Finite Group Databasebeta

← #29652309

Subgroups — #29652309

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455854C212yesmaximal, complemented
2#21449087C211yesmaximal, characteristic
3#21444861C212yesmaximal, complemented
4#21444844C212yesmaximal
5#21444793C212yesmaximal, complemented
6#21444776C212yesmaximal
7#21444318C212yesmaximal, complemented
8#21444133C211yesmaximal, characteristic
9C61×#3458C211yesmaximal, characteristic
10C2×C4×Dic122C2211yescharacteristic
11#16840095C2214yes—
12#16840054—22no—
13#16839969—22no—
14#16839917C2214yes—
15C4×#11758202—24no—
16#16839815C2212yes—
17#16839701C2211yescharacteristic
18#16839698C2211yescharacteristic
19#16839695—21no—
20#16839685—21no—
21#16839682C2211yescharacteristic
22#16839674C2211yescharacteristic
23C4×#11758196—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25C4×C22⋊Dic61—21no—
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#544—22no—
31C61×#544—22no—
32C61×#544—21no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—24no—
37C61×#533—21no—
38C61×#533C2212yes—
39C2×#117×C61—21no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C2×#11758202—24no—
43C4×Dic122—22no—
44C4×Dic122—22no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61C2311yescharacteristic
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54#14519710—24no—
55C4⋊#92418—22no—
56#14519701—24no—
57#14519700—24no—
58C22⋊#92418C2314yes—
59C4×#92418—22no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C23×C4×C61D411yescharacteristic
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—24no—
71C4⋊Q8×C61D412yes—
72C4⋊Q8×C61D412yes—
73#125×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76#125×C61—22no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61C2314yes—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C42×C61—22no—
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61C2×D414yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61Q1614yescomplemented
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C2×Dic122—22no—
127C2×Dic122—42no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61C2×D414yes—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C42×C61—24no—
147C42×C61—24no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152C22⋊Dic61—42no—
153C22⋊Dic61—42no—
154#11758202—48no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157#11758196—42no—
158#11758196—42no—
159#11758196—42no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61—42no—
163C2×#92418—24no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61C2×Q1612yes—
169C23×C61C22×D411yescharacteristic
170Q8×C61—44no—
171Q8×C61—82no—
172Q8×C61—82no—
173Q8×C61—82no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—24no—
177C2×C4×C61—24no—
178C2×C4×C61—24no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—24no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—82no—
187C2×C4×C61—82no—
188C2×C4×C61#12012yes—
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61C2×Q1612yes—
192C2×C4×C61C2×Q1612yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C22×D412yes—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207Dic122—24no—
208Dic122—42no—
209#92418—44no—
210#18125—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—22no—
214C22×C61—42no—
215C22×C61—41no—
216C22×C61—41no—
217C22×C61—81no—
218C22×C61C2×#12011yescharacteristic
219C22×C61C22×Q1611yescharacteristic
220C22×C61C2×#13711yescharacteristic
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—41no—
225C4×C61—44no—
226C4×C61—41no—
227C4×C61—44no—
228C4×C61#45112yes—
229C4×C61#45112yes—
230Dic61—42no—
231Dic61—42no—
232Dic61—81no—
233Dic61—81no—
234#3458D6111yescharacteristic
235#3104—612no—
236#2396—611no—
237#1617—612no—
238#1614—612no—
239#1605—612no—
240#1602—612no—
241#1503—612no—
242#1458—611no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—42no—
246C2×C61C2×#45111yescharacteristic
247C2×C61#301811yescharacteristic
248C2×C61#302511yescharacteristic
249#544—22no—
250#544—22no—
251#544—21no—
252#544C2×D6112yes—
253#543—21no—
254#543C2×D6112yes—
255#533—24no—
256#533—21no—
257#533C2×D6112yes—
258C2×C4×Q8—611no—
259C2×#117—21no—
260C2×#117C2×D6111yescharacteristic
261#484—614no—
262#477—1222no—
263#462—1222no—
264#451—614no—
265C4×Q16—1224no—
266#431—612no—
267#400—611no—
268#399—611no—
269#398—1221no—
270#396—1221no—
271#395—611no—
272#393—611no—
273C4×C4⋊C4—611no—
274C4×C4⋊C4—1221no—
275C4×C22⋊C4—1221no—
276#328—614no—
277#328—614no—
278#324—612no—
279#323—612no—
280C61#1812511yescharacteristic, complemented, Sylow
281C22×Q8—611no—
282C23×C4C22⋊D6111yescharacteristic
283C2×Q16—1224no—
284C4⋊Q8—24no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—24no—
288C4⋊Q8C22⋊D6112yes—
289C4⋊Q8C22⋊D6112yes—
290#125—24no—
291#125—22no—
292#125—24no—
293#125—22no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—24no—
301C22⋊Q8C22×D6114yes—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—1222no—
306C4×Q8—1222no—
307C4×Q8C22⋊D6114yes—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4C22⋊D6111yescharacteristic
327C2×C22⋊C4C22×D6111yescharacteristic
328C2×C42—22no—
329C2×C42—1221no—
330C2×C42—1221no—
331C2×C42C22⋊D6111yescharacteristic
332C2×C42C22×D6111yescharacteristic
333#107—1224no—
334C4⋊C8—1222no—
335#103—1224no—
336#103—1224no—
337C22⋊C8—614no—
338C4×C8—1222no—
339#95—611no—
340#95—611no—
341#95—1221no—
342#95—1221no—
343#95—1221no—
344#95—1221no—
345C24C2×C22⋊D6111yescharacteristic
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—1222no—
350C2×Q8—2442no—
351C2×Q8C2×C22⋊D6114yes—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4#1175820214yescomplemented
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C23×D6111yescharacteristic
368Q16—2448no—
369C2×C8—1224no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—1222no—
384C4⋊C4—2442no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
387C4⋊C4—2442no—
388C4⋊C4C2×C22⋊D6114yes—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—2442no—
394C22⋊C4—2442no—
395C42—24no—
396C42—21no—
397C42—24no—
398C42—21no—
399C42—2442no—
400C42—2441no—
401C42—2441no—
402C42C2×C22⋊D6112yes—
403C42C2×C22⋊D6112yes—
404C23—21no—
405C23—21no—
406C23—41no—
407C23—41no—
408C23C2×#1175820212yes—
409C23C22×C22⋊D6111yescharacteristic
410Q8—44no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—1224no—
415Q8—2442no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4#1451974612yes—
443C2×C4#1451974611yescharacteristic
444C2×C4#1451974611yescharacteristic
445C2×C4C2×#1175820212yes—
446C2×C4C2×#1175820212yes—
447C2×C4#1451980812yes—
448C2×C4C22×C22⋊D6112yes—
449C8—2444no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—81no—
457C22C2×#1451974611yescharacteristic
458C22C22×#1175820211yescharacteristic
459C22C2×#1451980811yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—44no—
465C4—41no—
466C4—44no—
467C4—4881no—
468C4—4881no—
469C4—2442no—
470C4—2442no—
471C4#1683991812yes—
472C4#1683991812yes—
473C2—21no—
474C2—21no—
475C2—42no—
476C2C2×#1683991811yescharacteristic
477C2#2145516811yescharacteristic
478C2#2145521111yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#18125
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21455854C2semidirect
3. #21444861C2semidirect
5. #21444793C2semidirect
7. #21444318C2semidirect
115. C22×C4×C61Q16semidirect
280. C61#18125semidirect
361. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214558542
#214490872
#214448612
#214448442
#214447932
#214447762
#214443182
#214441332
C61×#34582
#1812561