Finite Group Databasebeta

← #30617882

Subgroups — #30617882

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462266C211yesmaximal, characteristic
2#21462266C211yesmaximal, characteristic
3#21462009C214yesmaximal
4#21462008C214yesmaximal
5#21461717C214yesmaximal
6C2×#16840317C211yesmaximal, characteristic
7#16840707—21no—
8#16840707—21no—
9#16840706—21no—
10#16840706—21no—
11#16840705C2212yes—
12#16840705C2212yes—
13#16840704C2212yes—
14#16840704C2212yes—
15C4×C4⋊#6923—21no—
16C4×C4⋊#6923—21no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2212yes—
19#16840698—21no—
20#16840698—21no—
21#16840640C22116yes—
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, complemented
31C2×#14519768C412yescomplemented
32C2×#14519757C2212yes—
33C2×#14519753C2212yes—
34C2×Q8×Dic61C412yescomplemented
35C2×#14519742C412yescomplemented
36C2×#14519735C2212yes—
37C2×C42×D61C2211yescharacteristic
38C2×C4⋊Q8×C61C411yescharacteristic, complemented
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—22no—
41C2×C22⋊#6923C2314yes—
42C2×C22⋊#6923C2314yes—
43C2×C4×#6923—22no—
44C2×C4×#6923—22no—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—22no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53C22×Dic122C2×C412yes—
54C22×C4×D61C2311yescharacteristic
55C22×C4×D61C2311yescharacteristic
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⋊Dic61C2×C414yes—
88C2×#11758196C2×C414yes—
89C2×#11758198C2314yes—
90C2×C4×Dic61C2×C414yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2×C411yescharacteristic
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×D61C2411yescharacteristic
111C22×#6923—24no—
112C22×#6923—24no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—44no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C414yes—
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C414yes—
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C411yescharacteristic
124C2×C4×D61—24no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61C2×D411yescharacteristic
128C2×C4×D61C2×D411yescharacteristic
129C2×C4×D61C2×D411yescharacteristic
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61C2×D411yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×D411yescharacteristic
138C2×C4×D61C2×D411yescharacteristic
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—44no—
142C22⋊#6923—28no—
143C22⋊#6923—28no—
144C4⋊#6923—44no—
145C4⋊#6923—44no—
146C4×#6923—44no—
147C4×#6923—44no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C42×C61C22⋊C411yescharacteristic
153C42×C61C22⋊C411yescharacteristic
154C42×C61C22⋊C411yescharacteristic
155C42×C61C22⋊C411yescharacteristic
156#11758198—24no—
157#11758198—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164#11758196—24no—
165#11758196—24no—
166C4×Dic61—24no—
167C4×Dic61—24no—
168C4×Dic61C22⋊C411yescharacteristic
169C4×Dic61C22⋊C411yescharacteristic
170C4×Dic61C22⋊C411yescharacteristic
171C4×Dic61C22⋊C411yescharacteristic
172C4×Dic61C22⋊C411yescharacteristic
173C4×Dic61C22⋊C411yescharacteristic
174C4×Dic61C22⋊C411yescharacteristic
175C4×Dic61C22⋊C411yescharacteristic
176C4×Dic61C22⋊C411yescharacteristic
177C4×Dic61C22⋊C411yescharacteristic
178C4×Dic61C22⋊C411yescharacteristic
179C4×Dic61C22⋊C411yescharacteristic
180C23×C61C23×C411yescharacteristic
181C22×D61—41no—
182C22×D61—41no—
183C22×D61C22×D411yescharacteristic
184C22×D61C22×D411yescharacteristic
185C22×D612−1+414yes—
186C2×#6923—44no—
187C2×#6923—44no—
188C2×#6923—44no—
189C2×#6923—44no—
190C2×#6923—44no—
191C2×#6923—44no—
192Q8×C61—44no—
193Q8×C61—44no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—44no—
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C2×C22⋊C411yescharacteristic
201C2×C4×C61C2×C22⋊C411yescharacteristic
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—24no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—44no—
210C2×Dic61C2×C22⋊C411yescharacteristic
211C2×Dic61C2×C22⋊C411yescharacteristic
212C2×Dic61C2×C22⋊C411yescharacteristic
213C2×Dic61C2×C22⋊C411yescharacteristic
214C2×Dic61C2×C22⋊C411yescharacteristic
215C2×Dic61C2×C22⋊C411yescharacteristic
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—41no—
225C4×D61—41no—
226C4×D61—41no—
227C4×D61—41no—
228Dic122—44no—
229Dic122—44no—
230#21826—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61C22×C22⋊C411yescharacteristic
234C22×C61#51912yes—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—21no—
239C2×D61—44no—
240C2×D61#53612yes—
241C2×D61#53612yes—
242#6923—84no—
243#6923—84no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—44no—
249C4×C61#41012yes—
250C4×C61#41012yes—
251Dic61—21no—
252Dic61—21no—
253Dic61—21no—
254Dic61—21no—
255Dic61—21no—
256Dic61—21no—
257Dic61—44no—
258C2×#533—611no—
259#2283—611no—
260#2283—611no—
261#1996—614no—
262#1996—614no—
263#1820—614no—
264C2×C61—22no—
265C2×C61#231011yescharacteristic
266C2×C61C2×#41011yescharacteristic
267C2×C61C2×#41011yescharacteristic
268D61—22no—
269D61—22no—
270#533—1221no—
271#533—1221no—
272#533—1221no—
273#533—1221no—
274#533—1221no—
275#533—1221no—
276#533—1221no—
277#533—1221no—
278C2×C4⋊Q8—611no—
279C2×C4⋊Q8#692311yescharacteristic, complemented
280C2×#125—612no—
281C2×C22⋊Q8—612no—
282C2×C22⋊Q8—612no—
283C2×C4×Q8—612no—
284C2×C4×Q8—612no—
285C2×#117—612no—
286C22×C42—611no—
287#383—1221no—
288#383—1221no—
289#382—1221no—
290#382—1221no—
291#381—612no—
292#381—612no—
293#380—612no—
294#380—612no—
295C4×C4⋊C4—1221no—
296C4×C4⋊C4—1221no—
297C4×C22⋊C4—612no—
298C4×C22⋊C4—612no—
299#376—1221no—
300#376—1221no—
301#342—6116no—
302C61#2182611yescharacteristic, complemented, Sylow
303C22×Q8—612no—
304C22×Q8C2×#692312yes—
305C23×C4—611no—
306C23×C4—611no—
307C23×C4—611no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—1221no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8—1221no—
316#125—1222no—
317#125—1222no—
318#125—1222no—
319#125—1222no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C22⋊Q8—1222no—
326C22⋊Q8—1222no—
327C22⋊Q8—1222no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×Q8—1222no—
336#117—1222no—
337#117—1222no—
338#117—1222no—
339#117—1222no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—614no—
342C2×C4⋊C4—614no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4C2×#692312yes—
347C2×C4⋊C4C2×#692312yes—
348C2×C22⋊C4—614no—
349C2×C22⋊C4—614no—
350C2×C22⋊C4—614no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—611no—
362C2×C42—614no—
363C2×C42—1222no—
364C2×C42—1222no—
365C2×C42C2×#692311yescharacteristic
366#95—1222no—
367#95—1222no—
368#95—1222no—
369#95—1222no—
370#95—1222no—
371#95—1222no—
372#95—1222no—
373#95—1222no—
374C24—611no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—44no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—2444no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—614no—
397C22×C4—1224no—
398C22×C4—1224no—
399C22×C4—1224no—
400C22×C4—2442no—
401C22×C4—2442no—
402C22×C4C22×#692311yescharacteristic
403C22×C4C22×#692311yescharacteristic
404C22×C4C22×#692311yescharacteristic
405C22×C4C22×#692314yes—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—1224no—
411C4⋊C4—1224no—
412C4⋊C4—1224no—
413C4⋊C4—1224no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—2444no—
419C4⋊C4—2444no—
420C22⋊C4—1224no—
421C22⋊C4—1224no—
422C22⋊C4—1228no—
423C22⋊C4—1228no—
424C42—611no—
425C42—611no—
426C42—611no—
427C42—611no—
428C42—611no—
429C42—611no—
430C42—611no—
431C42—611no—
432C42—611no—
433C42—611no—
434C42—611no—
435C42—611no—
436C42—1224no—
437C42—1224no—
438C42—2444no—
439C42—2444no—
440C42C22⋊#692311yescharacteristic
441C42C22⋊#692311yescharacteristic
442C42C22⋊#692311yescharacteristic
443C42C22⋊#692311yescharacteristic
444C23—611no—
445C23—611no—
446C23—614no—
447C23—2441no—
448C23—2441no—
449C23C23×#692311yescharacteristic
450Q8—44no—
451Q8—44no—
452Q8—2444no—
453Q8—2444no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—44no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—611no—
464C2×C4—611no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1224no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2444no—
482C2×C4—2444no—
483C2×C4—2444no—
484C2×C4—2444no—
485C2×C4—2444no—
486C2×C4—2444no—
487C2×C4—2444no—
488C2×C4C2×C22⋊#692311yescharacteristic
489C2×C4C2×C22⋊#692311yescharacteristic
490C2×C4C2×C22⋊#692311yescharacteristic
491C2×C4C2×C22⋊#692311yescharacteristic
492C2×C4C2×C22⋊#692311yescharacteristic
493C2×C4C2×C22⋊#692311yescharacteristic
494C22—21no—
495C22—21no—
496C22—612no—
497C22—612no—
498C22—1221no—
499C22—1221no—
500C22—1221no—
501C22—1221no—
502C22—2444no—
503C22C22×C22⋊#692311yescharacteristic
504C22#1684083012yes—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—44no—
510C4—1221no—
511C4—1221no—
512C4—1221no—
513C4—1221no—
514C4—1221no—
515C4—1221no—
516C4—4884no—
517C4—4884no—
518C4—2444no—
519C4#1684074212yes—
520C4#1684074312yes—
521C2—22no—
522C2—1222no—
523C2—1222no—
524C2#2146229311yescharacteristic
525C2C2×#1684074211yescharacteristic
526C2C2×#1684074311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radical, cosocleC23×D61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#21826
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
30. C2×#14519777C4semidirect
31. C2×#14519768C4semidirect
34. C2×Q8×Dic61C4semidirect
35. C2×#14519742C4semidirect
38. C2×C4⋊Q8×C61C4semidirect
279. C2×C4⋊Q8#6923semidirect
302. C61#21826semidirect

Maximal subgroups

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

subgroupindex
#214622662
#214620092
#214620082
#214617172
C2×#168403172
#2182661