Finite Group Databasebeta

← #29654883

Subgroups — #29654883

Order 15616.

998 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21450992C211yesmaximal, characteristic, complemented
4#21444799C218yesmaximal, complemented
5#21444318C211yesmaximal, characteristic, complemented
6#21444314C211yesmaximal, characteristic, complemented
7#21444140C211yesmaximal, characteristic, complemented
8C61×#3506C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10#16840141—21no—
11#16840134—21no—
12#16840114—21no—
13#16840114—21no—
14#16839972—22no—
15#16839967—22no—
16#16839917C2214yes—
17#16839900C2214yes—
18#16839815C2211yescharacteristic
19#16839815C2211yescharacteristic
20#16839674C2214yes—
21#16839665—21no—
22#16839627—21no—
23#16839622—22no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839499C2214yes—
26#16839496C2214yes—
27#16839486C2214yes—
28C61×#559—22no—
29C61×#556—21no—
30C61×#555—21no—
31C61×#553—22no—
32C61×#550—21no—
33C61×#547—21no—
34C61×#538C2214yes—
35C61×#535—21no—
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#525—22no—
39C61×#520—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122C2312yes—
42C2×#11758202—24no—
43C2×#11758200—24no—
44#14519777—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C22⋊Dic61—24no—
52C2×C4×Dic61—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55#14519700—24no—
56#14519700—24no—
57C22⋊#92418C2312yes—
58C22⋊#92418C2312yes—
59C4×#92418—21no—
60C4×#92418—21no—
61#14519684—24no—
62#14519684C2312yes—
63C2×#41×C61—22no—
64C2×#41×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61D412yes—
67C4⋊Q8×C61D411yescharacteristic
68C4⋊Q8×C61D411yescharacteristic
69C4⋊Q8×C61D411yescharacteristic
70#127×C61—22no—
71#127×C61D411yescharacteristic
72#126×C61—24no—
73#125×C61—21no—
74#124×C61—22no—
75#124×C61—24no—
76#124×C61—21no—
77#123×C61—24no—
78#123×C61—24no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61C2312yes—
82#121×C61—22no—
83#121×C61—22no—
84#121×C61—22no—
85#121×C61—24no—
86#121×C61—21no—
87#121×C61C2312yes—
88#120×C61—24no—
89C4×Q8×C61—22no—
90C4×Q8×C61—21no—
91C4×Q8×C61D412yes—
92C4×D4×C61—22no—
93C4×D4×C61—24no—
94C4×D4×C61—24no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D412yes—
101#117×C61—22no—
102#117×C61—22no—
103#117×C61—21no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61C2312yes—
107C2×C42×C61C2311yescharacteristic
108C24×C61C2×D412yes—
109#41×C61—42no—
110#41×C61—42no—
111#41×C61—84no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—24no—
117C2×D4×C61—24no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—44no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—82no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D412yes—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C42×C61—24no—
159C42×C61—22no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C22⋊Dic61—84no—
165#11758202—84no—
166#11758200—84no—
167C4⋊Dic61—42no—
168C4⋊Dic61—41no—
169C4⋊Dic61—41no—
170C4⋊Dic61—41no—
171C4⋊Dic61—41no—
172C4×Dic61—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—42no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—84no—
193D4×C61—84no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—44no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—82no—
204C2×C4×C61—84no—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#13512yes—
210C2×C4×C61#13712yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—44no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218Dic122—44no—
219Dic122—44no—
220#92418—42no—
221#92418—42no—
222#18406—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—44no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—82no—
233C22×C61C2×#12011yescharacteristic
234C22×C61C2×#13511yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61#45212yes—
244C4×C61#45212yes—
245C4×C61#45512yes—
246Dic61—82no—
247Dic61—82no—
248#3506D6111yescharacteristic
249#3209—611no—
250#3206—611no—
251#2563—611no—
252#1606—618no—
253#1503—611no—
254#1502—611no—
255#1460—611no—
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—82no—
259C2×C61—82no—
260C2×C61C2×#45211yescharacteristic
261C2×C61C2×#45511yescharacteristic
262C2×C61#302711yescharacteristic
263#559—22no—
264#556—21no—
265#555—21no—
266#553—22no—
267#550—21no—
268#547—21no—
269#538C2×D6114yes—
270#535—21no—
271#533C2×D6111yescharacteristic
272#532C2×D6111yescharacteristic
273C2×C4⋊Q8—611no—
274#525—22no—
275#520—21no—
276C2×C4×D4C2×D6111yescharacteristic
277#494—1221no—
278#492—1221no—
279#487—1221no—
280#487—1221no—
281#462—1222no—
282#461—1222no—
283#451—614no—
284#448—614no—
285#431—611no—
286#431—611no—
287#393—614no—
288#390—1221no—
289#382—1221no—
290#380—1222no—
291C4×C4⋊C4—611no—
292#328—614no—
293#327—614no—
294#323—614no—
295C61#1840611yescharacteristic, complemented, Sylow
296C2×#41—22no—
297C2×#41—21no—
298C22×Q8—612no—
299C22×D4C22⋊D6111yescharacteristic
300C23×C4C22⋊D6112yes—
301C2×Q16—1224no—
302C2×SD16—1224no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8—1221no—
307C4⋊Q8C22⋊D6111yescharacteristic
308C4⋊Q8C22⋊D6111yescharacteristic
309C4⋊Q8C22⋊D6111yescharacteristic
310#127—22no—
311#127C22⋊D6111yescharacteristic
312#126—24no—
313#125—21no—
314#124—22no—
315#124—24no—
316#124—21no—
317#123—24no—
318#123—24no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8C22×D6112yes—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—21no—
327#121C22×D6112yes—
328#120—24no—
329C4×Q8—22no—
330C4×Q8—21no—
331C4×Q8C22⋊D6112yes—
332C4×D4—22no—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D6112yes—
341#117—22no—
342#117—22no—
343#117—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C22⋊C4—1224no—
350C2×C22⋊C4C22×D6112yes—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353C4⋊C8—1221no—
354C4⋊C8—1221no—
355#103—1224no—
356#103—1224no—
357C22⋊C8—612no—
358C22⋊C8—612no—
359C4×C8—1221no—
360C4×C8—1221no—
361#95—612no—
362#95—1224no—
363C24C2×C22⋊D6112yes—
364#41—42no—
365#41—42no—
366#41—84no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—4882no—
372C2×Q8C2×C22⋊D6112yes—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4C2×C22⋊D6112yes—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—44no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C2×C22⋊D6112yes—
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C23×D6111yescharacteristic
394Q16—4884no—
395SD16—4884no—
396C2×C8—1222no—
397C2×C8—1222no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—2442no—
408C4⋊C4—2441no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C4⋊C4C2×C22⋊D6112yes—
413C4⋊C4C2×C22⋊D6112yes—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—4884no—
423C42—24no—
424C42—22no—
425C42—2442no—
426C42C2×C22⋊D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
429C42C2×C22⋊D6111yescharacteristic
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23C22×C22⋊D6111yescharacteristic
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—2444no—
441Q8—2444no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—84no—
450D4—84no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—2442no—
463C2×C4—2444no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451974611yescharacteristic
468C2×C4#1451974611yescharacteristic
469C2×C4#1451974611yescharacteristic
470C2×C4#1451974611yescharacteristic
471C2×C4#1451979712yes—
472C2×C4#1451980712yes—
473C2×C4C22×C22⋊D6111yescharacteristic
474C2×C4C22×C22⋊D6111yescharacteristic
475C8—2442no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—44no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—82no—
487C22C2×#1451974611yescharacteristic
488C22C2×#1451979711yescharacteristic
489C22C2×#1451980711yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—4882no—
498C4—4882no—
499C4#1683992412yes—
500C4#1683992412yes—
501C4#1683994012yes—
502C2—21no—
503C2—21no—
504C2—82no—
505C2—82no—
506C2C2×#1683992411yescharacteristic
507C2C2×#1683994011yescharacteristic
508C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18406
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
2. #21456741C2semidirect
3. #21450992C2semidirect
4. #21444799C2semidirect
5. #21444318C2semidirect
6. #21444314C2semidirect
7. #21444140C2semidirect
295. C61#18406semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214567412
#214509922
#214447992
#214443182
#214443142
#214441402
C61×#35062
#1840661