Finite Group Databasebeta

← #30610211

Subgroups — #30610211

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic
2D61⋊#432C211yesmaximal, characteristic
3#21461097C214yesmaximal
4#21461092C214yesmaximal, complemented
5#21461082C214yesmaximal, complemented
6C2×#16840317C211yesmaximal, characteristic
7#16840785—21no—
8#16840785—21no—
9#16840784—21no—
10#16840783C2212yes—
11#16840782C2212yes—
12#16840782C2212yes—
13#16840780—21no—
14#16840779C2212yes—
15#16840767—21no—
16D61⋊(C4⋊C8)—21no—
17#16840733C2212yes—
18D61⋊(C22⋊C8)C2212yes—
19C4×#11758213—21no—
20#16840721—21no—
21#16840607C22116yes—
22#16840317—21no—
23#16840317—21no—
24#16840317—21no—
25#16840317—21no—
26#16840317—21no—
27#16840317—21no—
28#16840317—21no—
29#16840317—21no—
30C2×#14519777C411yescharacteristic
31C2×#14519768C2212yes—
32C2×#14519757C2212yes—
33C2×#14519753C412yes—
34C2×Q8×Dic61C2212yes—
35C2×#14519742C412yes—
36C2×#14519735C412yes—
37C2×C42×D61C2211yescharacteristic
38C2×C4⋊Q8×C61C411yescharacteristic
39C2×#11758213—22no—
40C2×D61⋊C8—22no—
41#14519844—22no—
42#14519844—22no—
43#14519844—22no—
44C4⋊#92420—22no—
45#14519834—22no—
46#14519834—22no—
47C22⋊#92420C2314yes—
48C22⋊#92420C2314yes—
49#14519832—22no—
50#14519832—22no—
51#14519832—22no—
52C4×#92420—22no—
53C22×Dic122C2×C412yes—
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519768—22no—
65#14519757—22no—
66#14519757—22no—
67#14519757—22no—
68#14519757—22no—
69#14519753—22no—
70#14519753—22no—
71#14519753—22no—
72#14519753—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84#14519735—22no—
85C2×C4⋊Dic61C2×C412yes—
86C2×C4⋊Dic61C2×C412yes—
87C2×C4⋊Dic61C2314yes—
88C2×#11758196C2×C414yes—
89C2×#11758198C2×C414yes—
90C2×C4×Dic61C2×C414yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C22×Q8×C61C2×C412yes—
103C4⋊Q8×C61—21no—
104C4⋊Q8×C61—21no—
105C4⋊Q8×C61—21no—
106C4⋊Q8×C61—21no—
107C2×C4⋊C4×C61C2×C412yes—
108C2×C4⋊C4×C61C2×C412yes—
109C2×C42×C61C2×C411yescharacteristic
110C23×D61C22×C411yescharacteristic
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—44no—
114C22×C4×C61C22×C414yes—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×Dic61C22×C411yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C414yes—
121C22×Dic61C2411yescharacteristic
122C2×C4×D61—24no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61C22⋊C411yescharacteristic
126C2×C4×D61C22⋊C411yescharacteristic
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C22⋊C411yescharacteristic
133C2×C4×D61C2×D411yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—44no—
140C2×#92420—24no—
141C2×#92420—24no—
142#11758213—44no—
143D61⋊C8—44no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C42×C61C22⋊C411yescharacteristic
149C42×C61C22⋊C411yescharacteristic
150C42×C61C22⋊C411yescharacteristic
151C42×C61C22⋊C411yescharacteristic
152#11758198—24no—
153#11758198—24no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—24no—
159C4⋊Dic61—24no—
160#11758196—24no—
161#11758196—24no—
162C4×Dic61—24no—
163C4×Dic61—24no—
164C4×Dic61C22⋊C411yescharacteristic
165C4×Dic61C22⋊C411yescharacteristic
166C4×Dic61C22⋊C411yescharacteristic
167C4×Dic61C22⋊C411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C4×Dic61C2×D411yescharacteristic
172C4×Dic61C2×D411yescharacteristic
173C4×Dic61C2×D411yescharacteristic
174C4×Dic61C2×D411yescharacteristic
175C4×Dic61C2×D411yescharacteristic
176C23×C61C23×C411yescharacteristic
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—41no—
180C22×D61C2×C22⋊C411yescharacteristic
181C22×D61C2×C22⋊C411yescharacteristic
182Q8×C61—44no—
183Q8×C61—44no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—28no—
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61C2×C22⋊C411yescharacteristic
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—24no—
198C2×Dic61—41no—
199C2×Dic61—44no—
200C2×Dic61C2×C22⋊C411yescharacteristic
201C2×Dic61C2×C22⋊C411yescharacteristic
202C2×Dic61C2×C22⋊C411yescharacteristic
203C2×Dic61C2×C22⋊C411yescharacteristic
204C2×Dic61C22×D411yescharacteristic
205C2×Dic61C22×D411yescharacteristic
206C4×D61—22no—
207C4×D61—22no—
208C4×D61—22no—
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—41no—
212C4×D61—41no—
213C4×D61—41no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61#10114yes—
218Dic122—44no—
219Dic122—44no—
220#92420—44no—
221#92420—44no—
222#17404—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61C22×C22⋊C411yescharacteristic
226C22×C61#56812yes—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—44no—
233C2×D61C2×#10112yes—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—44no—
239C4×C61#41012yes—
240C4×C61C2×#10112yes—
241Dic61—21no—
242Dic61—21no—
243Dic61—21no—
244Dic61—21no—
245Dic61—21no—
246Dic61—44no—
247Dic61C2×#10112yes—
248C2×#533—611no—
249#2920—611no—
250C2×#432—611no—
251#1472—614no—
252#1470—614no—
253#1467—614no—
254C2×C61—22no—
255C2×C61#287911yescharacteristic
256C2×C61C2×#41011yescharacteristic
257C2×C61C22×#10111yescharacteristic
258D61—22no—
259D61—41no—
260#533—1221no—
261#533—1221no—
262#533—1221no—
263#533—1221no—
264#533—1221no—
265#533—1221no—
266#533—1221no—
267#533—1221no—
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8#692311yescharacteristic
270C2×#125—612no—
271C2×C22⋊Q8—612no—
272C2×C22⋊Q8—612no—
273C2×C4×Q8—612no—
274C2×C4×Q8—612no—
275C2×#117—612no—
276C22×C42—611no—
277#432—612no—
278#432—612no—
279#432—612no—
280#432—1221no—
281#432—1221no—
282#432—1221no—
283#431—612no—
284#431—1221no—
285#423—1221no—
286C2×C4⋊C8—1221no—
287#407—612no—
288C2×C22⋊C8—612no—
289C4×M16—1221no—
290C2×#97—1221no—
291#324—6116no—
292C61#1740411yescharacteristic, complemented, Sylow
293C22×Q8—612no—
294C22×Q8C2×#692312yes—
295C23×C4—611no—
296C23×C4—611no—
297C23×C4—611no—
298C2×M16—1222no—
299C22×C8—1222no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8—1221no—
307C4⋊Q8—1221no—
308#125—1222no—
309#125—1222no—
310#125—1222no—
311#125—1222no—
312C22⋊Q8—1222no—
313C22⋊Q8—1222no—
314C22⋊Q8—1222no—
315C22⋊Q8—1222no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C4×Q8—1222no—
321C4×Q8—1222no—
322C4×Q8—1222no—
323C4×Q8—1222no—
324C4×Q8—1222no—
325C4×Q8—1222no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328#117—1222no—
329#117—1222no—
330#117—1222no—
331#117—1222no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—614no—
334C2×C4⋊C4—614no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4C2×#692312yes—
337C2×C4⋊C4C2×#692312yes—
338C2×C22⋊C4—614no—
339C2×C42—611no—
340C2×C42—611no—
341C2×C42—611no—
342C2×C42—611no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—614no—
351C2×C42C2×#692311yescharacteristic
352C4⋊C8—1222no—
353C4⋊C8—1222no—
354C4⋊C8—1222no—
355C4⋊C8—1222no—
356C22⋊C8—614no—
357C22⋊C8—614no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—1222no—
361#97—1222no—
362#97—1222no—
363C4×C8—1222no—
364C24—611no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—44no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8—2444no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—614no—
387C22×C4—1224no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4C22×#692314yes—
391C22×C4C22×#692311yescharacteristic
392C22×C4C22×#692311yescharacteristic
393C22×C4C22×#692311yescharacteristic
394M16—2444no—
395C2×C8—1224no—
396C2×C8—1224no—
397C2×C8—2444no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C22⋊C4—1224no—
411C22⋊C4—1224no—
412C42—611no—
413C42—611no—
414C42—611no—
415C42—611no—
416C42—611no—
417C42—611no—
418C42—611no—
419C42—611no—
420C42—611no—
421C42—611no—
422C42—611no—
423C42—611no—
424C42—1224no—
425C42—1224no—
426C42C22⋊#692311yescharacteristic
427C42C22⋊#692311yescharacteristic
428C42C22⋊#692311yescharacteristic
429C42C22⋊#692311yescharacteristic
430C23—611no—
431C23—611no—
432C23—1222no—
433C23—1222no—
434C23—2441no—
435C23C23×#692311yescharacteristic
436Q8—44no—
437Q8—44no—
438Q8—2444no—
439Q8—2444no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—611no—
446C2×C4—614no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—611no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1224no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2444no—
468C2×C4C2×C22⋊#692311yescharacteristic
469C2×C4C2×C22⋊#692311yescharacteristic
470C2×C4C2×C22⋊#692311yescharacteristic
471C2×C4C2×C22⋊#692311yescharacteristic
472C2×C4C2×C22⋊#692311yescharacteristic
473C2×C4C2×C22⋊#692311yescharacteristic
474C8—2444no—
475C8—2444no—
476C22—21no—
477C22—21no—
478C22—612no—
479C22—1221no—
480C22—1221no—
481C22—1221no—
482C22—1221no—
483C22—1221no—
484C22—2444no—
485C22C22×C22⋊#692311yescharacteristic
486C22#1684084012yes—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—44no—
492C4—612no—
493C4—1221no—
494C4—1221no—
495C4—1221no—
496C4—1221no—
497C4—1221no—
498C4—2444no—
499C4#1684074312yes—
500C4D61⋊#10112yes—
501C2—22no—
502C2—1222no—
503C2—2441no—
504C2#2146240511yescharacteristic
505C2C2×#1684074311yescharacteristic
506C2C2×D61⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#17404
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #21461092C2semidirect
5. #21461082C2semidirect
292. C61#17404semidirect

Maximal subgroups

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

subgroupindex
#214624982
D61⋊#4322
#214610972
#214610922
#214610822
C2×#168403172
#1740461