21446627C212yesmaximal
6#21445948C212yesmaximal, complemented
7#21445824C211yesmaximal, characteristic, complemented
8#21445820C211yesmaximal, characteristic
9#21445816C211yesmaximal, characteristic
10C4⋊C8×Dic61C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C2×#14519768C2212yes—
13C2×#14519766C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519701C2212yes—
17C2×#14519699C2212yes—
18C2×#14519697C2212yes—
19C2×C8×Dic61C2212yes—
20#16839669C2211yescharacteristic
21#16839663C2211yescharacteristic
22#16839661—21no—
23#16839661—21no—
24#16839658—21no—
25#16839658—21no—
26#16839652—21no—
27#16839652—21no—
28#16839632C2211yescharacteristic
29#16839627C2211yescharacteristic
30#16839627C2212yes—
31C4×#11758198—21no—
32C4×#11758198—21no—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C22×#92418C2312yes—
46#14519768—42no—
47#14519768—42no—
48#14519766—22no—
49#14519766Q814yes—
50C4×D244—22no—
51C4×D244—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2312yes—
57C2×#11758196—22no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C4⋊#92418—22no—
70#14519701—42no—
71#14519699—42no—
72#14519697—42no—
73C8×Dic61—24no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—22no—
79#14519684—22no—
80C22×Q8×C61C2311yescharacteristic
81C22×C8×C61C2312yes—
82C4×Q8×C61—22no—
83C4×Q8×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61—22no—
92#103×C61—42no—
93C23×D61—21no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61C2×Q812yes—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244C2×Q812yes—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61—42no—
131C42×C61—42no—
132C42×C61#4112yes—
133C42×C61#4112yes—
134#11758198—24no—
135#11758198—24no—
136#11758198—42no—
137#11758198—42no—
138#11758198—42no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—44no—
144C4⋊Dic61—42no—
145C4⋊Dic61—42no—
146#11758196—44no—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C4×Dic61—41no—
152C4×Dic61—41no—
153C4×Dic61C2×Q812yes—
154C4×Dic61#4112yes—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C23×C61C22×D411yescharacteristic
159C22×D61—22no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166Q8×C61—22no—
167Q8×C61—22no—
168Q8×C61—41no—
169Q8×C61—41no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61#12412yes—
182C2×C4×C61#12412yes—
183C2×C4×C61#13512yes—
184C2×C4×C61#13512yes—
185C2×C4×C61C22×Q811yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61H(5,𝔽2)11yescharacteristic
190C2×C4×C61H(5,𝔽2)11yescharacteristic
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61C22⋊Q812yes—
208C2×Dic61C22⋊Q812yes—
209D244—22no—
210D244—22no—
211D244—41no—
212D244—41no—
213C4×D61—84no—
214C8×C61—42no—
215#92418—42no—
216#29140—611nomaximal, Sylow
217C22×C61C2×C22⋊Q811yescharacteristic
218C22×C61C2×#12411yescharacteristic
219C22×C61#53511yescharacteristic
220C22×C61C2×#13511yescharacteristic
221C22×C61C2×#13511yescharacteristic
222C22×C61#57511yescharacteristic
223C22×C61#57511yescharacteristic
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—42no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—22no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61—82no—
235C4×C61#48814yes—
236C4×C61#55411yescharacteristic
237C4×C61#55411yescharacteristic
238C4×C61#55511yescharacteristic
239C4×C61#55511yescharacteristic
240Dic61—22no—
241Dic61—22no—
242Dic61—41no—
243Dic61—41no—
244Dic61—81no—
245Dic61—82no—
246#2859—611no—
247C4×#124—611no—
248#2026—612no—
249#2025—612no—
250#1958—612no—
251#1835—612no—
252#1803—611no—
253#1803D6111yescharacteristic, complemented
254#1802—611no—
255#1802—611no—
256C4×C4⋊C8—611no—
257C2×C61#256311yescharacteristic
258C2×C61#307611yescharacteristic
259C2×C61#307611yescharacteristic
260C2×C61C2×#48811yescharacteristic
261C2×C61#314211yescharacteristic
262C2×C61#321211yescharacteristic
263C2×C61#321211yescharacteristic
264D61—82no—
265C2×#125—612no—
266C2×#124—611no—
267C2×C4×Q8C2×D6111yescharacteristic
268C2×C4×D4—611no—
269C2×C4⋊C8—611no—
270C2×C4⋊C8C2×D6111yescharacteristic
271C2×#103—612no—
272C2×#103C2×D6112yes—
273C2×#102—612no—
274C2×#102—612no—
275C2×C4×C8—612no—
276#391—611no—
277#390—611no—
278#389—1221no—
279#389—1221no—
280#389—1221no—
281#389—1221no—
282#388—1221no—
283#388—1221no—
284#384—611no—
285#384C2×D6111yescharacteristic
286#382—611no—
287#382—612no—
288C4×C22⋊C4—1221no—
289C4×C22⋊C4—1221no—
290C43—611no—
291#340—612no—
292#340—612no—
293#340—612no—
294#340C2×D6112yes—
295#336—612no—
296C61#2914011yescharacteristic, complemented, Sylow
297C22×Q8C22×D6111yescharacteristic
298C22×D4—611no—
299C23×C4—1221no—
300C22×C8—612no—
301C22×C8C22×D6112yes—
302#125—2442no—
303#125—2442no—
304#124—614no—
305#124—1222no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×D4—1222no—
309C4×D4—1222no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C4⋊C4C22×D6112yes—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C42—21no—
327C2×C42—611no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—611no—
331C2×C42—611no—
332C2×C42—611no—
333C2×C42C22×D6111yescharacteristic
334C4⋊C8—22no—
335C4⋊C8—1222no—
336#103—42no—
337#103—2442no—
338#102—2442no—
339#102—2442no—
340C4×C8—1224no—
341#95—1221no—
342#95—1221no—
343#95—1221no—
344#95—1221no—
345#95—1222no—
346#95—1222no—
347C24—1221no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8Q8×D6112yes—
354C2×D4—612no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4D4×D6111yescharacteristic
373C22×C4D4×D6111yescharacteristic
374C22×C4C23×D6111yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—1222no—
379C2×C8—1222no—
380C2×C8—1222no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4—2442no—
393C4⋊C4—2444no—
394C4⋊C4—2444no—
395C4⋊C4—2442no—
396C22⋊C4—1224no—
397C22⋊C4—1224no—
398C22⋊C4—2442no—
399C22⋊C4—2442no—
400C22⋊C4—2442no—
401C42—42no—
402C42—612no—
403C42—612no—
404C42—1221no—
405C42—1221no—
406C42—1222no—
407C42—1222no—
408C42—2441no—
409C42—2441no—
410C42#1175822212yes—
411C42#1175822412yes—
412C23—1222no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23—2441no—
419C23C2×D4×D6111yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424D4—1222no—
425D4—1222no—
426D4—2441no—
427D4—2441no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—4884no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4#1451976312yes—
459C2×C4#1451976312yes—
460C2×C4#1451979812yes—
461C2×C4#1451980012yes—
462C2×C4C2×Q8×D6111yescharacteristic
463C2×C4#41×D6111yescharacteristic
464C2×C4C2×#1175822411yescharacteristic
465C2×C4C2×#1175822211yescharacteristic
466C2×C4#1451982211yescharacteristic
467C2×C4#1451982311yescharacteristic
468C8—42no—
469C8—2442no—
470C22—4881no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—2442no—
475C22—2442no—
476C22—2442no—
477C22C22⋊Q8×D6111yescharacteristic
478C22C2×#1451976311yescharacteristic
479C22#1684032811yescharacteristic
480C22C2×#1451979811yescharacteristic
481C22C2×#1451980011yescharacteristic
482C22#1684055811yescharacteristic
483C22#1684056111yescharacteristic
484C4—22no—
485C4—42no—
486C4—81no—
487C4—82no—
488C4—1222no—
489C4—1222no—
490C4—4881no—
491C4—4882no—
492C4—2441no—
493C4—2441no—
494C4#1684011814yes—
495C4#1684046711yescharacteristic
496C4#1684047011yescharacteristic
497C4#1684047511yescharacteristic
498C4#1684047611yescharacteristic
499C2—4882no—
500C2#2145098911yescharacteristic
501C2#2145560211yescharacteristic
502C2#2145561011yescharacteristic
503C2C2×#1684011811yescharacteristic
504C2#2145621711yescharacteristic
505C2#2145679911yescharacteristic
506C2#2145680011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1803D61
hypocenterC61#29140
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. #21454180C2semidirect
3. #21446968C2semidirect
6. #21445948C2semidirect
7. #21445824C2semidirect
10. C4⋊C8×Dic61C2semidirect
11. C61×#1803C2semidirect
253. #1803D61semidirect
296. C61#29140semidirect

Maximal subgroups

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

subgroupindex
#214541802
C4×#145197662
#214469682
#214469642
#214466272
#214459482
#214458242
#214458202
#214458162
C4⋊C8×Dic612
C61×#18032
#2914061