Finite Group Databasebeta

← #30620421

Subgroups — #30620421

Order 15616.

1114 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#21462328C212yesmaximal, complemented
2#21462009C214yesmaximal, complemented
3#21462008C212yesmaximal, complemented
4#21461681C212yesmaximal
5#21461679C211yesmaximal, characteristic
6#21461566C212yesmaximal
7C4×#14519835C211yesmaximal, characteristic
8C2×#16840470C211yesmaximal, characteristic
9C2×#14519835C2211yescharacteristic
10C2×#14519835C2212yes—
11C2×#14519835C2211yescharacteristic
12#16840713—22no—
13#16840711—24no—
14#16840704C2212yes—
15#16840704C2214yes—
16C4⋊C4×#6923—22no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2214yes—
19#16840640C2212yes—
20#16840640C2214yes—
21#16840640C2212yes—
22#16840640C2214yes—
23#16840470—24no—
24#16840470—22no—
25#16840470—22no—
26C2×#14519776C411yescharacteristic, complemented
27C2×#14519767C412yescomplemented
28C2×#14519757C2212yes—
29C2×#14519754C412yescomplemented
30C2×#14519753C2211yescharacteristic
31C2×Q8×Dic61C411yescharacteristic, complemented
32C2×C4×C22⋊D61C2211yescharacteristic
33C2×#14519737C2212yes—
34C22×C4⋊Dic61C411yescharacteristic, complemented
35C2×C4⋊C4×D61C2211yescharacteristic
36C2×C22⋊Q8×C61C411yescharacteristic, complemented
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C22⋊#6923C2314yes—
40C2×C22⋊#6923C2312yes—
41C2×C22⋊#6923C2312yes—
42C2×C4×#6923—22no—
43C2×C4×#6923—24no—
44#14519835—22no—
45#14519835—24no—
46#14519835Q814yescomplemented
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—24no—
51C22×Dic122C2×C412yes—
52C22×C22⋊D61C2311yescharacteristic
53C23×Dic61C2×C412yes—
54C22×C4×D61C2311yescharacteristic
55C22×C4×D61C2312yes—
56#14519776—22no—
57#14519776—22no—
58#14519767—24no—
59#14519767—24no—
60#14519757—24no—
61#14519757—24no—
62#14519754—24no—
63#14519754—24no—
64#14519753—22no—
65#14519753—22no—
66Q8×Dic61—22no—
67Q8×Dic61—21no—
68Q8×Dic61—21no—
69C4×C22⋊D61—22no—
70C4×C22⋊D61—22no—
71#14519737—24no—
72#14519737—24no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×C4⋊Dic61C2×C412yes—
78C2×C4⋊Dic61C2×C412yes—
79C2×C4⋊Dic61C2×C411yescharacteristic
80C2×#11758196C2×C412yes—
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2×C412yes—
83C2×#11758196C2×C412yes—
84C4⋊C4×D61D414yes—
85C4⋊C4×D61D412yes—
86C4⋊C4×D61D412yes—
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×#11758198C2312yes—
89C2×#11758198C2311yescharacteristic
90C2×C4×Dic61C2×C412yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C22×Q8×C61C2×C411yescharacteristic
93C23×C4×C61C2×C411yescharacteristic
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—22no—
96C2×C4⋊C4×C61C2×C412yes—
97C2×C4⋊C4×C61C2×C411yescharacteristic
98C2×C22⋊C4×C61C2×C412yes—
99C24×C61C22×C411yescharacteristic
100C23×D61C2411yescharacteristic
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—24no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—42no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—42no—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C412yes—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C412yes—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—22no—
116C2×C22⋊D61C2×Q812yes—
117C2×C22⋊D61#4112yes—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—82no—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C412yes—
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C411yescharacteristic
126C2×C4×D61—42no—
127C2×C4×D61—82no—
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D414yes—
131C2×C4×D61C2×D414yes—
132C2×Dic122—24no—
133C2×Dic122—82no—
134C22⋊#6923—22no—
135C22⋊#6923—22no—
136C22⋊#6923—24no—
137C22⋊#6923—24no—
138C22⋊#6923—22no—
139C22⋊#6923C2×Q814yes—
140C4⋊#6923—44no—
141C4⋊#6923—84no—
142C4×#6923—44no—
143C4×#6923—84no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61C22⋊C412yes—
146C4⋊C4×C61C22⋊C412yes—
147C22⋊C4×C61—24no—
148C22⋊Dic61—22no—
149#11758198—24no—
150#11758198—22no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61C22⋊C412yes—
158C4⋊Dic61C22⋊C411yescharacteristic
159C4⋊Dic61C22⋊C411yescharacteristic
160#11758196—24no—
161#11758196—22no—
162#11758196—24no—
163#11758196C22⋊C414yes—
164#11758196C22⋊C414yes—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61C4×Q812yes—
174C23×C61C23×C411yescharacteristic
175C22×D61—41no—
176C22×D61—41no—
177C22×D61C22×D412yes—
178C22×D61C22×Q811yescharacteristic
179C22×D61C2×#4111yescharacteristic
180C22×D612−1+412yes—
181C2×#6923—24no—
182C2×#6923—42no—
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—44no—
186C2×#6923—44no—
187C2×#6923—42no—
188C2×#6923—42no—
189C2×#6923—44no—
190Q8×C61—42no—
191Q8×C61—42no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C411yescharacteristic
201C2×C4×C61C2×C22⋊C414yes—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C22⋊D61—24no—
204C22⋊D61—24no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—82no—
214C2×Dic61—82no—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C4×Q812yes—
219C4×D61—24no—
220C4×D61—44no—
221C4×D61—44no—
222C4×D61—42no—
223Dic122—84no—
224#22513—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—22no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61C2×C4×Q811yescharacteristic
234C22×C61#51911yescharacteristic
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—42no—
240C2×D61#54812yes—
241C2×D61D4×Q812yes—
242#6923—44no—
243#6923—82no—
244#6923—84no—
245C4×C61—21no—
246C4×C61—24no—
247C4×C61—41no—
248C4×C61—42no—
249C4×C61#41012yes—
250Dic61—22no—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—41no—
255Dic61—42no—
256C2×#554—611no—
257#2345—612no—
258#1996—612no—
259#1996—614no—
260#1799—611no—
261#1799—612no—
262C4×#99—611no—
263C4×#99—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—42no—
267C2×C61Q8×C22⋊C411yescharacteristic
268C2×C61C2×#41011yescharacteristic
269C2×C61#288311yescharacteristic
270D61—22no—
271D61—22no—
272#554—1222no—
273#554—1224no—
274#554—1222no—
275C2×C4⋊Q8—611no—
276C2×#125—612no—
277C2×C22⋊Q8—611no—
278C2×C22⋊Q8—612no—
279C2×C22⋊Q8—612no—
280C2×C22⋊Q8#692311yescharacteristic, complemented
281C2×C4×Q8—611no—
282C2×C4×D4—611no—
283C2×C4×D4—612no—
284C22×C4⋊C4—611no—
285C22×C4⋊C4—611no—
286C2×#99—611no—
287C2×#99—612no—
288C2×#99—611no—
289#389—1222no—
290#387—1224no—
291#380—614no—
292#380—612no—
293C4×C4⋊C4—1222no—
294C4×C22⋊C4—612no—
295C4×C22⋊C4—614no—
296#342—612no—
297#342—614no—
298#342—612no—
299#342—614no—
300C61#2251311yescharacteristic, complemented, Sylow
301C22×Q8—612no—
302C22×Q8C2×#692311yescharacteristic
303C22×D4—611no—
304C23×C4—612no—
305C23×C4—611no—
306C23×C4—612no—
307C23×C4C2×#692311yescharacteristic
308C4⋊Q8—1222no—
309C4⋊Q8—1222no—
310#125—1224no—
311#125—1224no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C22⋊Q8—1224no—
317C22⋊Q8—1224no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C4×Q8—1222no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×D4—1224no—
324C4×D4—1224no—
325C4×D4—1222no—
326C4×D4—1222no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—614no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1224no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4C2×#692312yes—
344C2×C4⋊C4C2×#692311yescharacteristic
345C2×C22⋊C4—612no—
346C2×C22⋊C4—611no—
347C2×C22⋊C4—611no—
348C2×C22⋊C4—612no—
349C2×C22⋊C4—614no—
350C2×C22⋊C4—612no—
351C2×C22⋊C4C2×#692312yes—
352C2×C42—612no—
353C2×C42—611no—
354C2×C42—1224no—
355C2×C42—1222no—
356#99—614no—
357#99—1222no—
358#99—1224no—
359#95—1222no—
360#95—1224no—
361#95—1224no—
362#95—1222no—
363C24—611no—
364C24C22×#692311yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—42no—
368C2×Q8—1224no—
369C2×Q8—4882no—
370C2×D4—612no—
371C2×D4—612no—
372C2×D4—1222no—
373C2×D4—1222no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—611no—
378C22×C4—612no—
379C22×C4—611no—
380C22×C4—612no—
381C22×C4—612no—
382C22×C4—614no—
383C22×C4—614no—
384C22×C4—611no—
385C22×C4—612no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1224no—
390C22×C4—1222no—
391C22×C4—4882no—
392C22×C4—4882no—
393C22×C4—2442no—
394C22×C4C22×#692311yescharacteristic
395C22×C4C22×#692312yes—
396C22×C4C22×#692311yescharacteristic
397C22×C4C22×#692312yes—
398C4⋊C4—24no—
399C4⋊C4—612no—
400C4⋊C4—611no—
401C4⋊C4—611no—
402C4⋊C4—614no—
403C4⋊C4—614no—
404C4⋊C4—1222no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1224no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1224no—
413C4⋊C4—4884no—
414C4⋊C4—2444no—
415C4⋊C4C22⋊#692312yes—
416C4⋊C4C22⋊#692312yes—
417C22⋊C4—24no—
418C22⋊C4—614no—
419C22⋊C4—1224no—
420C22⋊C4—1222no—
421C22⋊C4—1222no—
422C22⋊C4—1222no—
423C22⋊C4—1224no—
424C22⋊C4—1222no—
425C22⋊C4—1224no—
426C22⋊C4—1222no—
427C42—1222no—
428C42—1222no—
429C42—1221no—
430C42—1221no—
431C42—4884no—
432C42—2444no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—612no—
438C23—611no—
439C23—611no—
440C23—612no—
441C23—2441no—
442C23—2441no—
443C23Q8×#692312yes—
444C23C23×#692311yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—4884no—
448D4—1224no—
449D4—1224no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—612no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—612no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1221no—
467C2×C4—1222no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4—2444no—
471C2×C4—2444no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2444no—
476C2×C4—2442no—
477C2×C4—2444no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2444no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4C2×C22⋊#692311yescharacteristic
486C2×C4C2×C22⋊#692314yes—
487C2×C4C2×C22⋊#692311yescharacteristic
488C22—21no—
489C22—21no—
490C22—22no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—612no—
496C22—612no—
497C22—1222no—
498C22—1222no—
499C22—2441no—
500C22—2441no—
501C22—2442no—
502C22C22×C22⋊#692311yescharacteristic
503C22C2×Q8×#692311yescharacteristic
504C22#1684083011yescharacteristic
505C4—21no—
506C4—24no—
507C4—41no—
508C4—42no—
509C4—1222no—
510C4—1222no—
511C4—1222no—
512C4—1222no—
513C4—4884no—
514C4—4882no—
515C4—2444no—
516C4—2442no—
517C4—2441no—
518C4#1684074212yes—
519C2—21no—
520C2—21no—
521C2—42no—
522C2—1222no—
523C2—1222no—
524C2Q8×C22⋊#692311yescharacteristic
525C2C2×#1684074211yescharacteristic
526C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#22513
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462328C2semidirect
2. #21462009C2semidirect
3. #21462008C2semidirect
26. C2×#14519776C4semidirect
27. C2×#14519767C4semidirect
29. C2×#14519754C4semidirect
31. C2×Q8×Dic61C4semidirect
34. C22×C4⋊Dic61C4semidirect
36. C2×C22⋊Q8×C61C4semidirect
46. #14519835Q8semidirect
280. C2×C22⋊Q8#6923semidirect
300. C61#22513semidirect

Maximal subgroups

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

subgroupindex
#214623282
#214620092
#214620082
#214616812
#214616792
#214615662
C4×#145198352
C2×#168404702
#2251361