Finite Group Databasebeta

← #30611623

Subgroups — #30611623

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840783C212yesmaximal, complemented
2#21461310C212yesmaximal, complemented
3#21461307C212yesmaximal, complemented
4#21461297C214yesmaximal, complemented
5#21461295C214yesmaximal, complemented
6#21459111C211yesmaximal, characteristic
7#16840783—22no—
8#16840783—22no—
9#16840783C2214yes—
10#16840783C2214yes—
11C2×C4⋊#92420—22no—
12C2×C22⋊#92420—22no—
13C2×#14519832C2214yes—
14#16840619C2218yescomplemented
15#16840618C2214yescomplemented
16#16840617C2214yescomplemented
17#16840315C2211yescharacteristic
18#16840314C412yes—
19#16840308C2211yescharacteristic
20#16840307C412yes—
21#16840298C2212yes—
22#16840293C2212yes—
23#16840292—22no—
24#16840289—22no—
25#16840273C411yescharacteristic
26#16840268—22no—
27#41×Dic61C411yescharacteristic
28#16840230C411yescharacteristic
29C4×#11758224—22no—
30C22×C4×Dic61C2211yescharacteristic
31C61×#525C411yescharacteristic
32C22×#92420—24no—
33C4⋊#92420—24no—
34C4⋊#92420—24no—
35C22⋊#92420—24no—
36C22⋊#92420—24no—
37#14519832—24no—
38#14519832C2318yes—
39C2×#11758222C2×C411yescharacteristic
40C2×#11758224—22no—
41C23×Dic61D412yes—
42#14519776C2×C412yes—
43#14519775C2×C411yescharacteristic
44#14519773C2×C411yescharacteristic
45#14519772—22no—
46#14519770—22no—
47#14519768C2312yes—
48#14519767—22no—
49#14519766C2311yescharacteristic
50#14519764C2311yescharacteristic
51#14519763—22no—
52#14519761—22no—
53#14519760—21no—
54#14519758—22no—
55#14519756C2×C412yes—
56#14519754—22no—
57#14519754—22no—
58#14519751C2×C412yes—
59#14519749—22no—
60#14519749—22no—
61#14519746—21no—
62Q8×Dic61C2×C412yes—
63C4×Dic122—22no—
64#14519742—22no—
65#14519742C2×C412yes—
66D4×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61C2×C412yes—
69#14519739C2×C412yes—
70C4×C22⋊D61—22no—
71C4×C22⋊D61—22no—
72#14519737—22no—
73#14519737—22no—
74#14519737—22no—
75#14519736—21no—
76#14519736C2×C411yescharacteristic
77#14519735—22no—
78#14519734—21no—
79#14519733—22no—
80#14519733—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2×C411yescharacteristic
83C2×C4×Dic61C2×C412yes—
84C2×C4×Dic61D412yes—
85C2×C4×Dic61D412yes—
86C2×C4×Dic61D412yes—
87C2×C4×Dic61C2312yes—
88C2×C4×Dic61C2311yescharacteristic
89C42×D61—21no—
90C2×#41×C61C2×C411yescharacteristic
91C23×C4×C61C2×C411yescharacteristic
92#123×C61—21no—
93C22⋊Q8×C61C2×C412yes—
94#121×C61C2×C412yes—
95#120×C61—21no—
96C4×D4×C61—22no—
97#117×C61C2×C411yescharacteristic
98C24×C61C22⋊C411yescharacteristic
99#41×C61—42no—
100C2×Q8×C61C22×C411yescharacteristic
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61C22×C411yescharacteristic
104C22×C4×C61—21no—
105C22×C4×C61—42no—
106C22×C4×C61C22⋊C412yes—
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C412yes—
110C2×C22⋊D61—22no—
111C2×C22⋊D61—21no—
112C2×C22⋊D61—22no—
113C22×Dic61—22no—
114C22×Dic61—44no—
115C22×Dic61C22⋊C412yes—
116C22×Dic61C22⋊C412yes—
117C22×Dic61C22⋊C412yes—
118C22×Dic61C2×D412yes—
119C22×Dic61C2×D412yes—
120C22×Dic61C2×D412yes—
121#11758224—48no—
122#11758222—42no—
123C2×D244C22×C411yescharacteristic
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×Dic122—22no—
127C2×Dic122C22×C411yescharacteristic
128C2×#92420—28no—
129C2×#92420—48no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61C22×C412yes—
132C22⋊C4×C61—21no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—22no—
135C42×C61—21no—
136C22⋊Dic61—22no—
137C22⋊Dic61—21no—
138C22⋊Dic61—22no—
139C22⋊Dic61—22no—
140#11758198—22no—
141#11758198—22no—
142#11758198—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61C22×C412yes—
146#11758196—22no—
147#11758196—22no—
148#11758196—22no—
149#11758196—22no—
150#11758196—22no—
151C4×Dic61—21no—
152C4×Dic61—22no—
153C4×Dic61—22no—
154C4×Dic61C22⋊C411yescharacteristic
155C4×Dic61C22⋊C411yescharacteristic
156C4×Dic61C22⋊C412yes—
157C4×Dic61C22⋊C412yes—
158C4×Dic61C22×C412yes—
159C4×Dic61C22×C411yescharacteristic
160C4×Dic61C2×D412yes—
161C4×Dic61C2×D412yes—
162C4×Dic61C2×D412yes—
163C4×Dic61C2411yescharacteristic
164C23×C61—24no—
165C23×C61—21no—
166C23×C61—41no—
167C23×C61C2×C22⋊C411yescharacteristic
168C23×C61C2×C22⋊C412yes—
169C22×D61—21no—
170Q8×C61—41no—
171D4×C61—41no—
172D4×C61—42no—
173D4×C61—44no—
174C2×C4×C61—22no—
175C2×C4×C61—24no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61C2×C22⋊C412yes—
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61C23×C411yescharacteristic
184C22⋊D61—44no—
185C22⋊D61—42no—
186C22⋊D61—44no—
187C2×Dic61—24no—
188C2×Dic61—24no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—44no—
196C2×Dic61C2×C22⋊C412yes—
197C2×Dic61C2×C22⋊C412yes—
198C2×Dic61C2×C22⋊C412yes—
199C2×Dic61C22×D412yes—
200D244—41no—
201C4×D61—44no—
202C4×D61—42no—
203Dic122—44no—
204Dic122—41no—
205#92420—48no—
206#17678—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—21no—
210C22×C61—22no—
211C22×C61—44no—
212C22×C61—41no—
213C22×C61—42no—
214C22×C61#42114yes—
215C22×C61C22×C22⋊C411yescharacteristic
216C2×D61—41no—
217C2×D61—42no—
218C4×C61—21no—
219C4×C61—22no—
220C4×C61—41no—
221C4×C61—42no—
222C4×C61#41312yes—
223Dic61—22no—
224Dic61—22no—
225Dic61—22no—
226Dic61—22no—
227Dic61—41no—
228Dic61—42no—
229#3444—611no—
230C2×#432—612no—
231#1537—612no—
232#1536—612no—
233#1532—614no—
234#1532—614no—
235C2×C61—24no—
236C2×C61—41no—
237C2×C61—81no—
238C2×C61C2×#41311yescharacteristic
239C2×C61C2×#42112yes—
240D61—81no—
241#533—611no—
242#533—612no—
243#532—612no—
244#532—611no—
245#529—612no—
246#529—612no—
247#529—1222no—
248#529—1222no—
249#525—611no—
250#525—1222no—
251#525#692311yescharacteristic
252C4×#41—611no—
253C4×#41—611no—
254C4×#41—1222no—
255C22×C42—611no—
256#432—614no—
257#432—614no—
258#432—1222no—
259#432—1222no—
260C2×C4⋊C8—1222no—
261C2×C22⋊C8—1222no—
262C2×#97—614no—
263#330—618no—
264#329—614no—
265#329—614no—
266C61#1767811yescharacteristic, complemented, Sylow
267C2×#41—611no—
268C2×#41—1222no—
269C2×#41C2×#692311yescharacteristic
270C23×C4—612no—
271C23×C4C2×#692311yescharacteristic
272C22×C8—1224no—
273C4⋊Q8—612no—
274C4⋊Q8—611no—
275#127—611no—
276#126—1222no—
277#126—1222no—
278#125—612no—
279#125—1222no—
280#124—611no—
281#124—611no—
282#124—1222no—
283#123—21no—
284#123—1222no—
285#123—1222no—
286#123—1221no—
287C22⋊Q8—612no—
288C22⋊Q8—1222no—
289C22⋊Q8—1222no—
290C22⋊Q8C2×#692312yes—
291#121—612no—
292#121—1222no—
293#121—1222no—
294#121C2×#692312yes—
295#120—21no—
296#120—1221no—
297C4×Q8—612no—
298C4×Q8—612no—
299C4×Q8—1222no—
300C4×Q8—1222no—
301C4×D4—22no—
302C4×D4—612no—
303C4×D4—612no—
304C4×D4—1222no—
305C4×D4—1222no—
306C4×D4—1222no—
307C4×D4—1222no—
308C4×D4—1222no—
309C4×D4—1222no—
310C4×D4—1222no—
311#117—611no—
312#117—1221no—
313#117—1222no—
314#117—1222no—
315#117—1221no—
316#117—1222no—
317#117C2×#692311yescharacteristic
318C2×C42—612no—
319C2×C42—612no—
320C2×C42—612no—
321C2×C42—611no—
322C2×C42—612no—
323C2×C42—612no—
324C2×C42—611no—
325C2×C42—1221no—
326C2×C42—1221no—
327C4⋊C8—1224no—
328C4⋊C8—1224no—
329C22⋊C8—1224no—
330C22⋊C8—1224no—
331#97—618no—
332#97—1224no—
333C24C22⋊#692311yescharacteristic
334#41—42no—
335#41—2442no—
336#41—2448no—
337C2×Q8—611no—
338C2×Q8—1222no—
339C2×Q8C22×#692311yescharacteristic
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—611no—
343C2×D4—1222no—
344C2×D4—1222no—
345C2×D4—1221no—
346C2×D4C22×#692311yescharacteristic
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—612no—
353C22×C4—612no—
354C22×C4—612no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1221no—
358C22×C4—2444no—
359C22×C4C22⋊#692312yes—
360C22×C4C22⋊#692311yescharacteristic
361C22×C4C22×#692311yescharacteristic
362C22×C4C22×#692312yes—
363C2×C8—1228no—
364C2×C8—2448no—
365C4⋊C4—22no—
366C4⋊C4—612no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4C22×#692312yes—
375C22⋊C4—22no—
376C22⋊C4—21no—
377C22⋊C4—22no—
378C22⋊C4—1222no—
379C22⋊C4—1222no—
380C22⋊C4—1222no—
381C22⋊C4—1222no—
382C22⋊C4—1221no—
383C22⋊C4—1222no—
384C22⋊C4—1222no—
385C42—21no—
386C42—612no—
387C42—612no—
388C42—612no—
389C42—611no—
390C42—611no—
391C42—612no—
392C42—612no—
393C42—611no—
394C42—612no—
395C42—611no—
396C42—1222no—
397C42—1222no—
398C42—1221no—
399C23—24no—
400C23—21no—
401C23—41no—
402C23—1221no—
403C23C2×C22⋊#692311yescharacteristic
404C23C2×C22⋊#692312yes—
405Q8—41no—
406Q8—2441no—
407Q8—2444no—
408D4—41no—
409D4—42no—
410D4—44no—
411D4—2441no—
412D4—2444no—
413D4—2442no—
414D4—2444no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—612no—
423C2×C4—612no—
424C2×C4—612no—
425C2×C4—612no—
426C2×C4—1224no—
427C2×C4—1224no—
428C2×C4—1222no—
429C2×C4—1221no—
430C2×C4—2444no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2444no—
436C2×C4—2442no—
437C2×C4C2×C22⋊#692312yes—
438C2×C4C2×C22⋊#692311yescharacteristic
439C2×C4C23×#692311yescharacteristic
440C8—2448no—
441C22—22no—
442C22—22no—
443C22—21no—
444C22—22no—
445C22—44no—
446C22—41no—
447C22—42no—
448C22—2441no—
449C22—2442no—
450C22#1684075914yes—
451C22C22×C22⋊#692311yescharacteristic
452C4—22no—
453C4—21no—
454C4—41no—
455C4—42no—
456C4—1222no—
457C4—1222no—
458C4—1222no—
459C4—1222no—
460C4—2441no—
461C4—2442no—
462C4#1684074912yes—
463C2—24no—
464C2—41no—
465C2—81no—
466C2—4881no—
467C2C2×#1684074911yescharacteristic
468C2C2×#1684075912yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#525C4
socleC22×C61C22×C22⋊C4
prime radical#21459111C2
cosocleC4×Dic61C24
hypercenter#525#6923
hypocenterC61#17678
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16840783C2semidirect
2. #21461310C2semidirect
3. #21461307C2semidirect
4. #21461297C2semidirect
5. #21461295C2semidirect
14. #16840619C22semidirect
15. #16840618C22semidirect
16. #16840617C22semidirect
266. C61#17678semidirect

Maximal subgroups

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

subgroupindex
C2×#168407832
#214613102
#214613072
#214612972
#214612952
#214591112
#1767861