Finite Group Databasebeta

← #29597310

Subgroups — #29597310

Order 15616.

644 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454881C211yesmaximal, characteristic, complemented
2#21447460C211yesmaximal, characteristic, complemented
3#21445783C211yesmaximal, characteristic, complemented
4#21445770C211yesmaximal, characteristic, complemented
5#21443947C211yesmaximal, characteristic, complemented
6#21443939C211yesmaximal, characteristic, complemented
7C61×#3025C211yesmaximal, characteristic
8#16839840—21no—
9#16839840—21no—
10#16839834—21no—
11#16839834—21no—
12#16839823—21no—
13#16839779—21no—
14#16839758—21no—
15#16839746—21no—
16#16839746C2211yescharacteristic
17#16839736C2211yescharacteristic
18#16839729—21no—
19#16839729C2211yescharacteristic
20#16839729C2211yescharacteristic
21#16839726—21no—
22#16839552—21no—
23#16839549—21no—
24C61×#585C2211yescharacteristic
25C61×#525—21no—
26C61×#525C2211yescharacteristic
27C61×#458—21no—
28C61×#458C412yescomplemented
29C61×B(4,𝔽2)—21no—
30C61×B(4,𝔽2)C412yescomplemented
31C61×#410—21no—
32C61×#410C2211yescharacteristic
33C2×#11758194—21no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758194D411yescharacteristic, complemented
37C2×#11758194D411yescharacteristic, complemented
38C22×#92418D411yescharacteristic, complemented
39C22×#92418D411yescharacteristic, complemented
40#14519736—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43#14519704—42no—
44#14519701—41no—
45#14519698—41no—
46#14519696—21no—
47#14519696—41no—
48#14519694—21no—
49#14519694—41no—
50#14519692—21no—
51#14519692—21no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—42no—
59#14519688—21no—
60C4×#92418—21no—
612−1+4×C61—21no—
622−1+4×C61—41no—
63H(5,𝔽2)×C61—21no—
64H(5,𝔽2)×C61—21no—
65H(5,𝔽2)×C61—41no—
66H(5,𝔽2)×C61C2×C412yes—
67C2×#41×C61—21no—
68C2×#41×C61—21no—
69C2×#41×C61—21no—
70C2×#41×C61—42no—
71C2×#41×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73#123×C61—22no—
74#123×C61C2×C411yescharacteristic
75#123×C61D411yescharacteristic
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—41no—
78#121×C61—21no—
79#121×C61—41no—
80#120×C61—22no—
81#120×C61C2×C411yescharacteristic
82#120×C61D411yescharacteristic
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61—42no—
86#117×C61—21no—
87#117×C61D411yescharacteristic
88#99×C61—21no—
89#99×C61—22no—
90#99×C61—22no—
91#99×C61C2×C412yes—
92C24×C61C22⋊C411yescharacteristic
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—41no—
96#41×C61—41no—
97#41×C61—42no—
98#41×C61—41no—
99#41×C61—41no—
100#41×C61—41no—
101#41×C61—41no—
102#41×C61—41no—
103#41×C61—42no—
104#41×C61—42no—
105#41×C61—42no—
106#41×C61—41no—
107#41×C61—41no—
108#41×C61—41no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×Q8×C61#4111yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—41no—
125C2×D4×C61—42no—
126C2×D4×C61C22×C411yescharacteristic
127C2×D4×C61C2×D411yescharacteristic
128C2×D4×C61#4111yescharacteristic
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—41no—
134C22×C4×C61—41no—
135C22×C4×C61—41no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—42no—
145C22⋊C4×C61—21no—
146C22⋊C4×C61—21no—
147C22⋊C4×C61—21no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C22⋊C4×C61C22⋊C411yescharacteristic
153C22⋊C4×C61C22⋊C411yescharacteristic
154C42×C61—21no—
155C42×C61—41no—
156C22⋊Dic61—41no—
157C4⋊Dic61—41no—
158C4×Dic61—41no—
159#11758194—41no—
160#11758194—41no—
161#11758194—41no—
162#11758194—41no—
163#11758194—81no—
164#11758194—81no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—41no—
170C2×#92418—41no—
171C2×#92418—41no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—21no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—81no—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61C4×D411yescharacteristic
180C23×C61#12111yescharacteristic
181Q8×C61—22no—
182Q8×C61—41no—
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—22no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—41no—
197D4×C61—81no—
198D4×C61—81no—
199D4×C61—84no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—81no—
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12311yescharacteristic
225C2×Dic61—81no—
226#92418—41no—
227#92418—41no—
228#92418—41no—
229#92418—41no—
230#92418—81no—
231#13046—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—41no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—81no—
246C22×C61—161no—
247C22×C61#38611yescharacteristic
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61—41no—
255C4×C61—82no—
256C4×C61B(4,𝔽2)11yescharacteristic
257C4×C61B(4,𝔽2)11yescharacteristic
258Dic61—161no—
259#3025D6111yescharacteristic
260#2978—611no—
261#2120—611no—
262#1794—611no—
263#1791—611no—
264#1395—611no—
265#1391—611no—
266C2×C61—21no—
267C2×C61—41no—
268C2×C61—41no—
269C2×C61—41no—
270C2×C61—81no—
271C2×C61—82no—
272C2×C61#188911yescharacteristic
273#585C2×D6111yescharacteristic
274#525—21no—
275#525C2×D6111yescharacteristic
276#458—21no—
277#458Dic6112yescomplemented
278B(4,𝔽2)—21no—
279B(4,𝔽2)Dic6112yescomplemented
280#436—1221no—
281#436—1221no—
282#435—1221no—
283#435—1221no—
284#433—1221no—
285#421—1221no—
286#416—1221no—
287#413—611no—
288#413—1221no—
289#410—21no—
290#410—611no—
291#410C2×D6111yescharacteristic
292#408—611no—
293#408—611no—
294#408—1221no—
295#407—1221no—
296#352—1221no—
297AΣL(1,𝔽16)—1221no—
298C61#1304611yescharacteristic, complemented, Sylow
2992−1+4—21no—
3002−1+4—41no—
301H(5,𝔽2)—21no—
302H(5,𝔽2)—21no—
303H(5,𝔽2)—41no—
304H(5,𝔽2)C2×Dic6112yes—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—21no—
308C2×#41—42no—
309C2×#41C22×D6111yescharacteristic
310C23×C4C22⋊D6111yescharacteristic
311C2×M16—611no—
312C2×M16—611no—
313C2×M16—1221no—
314C2×M16—1221no—
315C2×M16—1221no—
316C22×C8—611no—
317C22×C8—611no—
318#123—22no—
319#123C2×Dic6111yescharacteristic
320#123C22⋊D6111yescharacteristic
321C22⋊Q8—21no—
322C22⋊Q8—41no—
323#121—21no—
324#121—41no—
325#120—22no—
326#120C2×Dic6111yescharacteristic
327#120C22⋊D6111yescharacteristic
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—42no—
331#117—21no—
332#117—1221no—
333#117C22⋊D6111yescharacteristic
334C4⋊C8—1221no—
335C4⋊C8—1221no—
336#104—2442no—
337#103—2441no—
338#102—2441no—
339#101—1221no—
340#101—2441no—
341#100—1221no—
342#100—2441no—
343#99—21no—
344#99—22no—
345#99—22no—
346#99—1221no—
347#99—1221no—
348#99C2×Dic6112yes—
349C22⋊C8—1221no—
350C22⋊C8—1221no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C22⋊C8—1221no—
354C22⋊C8—1221no—
355C22⋊C8—2442no—
356#97—1221no—
357C4×C8—1221no—
358C24C22⋊Dic6111yescharacteristic
359#41—22no—
360#41—22no—
361#41—41no—
362#41—41no—
363#41—42no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—42no—
370#41—42no—
371#41—42no—
372#41—41no—
373#41—41no—
374#41—41no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8#1175822411yescharacteristic
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4#1175822411yescharacteristic
393C2×D4C22×Dic6111yescharacteristic
394C2×D4C2×C22⋊D6111yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4C22⋊Dic6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406M16—4881no—
407M16—4881no—
408M16—2441no—
409M16—2441no—
410M16—2441no—
411M16—2441no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—2441no—
417C2×C8—2441no—
418C2×C8—2441no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—2441no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—22no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—2441no—
433C22⋊C4C22⋊Dic6111yescharacteristic
434C22⋊C4C22⋊Dic6111yescharacteristic
435C42—21no—
436C42—41no—
437C42—2441no—
438C23—22no—
439C23—22no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—81no—
444C23C2×C22⋊Dic6111yescharacteristic
445C23D4×Dic6111yescharacteristic
446C23#1451974911yescharacteristic
447Q8—22no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—41no—
463D4—81no—
464D4—81no—
465D4—84no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—81no—
487C2×C4—4881no—
488C2×C4D4×Dic6111yescharacteristic
489C2×C4#1451974511yescharacteristic
490C2×C4#1451974911yescharacteristic
491C2×C4#1451975811yescharacteristic
492C8—4881no—
493C8—2441no—
494C8—2441no—
495C8—2441no—
496C8—2441no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—82no—
509C22—82no—
510C22—81no—
511C22—161no—
512C22#1683964211yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—42no—
519C4—41no—
520C4—82no—
521C4—9761no—
522C4#1683994811yescharacteristic
523C4#1683994811yescharacteristic
524C2—21no—
525C2—41no—
526C2—41no—
527C2—41no—
528C2—81no—
529C2—82no—
530C2#2144622811yescharacteristic
subgroupG/H
centerC4#16839948
commutatorC2×D4×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3025C2
socleC2×C61#1889
cosocleC2×#41×C61C23
hypercenter#3025D61
hypocenterC61#13046
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #3025

Complements

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

NH ≅ G/Ntype
1. #21454881C2semidirect
2. #21447460C2semidirect
3. #21445783C2semidirect
4. #21445770C2semidirect
5. #21443947C2semidirect
6. #21443939C2semidirect
28. C61×#458C4semidirect
30. C61×B(4,𝔽2)C4semidirect
36. C2×#11758194D4semidirect
37. C2×#11758194D4semidirect
38. C22×#92418D4semidirect
39. C22×#92418D4semidirect
277. #458Dic61semidirect
279. B(4,𝔽2)Dic61semidirect
298. C61#13046semidirect

Maximal subgroups

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

subgroupindex
#214548812
#214474602
#214457832
#214457702
#214439472
#214439392
C61×#30252
#1304661