Finite Group Databasebeta

← #30598197

Subgroups — #30598197

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461121C212yesmaximal, complemented
2#21461118C212yesmaximal, complemented
3#21461050C211yesmaximal, characteristic, complemented
4#21461050C211yesmaximal, characteristic, complemented
5C2×#16840245C211yesmaximal, characteristic
6#16840778C2212yes—
7#16840778C2212yes—
8Q8⋊#92420—24no—
9D4⋊#92420—24no—
10#16840607—21no—
11#16840607—21no—
12#16840605—21no—
13#16840605—21no—
14#16840245—24no—
15#16840245—24no—
16C2×Q8×Dic61C411yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61—21no—
19C2×D4×Dic61—21no—
20C2×D4×Dic61C411yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519736—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61—21no—
25C22×C4⋊Dic61C411yescharacteristic
26C22×#41×C61C411yescharacteristic
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C4×#92420—22no—
32C4×#92420—22no—
33C23×Dic61—21no—
34C23×Dic61—21no—
35C23×Dic61—21no—
36Q8×Dic61—22no—
37Q8×Dic61—22no—
38Q8×Dic61—22no—
39Q8×Dic61D414yes—
40D4×Dic61—28no—
41D4×Dic61—28no—
42D4×Dic61—22no—
43D4×Dic61—22no—
44D4×Dic61—22no—
45D4×Dic61D414yes—
46#14519736—28no—
47#14519736—22no—
48#14519736—22no—
49C2×C4⋊Dic61—28no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61D411yescharacteristic
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61C814yescomplemented
73C22×Q8×C61C2×C411yescharacteristic
74C22×D4×C61—21no—
75C22×D4×C61C2×C411yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61C2×C411yescharacteristic
78C24×C61—21no—
79C24×C61C22⋊C411yescharacteristic
80#41×C61—28no—
81#41×C61—28no—
82#41×C61—28no—
83#41×C61—28no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61C22⋊C412yes—
88C2×Q8×C61C2×C812yes—
89C2×Q8×C61M1612yes—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61C22⋊C412yes—
97C2×D4×C61C2×C812yes—
98C2×D4×C61M1612yes—
99C22×C4×C61—24no—
100C22×C4×C61—24no—
101C22×C4×C61—21no—
102C22×C4×C61—24no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C2×C812yes—
109C22×C4×C61M1612yes—
110C22×C4×C61C22×C411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—24no—
114C22×Dic61—24no—
115C22×Dic61—24no—
116C22×Dic61—82no—
117C22×Dic61—82no—
118C22×Dic61—82no—
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×#92420—42no—
124C2×#92420—42no—
125C22⋊Dic61—24no—
126C22⋊Dic61—24no—
127C22⋊Dic61—22no—
128C22⋊Dic61—22no—
129C4⋊Dic61—24no—
130C4⋊Dic61—24no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—22no—
134C4⋊Dic61—24no—
135C4⋊Dic61—22no—
136C4⋊Dic61C22⋊C412yes—
137C4⋊Dic61C22⋊C412yes—
138C4×Dic61—24no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141C4×Dic61—24no—
142C4×Dic61C2×D412yes—
143C4×Dic61C2×D412yes—
144C23×C61—21no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—21no—
148C23×C61—22no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61C22⋊C812yes—
154C23×C61C2×C22⋊C411yescharacteristic
155Q8×C61—22no—
156Q8×C61—22no—
157Q8×C61—22no—
158Q8×C61C22⋊C814yes—
159D4×C61—216no—
160D4×C61—22no—
161D4×C61—22no—
162D4×C61—22no—
163D4×C61C22⋊C814yes—
164C2×C4×C61—216no—
165C2×C4×C61—22no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61C22⋊C812yes—
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C2×C22⋊C411yescharacteristic
177C2×C4×C61C22×C811yescharacteristic
178C2×C4×C61C2×M1611yescharacteristic
179C2×C4×C61C2×M1611yescharacteristic
180C2×C4×C61C2×M1611yescharacteristic
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—84no—
192C2×Dic61—84no—
193C2×Dic61—84no—
194C2×Dic61#12012yes—
195C2×Dic61#12012yes—
196#92420—82no—
197#92420—82no—
198#8324—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—22no—
202C22×C61—24no—
203C22×C61—42no—
204C22×C61—44no—
205C22×C61—42no—
206C22×C61—44no—
207C22×C61—41no—
208C22×C61—42no—
209C22×C61—41no—
210C22×C61—42no—
211C22×C61#37911yescharacteristic
212C22×C61C2×C22⋊C811yescharacteristic
213C22×C61#40711yescharacteristic
214C4×C61—22no—
215C4×C61—24no—
216C4×C61C2×C22⋊C811yescharacteristic
217C4×C61C2×C22⋊C811yescharacteristic
218C4×C61#40711yescharacteristic
219C4×C61#40711yescharacteristic
220Dic61—22no—
221Dic61—22no—
222Dic61—42no—
223Dic61—42no—
224Dic61—41no—
225Dic61—41no—
226C2×#520—611no—
227#1479—612no—
228#1478—612no—
229#1458—611no—
230#1458—611no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—42no—
234C2×C61—44no—
235C2×C61#177911yescharacteristic
236C2×C61#179511yescharacteristic
237C2×C61#179611yescharacteristic
238C22×#41#692311yescharacteristic
239#520—1224no—
240#520—1224no—
241C2×C4×Q8—611no—
242C2×C4×Q8—611no—
243C2×C4×D4—611no—
244C2×C4×D4—611no—
245C2×C4×D4—1221no—
246C2×C4×D4—1221no—
247C2×#117—611no—
248C2×#117—1221no—
249C22×C4⋊C4—611no—
250C22×C4⋊C4—1221no—
251#431—612no—
252#431—612no—
253Q8⋊C8—1224no—
254D4⋊C8—1224no—
255#324—1221no—
256#324—1221no—
257#323—1221no—
258#323—1221no—
259C61#832411yescharacteristic, complemented, Sylow
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41#9242014yescomplemented
266C22×Q8C2×#692311yescharacteristic
267C22×D4—21no—
268C22×D4C2×#692311yescharacteristic
269C23×C4—21no—
270C23×C4—1221no—
271C23×C4—1221no—
272C23×C4—1221no—
273C23×C4C2×#692311yescharacteristic
274C4×Q8—614no—
275C4×Q8—1222no—
276C4×Q8—1222no—
277C4×Q8—1222no—
278C4×D4—614no—
279C4×D4—1222no—
280C4×D4—1222no—
281C4×D4—1228no—
282C4×D4—1228no—
283C4×D4—1222no—
284#117—1222no—
285#117—1222no—
286#117—1228no—
287C2×C4⋊C4—611no—
288C2×C4⋊C4—611no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—611no—
291C2×C4⋊C4—1222no—
292C2×C4⋊C4—1222no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1228no—
296C2×C4⋊C4—1221no—
297C2×C22⋊C4—611no—
298C2×C22⋊C4—611no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C42—611no—
302C2×C42—611no—
303C2×C42—1221no—
304C2×C42—1221no—
305C4⋊C8—1222no—
306C4⋊C8—1222no—
307C22⋊C8—1222no—
308C22⋊C8—1222no—
309C4×C8—1222no—
310C4×C8—1222no—
311C24—21no—
312C24C22⋊#692311yescharacteristic
313#41—28no—
314#41—28no—
315#41—28no—
316#41—28no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C2×#9242012yes—
321C2×Q8#1175821712yes—
322C2×Q8C22⋊#692312yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4C2×#9242012yes—
330C2×D4#1175821712yes—
331C2×D4C22⋊#692312yes—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—611no—
341C22×C4—611no—
342C22×C4—611no—
343C22×C4—611no—
344C22×C4—1224no—
345C22×C4—1224no—
346C22×C4—1224no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—4882no—
350C22×C4—4882no—
351C22×C4—4882no—
352C22×C4C2×#9242012yes—
353C22×C4#1175821712yes—
354C22×C4C22⋊#692311yescharacteristic
355C22×C4C22×#692311yescharacteristic
356C2×C8—2442no—
357C2×C8—2442no—
358C4⋊C4—612no—
359C4⋊C4—612no—
360C4⋊C4—1222no—
361C4⋊C4—1222no—
362C4⋊C4—1224no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C22⋊C4—1224no—
370C22⋊C4—1224no—
371C42—612no—
372C42—612no—
373C42—1224no—
374C42—1224no—
375C42—1221no—
376C42—1221no—
377C23—22no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23C22⋊#9242012yes—
387C23C2×C22⋊#692311yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8C22⋊#9242014yes—
392D4—216no—
393D4—22no—
394D4—22no—
395D4—22no—
396D4C22⋊#9242014yes—
397C2×C4—216no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—1221no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—1221no—
413C2×C4—4884no—
414C2×C4—4884no—
415C2×C4—4884no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4C22⋊#9242012yes—
423C2×C4C2×C22⋊#692311yescharacteristic
424C2×C4C2×C22⋊#692311yescharacteristic
425C2×C4C22×#9242011yescharacteristic
426C2×C4C2×#1175821711yescharacteristic
427C2×C4C2×#1175821711yescharacteristic
428C2×C4C2×#1175821711yescharacteristic
429C8—4882no—
430C8—4882no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—24no—
435C22—42no—
436C22—44no—
437C22—44no—
438C22—42no—
439C22—41no—
440C22—42no—
441C22—41no—
442C22—42no—
443C22#1684070311yescharacteristic
444C22C2×C22⋊#9242011yescharacteristic
445C22#1684073211yescharacteristic
446C4—22no—
447C4—24no—
448C4—1222no—
449C4—1222no—
450C4—2441no—
451C4—2442no—
452C4—2442no—
453C4—2441no—
454C4C2×C22⋊#9242011yescharacteristic
455C4C2×C22⋊#9242011yescharacteristic
456C4#1684073211yescharacteristic
457C4#1684073211yescharacteristic
458C2—21no—
459C2—21no—
460C2—44no—
461C2—42no—
462C2#2146164111yescharacteristic
463C2#2146167011yescharacteristic
464C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC22×#41×C61C4
socleC22×C61#379
cosocleC2×C4×Dic61C23
hypercenterC22×#41#6923
hypocenterC61#8324
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #21461121C2semidirect
2. #21461118C2semidirect
3. #21461050C2semidirect
4. #21461050C2semidirect
72. C2×#41×C61C8semidirect
259. C61#8324semidirect
265. C2×#41#92420semidirect

Maximal subgroups

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

subgroupindex
#214611212
#214611182
#214610502
C2×#168402452
#832461