Finite Group Databasebeta

← #29734507

Subgroups — #29734507

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456366C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21450992C211yesmaximal, characteristic, complemented
4#21447026C214yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446052C214yesmaximal, complemented
8C4×#14519694C211yesmaximal, characteristic, complemented
9C61×#3506C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840165—21no—
13#16840158—21no—
14#16840121—22no—
15#16840112—21no—
16#16840108—21no—
17#16839805C2214yes—
18C2×#14519704C2214yes—
19#16839763C2214yes—
20C2×#14519694C2212yes—
21C4×#11758194C2211yescharacteristic
22C4×#11758194C2211yescharacteristic
23#16839674—22no—
24#16839665C2212yes—
25#16839627C2212yes—
26#16839622—22no—
27C4×C4⋊Dic61C2211yescharacteristic
28#16839531C2212yes—
29#16839521C2214yes—
30C61×#559—22no—
31C61×#556—21no—
32C61×#555—21no—
33C61×#553—22no—
34C61×#550—21no—
35C61×#547—21no—
36C61×#538—22no—
37C61×#535C2212yes—
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#525—22no—
41C61×#520C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C2×#11758194C2312yes—
45C2×#11758194C2312yes—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×C22⋊Dic61—24no—
54C2×C4×Dic61C2312yes—
55#14519710—24no—
56#14519708—24no—
57#14519704—28no—
58#14519701—24no—
59#14519698—24no—
60#14519694—24no—
61#14519688—21no—
62#14519688—21no—
63C4×#92418—21no—
64C4×#92418—21no—
65#14519684—21no—
66#14519684—24no—
67C2×#41×C61—22no—
68C2×#41×C61C2312yes—
69C22×D4×C61C2311yescharacteristic
70C23×C4×C61D412yes—
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61D411yescharacteristic
74#127×C61—22no—
75#127×C61D411yescharacteristic
76#126×C61—24no—
77#125×C61D412yes—
78#124×C61—24no—
79#124×C61—22no—
80#124×C61D412yes—
81#123×C61—24no—
82#123×C61—24no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—21no—
86#121×C61—24no—
87#121×C61—22no—
88#121×C61—21no—
89#121×C61—21no—
90#121×C61—22no—
91#121×C61—22no—
92#120×C61—24no—
93C4×Q8×C61—22no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—24no—
97C4×D4×C61—22no—
98C4×D4×C61—24no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105#117×C61—22no—
106#117×C61—22no—
107#117×C61C2312yes—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61D412yes—
111C2×C42×C61C2311yescharacteristic
112C24×C61C2×D412yes—
113#41×C61—24no—
114#41×C61—24no—
115#41×C61—84no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61C2×D412yes—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—44no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—22no—
139C22×Dic61—22no—
140C2×Dic122—41no—
141C2×Dic122—41no—
142C2×Dic122—82no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C42×C61—24no—
163C42×C61—22no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C22⋊Dic61—84no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61—41no—
173C4⋊Dic61—41no—
174C4×Dic61—24no—
175#11758194—22no—
176#11758194—22no—
177#11758194—22no—
178#11758194#4114yescomplemented
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61C22×D411yescharacteristic
188Q8×C61—24no—
189Q8×C61—42no—
190Q8×C61—81no—
191D4×C61—24no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—81no—
198D4×C61—84no—
199D4×C61—84no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61—42no—
209C2×C4×C61—82no—
210C2×C4×C61—84no—
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C612−1+412yes—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—44no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224Dic122—82no—
225Dic122—82no—
226#92418—24no—
227#92418—42no—
228#25693—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—44no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61C2×#12011yescharacteristic
240C22×C61#53211yescharacteristic
241C22×C61#53611yescharacteristic
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61#45512yes—
250C4×C61#54712yes—
251C4×C61#54812yes—
252Dic61—82no—
253Dic61—82no—
254#3506D6111yescharacteristic
255#3165—611no—
256#3141—611no—
257#2563—611no—
258#2037—614no—
259#1986—611no—
260#1980—611no—
261#1860—614no—
262C4×#100—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—82no—
266C2×C61—82no—
267C2×C61#243211yescharacteristic
268C2×C61C2×#45511yescharacteristic
269C2×C61#302111yescharacteristic
270#559—22no—
271#556—21no—
272#555—21no—
273#553—22no—
274#550—21no—
275#547—21no—
276#538—22no—
277#535C2×D6112yes—
278#533C2×D6111yescharacteristic
279#532C2×D6111yescharacteristic
280C2×C4⋊Q8—611no—
281#525—22no—
282#520C2×D6112yes—
283C2×C4×D4C2×D6111yescharacteristic
284#501—1222no—
285#500—1221no—
286#498—1221no—
287#489—1222no—
288#487—1221no—
289#486—1221no—
290#428—614no—
291C2×#104—614no—
292#417—614no—
293C2×#100—612no—
294C4×M16—611no—
295C4×M16—611no—
296#393—1222no—
297#390—612no—
298#382—612no—
299#380—1222no—
300C4×C4⋊C4—611no—
301#343—612no—
302#339—614no—
303C61#2569311yescharacteristic, complemented, Sylow
304C2×#41—22no—
305C2×#41C22×D6112yes—
306C22×Q8—1221no—
307C22×D4C22×D6111yescharacteristic
308C23×C4C22⋊D6112yes—
309C2×M16—612no—
310C2×M16—612no—
311C4⋊Q8—1221no—
312C4⋊Q8—1221no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8C22⋊D6111yescharacteristic
316C4⋊Q8C22⋊D6111yescharacteristic
317C4⋊Q8C22⋊D6111yescharacteristic
318#127—22no—
319#127C22⋊D6111yescharacteristic
320#126—24no—
321#125C22⋊D6112yes—
322#124—24no—
323#124—22no—
324#124C22⋊D6112yes—
325#123—24no—
326#123—24no—
327C22⋊Q8—22no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330#121—22no—
331#121—24no—
332#121—21no—
333#121—21no—
334#121—22no—
335#121—22no—
336#120—24no—
337C4×Q8—22no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×D4—24no—
341C4×D4—22no—
342C4×D4—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349#117—22no—
350#117—22no—
351#117C22×D6112yes—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—1224no—
358C2×C22⋊C4C22⋊D6112yes—
359C2×C42—612no—
360C2×C42C22×D6111yescharacteristic
361#107—1224no—
362#106—1224no—
363#104—1228no—
364#103—1224no—
365#102—1224no—
366#100—1224no—
367#97—1221no—
368#97—1221no—
369C4×C8—1221no—
370C4×C8—1221no—
371#95—1221no—
372#95—1224no—
373C24C2×C22⋊D6112yes—
374#41—24no—
375#41—24no—
376#41—84no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—4882no—
380C2×Q8—2441no—
381C2×Q8—2441no—
382C2×Q8C2×C22⋊D6112yes—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×C22⋊D6112yes—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6112yes—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404M16—614no—
405M16—1222no—
406M16—1222no—
407M16—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C4⋊C4—2442no—
422C4⋊C4—2441no—
423C4⋊C4—2441no—
424C4⋊C4C2×C22⋊D6112yes—
425C4⋊C4C2×C22⋊D6112yes—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—4884no—
435C42—24no—
436C42—22no—
437C42—1224no—
438C42C2×C22⋊D6111yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C2×C22⋊D6111yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—42no—
447C23—42no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—24no—
450Q8—42no—
451Q8—81no—
452Q8—4882no—
453Q8—4882no—
454D4—24no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—81no—
461D4—84no—
462D4—84no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—84no—
474C2×C4—2442no—
475C2×C4—2444no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4#1451974611yescharacteristic
480C2×C4#1451974611yescharacteristic
481C2×C4#1451974611yescharacteristic
482C2×C4#1451974611yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4#41×D6112yes—
486C2×C4#1451982412yes—
487C8—1224no—
488C8—2442no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22C2×#1451974611yescharacteristic
500C22#1684030811yescharacteristic
501C22#1684033311yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—4882no—
510C4—4882no—
511C4#1683994012yes—
512C4#1684041112yes—
513C4#1684041912yes—
514C2—21no—
515C2—21no—
516C2—82no—
517C2—82no—
518C2#2144944311yescharacteristic
519C2C2×#1683994011yescharacteristic
520C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#25693
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
2. #21456212C2semidirect
3. #21450992C2semidirect
4. #21447026C2semidirect
5. #21446764C2semidirect
6. #21446739C2semidirect
7. #21446052C2semidirect
8. C4×#14519694C2semidirect
178. #11758194#41semidirect
303. C61#25693semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214562122
#214509922
#214470262
#214467642
#214467392
#214460522
C4×#145196942
C61×#35062
#2569361