Finite Group Databasebeta

← #29665196

Subgroups — #29665196

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic
2C2×#16840039C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic
4#21445267C214yesmaximal, complemented
5#21445181C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3209C211yesmaximal, characteristic, complemented
10C2×C4×D244C2211yescharacteristic
11#16840059—22no—
12#16840049C2212yes—
13#16840039—22no—
14#16840039C2212yes—
15#16840039C2214yes—
16C8×D244C2214yes—
17#16839827—22no—
18#16839815C2211yescharacteristic
19#16839805—21no—
20C2×#14519710—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839766—21no—
24C2×#14519697—21no—
25C22⋊C8×D61—22no—
26#16839515C2214yescomplemented
27#16839511C2212yescomplemented
28#16839508C2212yescomplemented
29#16839508C2214yescomplemented
30C61×#533C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#494C2212yescomplemented
33C61×#487C2212yescomplemented
34C61×#462—22no—
35C61×#451—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C2×#11758194—21no—
40C22×#92418—21no—
41C2×C8×D61—24no—
42C4×D244—21no—
43C4×D244—21no—
44C4×D244D412yes—
45C4×D244C2312yes—
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758198—21no—
48#14519710—24no—
49#14519710—22no—
50#14519708—22no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56#14519697—24no—
57#14519697—22no—
58#14519697—22no—
59C22⋊#92418—21no—
60#14519689—24no—
61C4×#92418—21no—
62C22×Q8×C61—21no—
63C2×Q16×C61—24no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61C2312yes—
68C4⋊Q8×C61C2311yescharacteristic
69C4⋊Q8×C61C2311yescharacteristic
70#125×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C4×Q8×C61—21no—
74C4×Q8×C61—21no—
75#117×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61D412yes—
80#103×C61—24no—
81C22⋊C8×C61D412yes—
82C4×C8×C61C2312yes—
83C23×D61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61—82no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C2×D244—21no—
95C2×D244—22no—
96C2×D244Q1614yescomplemented
97C2×D244C2×D412yes—
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—82no—
101Q16×C61—84no—
102C2×C8×C61C2×D414yes—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C42×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758198—22no—
120#11758198—22no—
121C4⋊Dic61—21no—
122C4⋊Dic61C2×D412yes—
123#11758194—42no—
124C2×#92418—22no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—42no—
128C8×D61—84no—
129C23×C61#12111yescharacteristic
130C22×D61—22no—
131C22×D61—24no—
132C22×D61—81no—
133C22×D61—81no—
134Q8×C61—42no—
135Q8×C61—44no—
136Q8×C61—82no—
137Q8×C61—81no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—82no—
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61C2×Q1612yes—
148C2×C4×C61#13512yes—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—81no—
154D244—22no—
155D244—24no—
156D244C2×Q1614yes—
157C4×D61—44no—
158C4×D61—82no—
159C8×C61—24no—
160#92418—42no—
161#92418—42no—
162#92418—42no—
163#19387—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#46712yes—
167C22×C61C2×#12111yescharacteristic
168C2×D61—24no—
169C2×D61—42no—
170C2×D61—42no—
171C2×D61—84no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—82no—
176C4×C61—81no—
177C4×C61—81no—
178C4×C61#46712yes—
179C4×C61C22×Q1611yescharacteristic
180C4×C61C2×#13511yescharacteristic
181C4×C61#57612yes—
182Dic61—42no—
183#3209D6111yescharacteristic, complemented
184#3077—611no—
185C2×#474—611no—
186#2972—611no—
187#1683—614no—
188#1669—614no—
189#1571—611no—
190#1549—611no—
191#1542—611no—
192C2×C61—22no—
193C2×C61#304111yescharacteristic
194C2×C61C2×#46711yescharacteristic
195C2×C61#306911yescharacteristic
196D61—44no—
197#533C2×D6111yescharacteristic
198C2×C4⋊Q8C2×D6111yescharacteristic
199C2×C4×D4—611no—
200#494C2×D6112yescomplemented
201#487C2×D6112yescomplemented
202#478—1222no—
203#476—612no—
204#474—612no—
205#474—614no—
206#474—1222no—
207#462—22no—
208#451—22no—
209C8×D4—614no—
210C8×D4—1222no—
211#431—611no—
212#431C2×D6111yescharacteristic
213#428—1221no—
214C2×#107—1221no—
215#423—611no—
216C2×C4⋊C8—611no—
217#418—1221no—
218C2×#102—1221no—
219C2×C22⋊C8—1222no—
220#335—614no—
221#333—612no—
222#332—612no—
223#332—614no—
224C61#1938711yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×D4—611no—
227C23×C4—1221no—
228C2×Q16—24no—
229C2×M16—1221no—
230C22×C8—1221no—
231C22×C8—1224no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C22×D6112yes—
236C4⋊Q8C22×D6111yescharacteristic
237C4⋊Q8C22×D6111yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—612no—
244C4×D4—612no—
245C4×D4—1221no—
246C4×D4—1221no—
247#117—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—611no—
251C2×C22⋊C4—1221no—
252C2×C42C22×D6111yescharacteristic
253#107—1224no—
254#107—1222no—
255#106—1222no—
256C4⋊C8—612no—
257C4⋊C8—612no—
258C4⋊C8—612no—
259C4⋊C8—1221no—
260C4⋊C8—1221no—
261C4⋊C8C22⋊D6112yes—
262#103—24no—
263#102—1224no—
264#102—1222no—
265#102—1222no—
266C22⋊C8—1221no—
267C22⋊C8—1224no—
268C22⋊C8C22⋊D6112yes—
269C4×C8—1221no—
270C4×C8C22×D6112yes—
271C24—1221no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—612no—
278C2×D4—614no—
279C2×D4—1221no—
280C2×D4—1222no—
281C22×C4—41no—
282C22×C4—1221no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—4882no—
286C22×C4D4×D6111yescharacteristic
287C22×C4#1175822411yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289Q16—84no—
290M16—2442no—
291C2×C8—1222no—
292C2×C8—1222no—
293C2×C8—1222no—
294C2×C8—4884no—
295C2×C8—2442no—
296C2×C8C2×C22⋊D6114yes—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—612no—
308C4⋊C4—1221no—
309C22⋊C4—41no—
310C22⋊C4—1222no—
311C22⋊C4—1222no—
312C42—41no—
313C42D4×D6111yescharacteristic
314C42#1175822411yescharacteristic
315C42C2×C22⋊D6111yescharacteristic
316C42C23×D6111yescharacteristic
317C23—1224no—
318C23—1222no—
319C23—4881no—
320C23—4881no—
321C23#1451974811yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—614no—
327D4—1224no—
328D4—1222no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—1222no—
336C2×C4—4882no—
337C2×C4—4881no—
338C2×C4—2444no—
339C2×C4#1451974811yescharacteristic
340C2×C4#1451974811yescharacteristic
341C2×C4#1451974811yescharacteristic
342C2×C4Q16×D6112yes—
343C2×C4#1451980012yes—
344C2×C4C2×D4×D6111yescharacteristic
345C2×C4C22×C22⋊D6111yescharacteristic
346C2×C4C2×#1175822411yescharacteristic
347C8—24no—
348C8—2442no—
349C8—2442no—
350C8—2442no—
351C22—21no—
352C22—21no—
353C22—1224no—
354C22—4884no—
355C22—2442no—
356C22—2442no—
357C22#1684000012yes—
358C22C2×#1451974811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—2442no—
366C4#1684000012yes—
367C4C2×Q16×D6111yescharacteristic
368C4C2×#1451980011yescharacteristic
369C4#1684056512yes—
370C2—22no—
371C2—2444no—
372C2#2145531811yescharacteristic
373C2C2×#1684000011yescharacteristic
374C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3209D61
hypocenterC61#19387
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #21445267C2semidirect
5. #21445181C2semidirect
6. #21444594C2semidirect
7. #21444504C2semidirect
8. #21444476C2semidirect
9. C61×#3209C2semidirect
26. #16839515C22semidirect
27. #16839511C22semidirect
28. #16839508C22semidirect
29. #16839508C22semidirect
32. C61×#494C22semidirect
33. C61×#487C22semidirect
96. C2×D244Q16semidirect
183. #3209D61semidirect
200. #494C2×D61semidirect
201. #487C2×D61semidirect
224. C61#19387semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168400392
#214548422
#214452672
#214451812
#214445942
#214445042
#214444762
C61×#32092
#1938761