Finite Group Databasebeta

← #29575125

Subgroups — #29575125

Order 15616.

676 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455037C211yesmaximal, characteristic
2#21454864C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic, complemented
4#21444311C211yesmaximal, characteristic, complemented
5#21443707C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#1780C211yesmaximal, characteristic, complemented
8C2×#14519797—21no—
9#16840273C2211yescharacteristic
10#16839924—21no—
11#16839918C411yescharacteristic, complemented
12#16839913C411yescharacteristic, complemented
13#16839907—21no—
14#16839901C411yescharacteristic, complemented
15#16839896C411yescharacteristic, complemented
16#16839837—21no—
17#16839836—21no—
18#16839833—21no—
19C8×C22⋊D61—21no—
20#16839820—21no—
21C2×#14519704—21no—
22C2×#14519704—21no—
23#16839758—21no—
24#16839727—21no—
25#16839726C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C4×C22⋊Dic61C2211yescharacteristic
28#16839500—21no—
29#16839497—21no—
30C22×#41×C61C2211yescharacteristic
31C2×C22⋊C8×C61—21no—
32C2×C22⋊C8×C61—21no—
33C2×C22⋊C8×C61C2211yescharacteristic
34C2×C22⋊C8×C61C2211yescharacteristic
35C2×#11758222D411yescharacteristic
36#14519797—42no—
37#14519797—42no—
38C2×#11758202—21no—
39C2×#11758200—21no—
40C2×#11758201—21no—
41C2×#11758199—21no—
42C2×#11758194D411yescharacteristic
43C2×#11758194D411yescharacteristic
44C2×#11758189—21no—
45C22×#92418—21no—
46C2×C8×D61—21no—
47#14519760—21no—
48#14519756C2×C411yescharacteristic
49#14519756D411yescharacteristic
50#14519751C2×C411yescharacteristic
51#14519751D411yescharacteristic
52#14519746—21no—
53C4×C22⋊D61—21no—
54C4×C22⋊D61—21no—
55#14519736D411yescharacteristic
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—21no—
58#14519706—21no—
59#14519706—21no—
60#14519704—42no—
61#14519704—42no—
62#14519701—21no—
63#14519701—21no—
64#14519698—21no—
65#14519698—21no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418C2×C411yescharacteristic
70C22⋊#92418C2×C411yescharacteristic
71#14519689—21no—
72#14519689—21no—
73#14519687—21no—
74C8×Dic61—21no—
75#14519684—21no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—21no—
79C2×#41×C61—21no—
80C2×#41×C61—42no—
81C2×#41×C61—42no—
82C22×Q8×C61C2×C411yescharacteristic
83C22×D4×C61—21no—
84C22×D4×C61C2×C411yescharacteristic
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—21no—
88C22×C8×C61D411yescharacteristic
89C22×C8×C61D411yescharacteristic
90C22⋊C8×C61—21no—
91C22⋊C8×C61—21no—
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61—42no—
95C22⋊C8×C61—42no—
96C24×C61—21no—
97C24×C61#4111yescharacteristic
98#41×C61—22no—
99#41×C61—22no—
100#41×C61—22no—
101#41×C61—22no—
102#41×C61—41no—
103#41×C61—41no—
104#41×C61—41no—
105#41×C61—41no—
106#41×C61—42no—
107#41×C61—42no—
108#41×C61—42no—
109#41×C61—42no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—21no—
138C22×Dic61—41no—
139#11758222—41no—
140#11758222—41no—
141C2×D244C22⋊C411yescharacteristic
142C2×C4×D61—21no—
143C2×Dic122C22⋊C411yescharacteristic
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—41no—
147C2×C8×C61—41no—
148C2×C8×C61—41no—
149C2×C8×C61—41no—
150C2×C8×C61—41no—
151C22⋊Dic61—21no—
152C22⋊Dic61—21no—
153C22⋊Dic61—21no—
154C22⋊Dic61—22no—
155C22⋊Dic61—41no—
156#11758202—42no—
157#11758201—42no—
158#11758200—42no—
159#11758199—42no—
160#11758198—21no—
161#11758198—21no—
162C4⋊Dic61C22⋊C411yescharacteristic
163C4⋊Dic61C22⋊C411yescharacteristic
164#11758196—21no—
165#11758196—21no—
166C4×Dic61—21no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169#11758194—41no—
170#11758194—41no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—41no—
176C2×#92418—41no—
177#11758189—81no—
178C8×D61—81no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61C4×D411yescharacteristic
191C23×C61#12111yescharacteristic
192C23×C61#12311yescharacteristic
193C22×D61—21no—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—41no—
197Q8×C61—41no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—42no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C22⋊D61—41no—
227C22⋊D61—41no—
228C22⋊D61—81no—
229C22⋊D61—81no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—42no—
234C2×Dic61—81no—
235C2×Dic61—81no—
236D244—41no—
237C4×D61—41no—
238C4×D61—41no—
239Dic122—41no—
240C8×C61—41no—
241C8×C61—41no—
242C8×C61—81no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#9037—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61#38611yescharacteristic
266C22×C61#44311yescharacteristic
267C22×C61#44411yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61B(4,𝔽2)11yescharacteristic
276C4×C61B(4,𝔽2)11yescharacteristic
277Dic61—41no—
278Dic61—41no—
279Dic61—41no—
280Dic61—82no—
281#3002—611no—
282#2976—611no—
283#1789—611no—
284#1780D6111yescharacteristic, complemented
285#1501—611no—
286#1298—611no—
287#1282—611no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—41no—
291C2×C61—41no—
292C2×C61—82no—
293C2×C61#187611yescharacteristic
294C2×C61#188111yescharacteristic
295C2×C61#188911yescharacteristic
296D61—81no—
297C22×#41C2×D6111yescharacteristic
298C2×#135—1221no—
299#525—611no—
300#452—1221no—
301#451—611no—
302#450—611no—
303#449—1221no—
304#448—611no—
305#447—611no—
306#436—1221no—
307#435—1221no—
308#435—1221no—
309C8×D4—1221no—
310#433—1221no—
311C2×#104—1221no—
312C2×#104—1221no—
313#416—1221no—
314#408—1221no—
315#407—611no—
316C2×C22⋊C8—21no—
317C2×C22⋊C8—21no—
318C2×C22⋊C8—611no—
319C2×C22⋊C8C2×D6111yescharacteristic
320C2×C22⋊C8C2×D6111yescharacteristic
321C4×C22⋊C4—611no—
322#328—1221no—
323#327—1221no—
324C61#903711yescharacteristic, complemented, Sylow
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—42no—
330C2×#41—42no—
331C2×#41—611no—
332C22×Q8C4×D6111yescharacteristic
333C22×D4—21no—
334C22×D4C4×D6111yescharacteristic
335C23×C4—21no—
336C23×C4C22×D6111yescharacteristic
337#135—2442no—
338#135—2442no—
339C2×Q16—1221no—
340C2×SD16—1221no—
341C2×SD16—1221no—
342C2×D8—1221no—
343C2×M16—611no—
344C2×M16—611no—
345C2×M16—1221no—
346C22×C8—21no—
347C22×C8—1221no—
348C22×C8—1221no—
349C22×C8C22⋊D6111yescharacteristic
350C22×C8C22⋊D6111yescharacteristic
351#123—1221no—
352C22⋊Q8—611no—
353C22⋊Q8—611no—
354#121—611no—
355#121—611no—
356#120—1221no—
357C4×D4—1221no—
358C4×D4—1221no—
359#117—611no—
360C2×C22⋊C4—1221no—
361C2×C42—1221no—
362C4⋊C8—1221no—
363C4⋊C8—1221no—
364#104—2442no—
365#104—2442no—
366#103—1221no—
367#103—1221no—
368#102—1221no—
369#102—1221no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376C22⋊C8—611no—
377C22⋊C8—611no—
378C22⋊C8—1221no—
379C22⋊C8—1221no—
380C22⋊C8—1221no—
381C22⋊C8—1221no—
382C22⋊C8—1221no—
383#97—1221no—
384C4×C8—1221no—
385#95—1221no—
386C24—21no—
387C24#1175822211yescharacteristic
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—42no—
397#41—42no—
398#41—42no—
399#41—42no—
400#41—2441no—
401#41—2441no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—611no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—611no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1221no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—1221no—
427C22×C4—2441no—
428C22×C4C2×C4×D6111yescharacteristic
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822411yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434Q16—2442no—
435SD16—2442no—
436SD16—2442no—
437D8—2442no—
438M16—4881no—
439M16—2441no—
440M16—2441no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—4881no—
453C2×C8—2441no—
454C2×C8—2441no—
455C4⋊C4—611no—
456C4⋊C4—611no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C22⋊C4—1221no—
460C22⋊C4—1221no—
461C22⋊C4—1221no—
462C22⋊C4—1221no—
463C22⋊C4—1221no—
464C22⋊C4—1222no—
465C22⋊C4—2441no—
466C42—1221no—
467C42—2441no—
468C42—2441no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—1221no—
481C23C4×C22⋊D6111yescharacteristic
482C23#1451975011yescharacteristic
483C23#1451975911yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—2441no—
489D4—22no—
490D4—22no—
491D4—41no—
492D4—41no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—4881no—
498D4—4881no—
499D4—2441no—
500D4—2441no—
501D4—2441no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—4881no—
522C2×C4—4881no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4C22⋊C4×D6111yescharacteristic
527C2×C4#1451973711yescharacteristic
528C2×C4#1451974511yescharacteristic
529C2×C4#1451974811yescharacteristic
530C8—41no—
531C8—41no—
532C8—81no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—2441no—
555C22—2441no—
556C22—2441no—
557C22#1683963811yescharacteristic
558C22#1683987711yescharacteristic
559C22#1683988211yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—4882no—
565C4—2441no—
566C4—2441no—
567C4—2441no—
568C4#1683994811yescharacteristic
569C4#1683994811yescharacteristic
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2—4881no—
576C2#2144614911yescharacteristic
577C2#2144618411yescharacteristic
578C2#2144622411yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1780C2
socleC22×C61#386
prime radical#21455037C2
cosocleC23×C4×C61C23
hypercenter#1780D61
hypocenterC61#9037
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #21454864C2semidirect
3. #21445763C2semidirect
4. #21444311C2semidirect
5. #21443707C2semidirect
6. #21443666C2semidirect
7. C61×#1780C2semidirect
11. #16839918C4semidirect
12. #16839913C4semidirect
14. #16839901C4semidirect
15. #16839896C4semidirect
284. #1780D61semidirect
324. C61#9037semidirect

Maximal subgroups

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

subgroupindex
#214550372
#214548642
#214457632
#214443112
#214437072
#214436662
C61×#17802
#903761