Finite Group Databasebeta

← #29665254

Subgroups — #29665254

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic
2C2×#16840049C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic
4#21445267C214yesmaximal, complemented
5#21445215C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3230C211yesmaximal, characteristic, complemented
10C2×C4×D244C2211yescharacteristic
11#16840059—22no—
12#16840049—22no—
13#16840049C2212yes—
14#16840049C2214yes—
15#16840039C2212yes—
16C8×D244C2214yes—
17#16839827—22no—
18#16839815C2211yescharacteristic
19#16839805—21no—
20C2×#14519708—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839766—21no—
24C2×#14519697—21no—
25C22⋊C8×D61—22no—
26#16839515C2214yescomplemented
27#16839511C2212yescomplemented
28#16839511C2214yescomplemented
29#16839508C2212yescomplemented
30C61×#533C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#500C2212yescomplemented
33C61×#487C2212yescomplemented
34C61×#484—22no—
35C61×#477—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—22no—
49#14519708—24no—
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—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61C2312yes—
67C4⋊Q8×C61C2311yescharacteristic
68C4⋊Q8×C61C2311yescharacteristic
69#125×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C4×Q8×C61—21no—
73C4×Q8×C61—21no—
74#117×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#107×C61—24no—
79C4⋊C8×C61D412yes—
80#103×C61—24no—
81C22⋊C8×C61D412yes—
82C4×C8×C61C2312yes—
83C23×D61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
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×D244Q1614yes—
97C2×D244C2×D412yes—
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—82no—
101C2×C8×C61C2×D414yes—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C42×C61C2411yescharacteristic
118#11758198—22no—
119#11758198—22no—
120C4⋊Dic61—21no—
121C4⋊Dic61C2×D412yes—
122#11758194—42no—
123C2×#92418—22no—
124C2×#92418—22no—
125C2×#92418—22no—
126C2×#92418—42no—
127C8×D61—84no—
128C23×C61#12111yescharacteristic
129C22×D61—22no—
130C22×D61—24no—
131C22×D61—81no—
132C22×D61—81no—
133Q8×C61—42no—
134Q8×C61—82no—
135Q8×C61—81no—
136Q8×C61—82no—
137C2×C4×C61—22no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—42no—
142C2×C4×C61—82no—
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61C2×Q1612yes—
147C2×C4×C61#13512yes—
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—81no—
153D244—22no—
154D244—24no—
155D244C2×Q1614yes—
156C4×D61—44no—
157C4×D61—82no—
158C8×C61—24no—
159#92418—42no—
160#92418—42no—
161#92418—42no—
162#19393—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61#46712yes—
166C22×C61C2×#12111yescharacteristic
167C2×D61—24no—
168C2×D61—42no—
169C2×D61—42no—
170C2×D61—84no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—81no—
175C4×C61—82no—
176C4×C61—81no—
177C4×C61#46712yes—
178C4×C61C22×Q1611yescharacteristic
179C4×C61C2×#13511yescharacteristic
180C4×C61#57712yes—
181Dic61—42no—
182#3230D6111yescharacteristic, complemented
183#3077—611no—
184C2×#476—611no—
185#2972—611no—
186#1683—614no—
187#1675—614no—
188#1571—611no—
189#1548—611no—
190#1542—611no—
191C2×C61—22no—
192C2×C61#304211yescharacteristic
193C2×C61C2×#46711yescharacteristic
194C2×C61#306811yescharacteristic
195D61—44no—
196#533C2×D6111yescharacteristic
197C2×C4⋊Q8C2×D6111yescharacteristic
198C2×C4×D4—611no—
199#500C2×D6112yescomplemented
200#487C2×D6112yescomplemented
201#484—22no—
202#478—1222no—
203#477—22no—
204#476—612no—
205#476—614no—
206#476—1222no—
207#474—612no—
208C8×D4—614no—
209C8×D4—1222no—
210#431—611no—
211#431C2×D6111yescharacteristic
212#428—1221no—
213C2×#106—1221no—
214#423—611no—
215C2×C4⋊C8—611no—
216#418—1221no—
217C2×#102—1221no—
218C2×C22⋊C8—1222no—
219#335—614no—
220#333—612no—
221#333—614no—
222#332—612no—
223C61#1939311yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×D4—611no—
226C23×C4—1221no—
227C2×M16—1221no—
228C22×C8—1221no—
229C22×C8—1224no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×D6112yes—
234C4⋊Q8C22×D6111yescharacteristic
235C4⋊Q8C22×D6111yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—612no—
242C4×D4—612no—
243C4×D4—1221no—
244C4×D4—1221no—
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C22⋊C4—1221no—
250C2×C42C22×D6111yescharacteristic
251#107—24no—
252#107—1222no—
253#106—1224no—
254#106—1222no—
255C4⋊C8—612no—
256C4⋊C8—612no—
257C4⋊C8—612no—
258C4⋊C8—1221no—
259C4⋊C8—1221no—
260C4⋊C8C22⋊D6112yes—
261#103—24no—
262#102—1224no—
263#102—1222no—
264#102—1222no—
265C22⋊C8—1221no—
266C22⋊C8—1224no—
267C22⋊C8C22⋊D6112yes—
268C4×C8—1221no—
269C4×C8C22×D6112yes—
270C24—1221no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—612no—
277C2×D4—614no—
278C2×D4—1221no—
279C2×D4—1222no—
280C22×C4—41no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—4882no—
285C22×C4D4×D6111yescharacteristic
286C22×C4#1175822411yescharacteristic
287C22×C4C2×C22⋊D6111yescharacteristic
288M16—2442no—
289C2×C8—1222no—
290C2×C8—1222no—
291C2×C8—1222no—
292C2×C8—4884no—
293C2×C8—2442no—
294C2×C8C2×C22⋊D6114yes—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—612no—
306C4⋊C4—1221no—
307C22⋊C4—41no—
308C22⋊C4—1222no—
309C22⋊C4—1222no—
310C42—41no—
311C42D4×D6111yescharacteristic
312C42#1175822411yescharacteristic
313C42C2×C22⋊D6111yescharacteristic
314C42C23×D6111yescharacteristic
315C23—1224no—
316C23—1222no—
317C23—4881no—
318C23—4881no—
319C23#1451974811yescharacteristic
320Q8—42no—
321Q8—81no—
322Q8—82no—
323Q8—82no—
324D4—614no—
325D4—1224no—
326D4—1222no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—1222no—
334C2×C4—4882no—
335C2×C4—4881no—
336C2×C4—2444no—
337C2×C4#1451974811yescharacteristic
338C2×C4#1451974811yescharacteristic
339C2×C4#1451974811yescharacteristic
340C2×C4Q16×D6112yes—
341C2×C4#1451980012yes—
342C2×C4C2×D4×D6111yescharacteristic
343C2×C4C22×C22⋊D6111yescharacteristic
344C2×C4C2×#1175822411yescharacteristic
345C8—24no—
346C8—2442no—
347C8—2442no—
348C8—2442no—
349C22—21no—
350C22—21no—
351C22—1224no—
352C22—4884no—
353C22—2442no—
354C22—2442no—
355C22#1684000012yes—
356C22C2×#1451974811yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—2442no—
364C4#1684000012yes—
365C4C2×Q16×D6111yescharacteristic
366C4C2×#1451980011yescharacteristic
367C4#1684057012yes—
368C2—22no—
369C2—2444no—
370C2#2145532711yescharacteristic
371C2C2×#1684000011yescharacteristic
372C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3230D61
hypocenterC61#19393
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #21445267C2semidirect
5. #21445215C2semidirect
6. #21444594C2semidirect
7. #21444500C2semidirect
8. #21444476C2semidirect
9. C61×#3230C2semidirect
26. #16839515C22semidirect
27. #16839511C22semidirect
28. #16839511C22semidirect
29. #16839508C22semidirect
32. C61×#500C22semidirect
33. C61×#487C22semidirect
182. #3230D61semidirect
199. #500C2×D61semidirect
200. #487C2×D61semidirect
223. C61#19393semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168400492
#214548422
#214452672
#214452152
#214445942
#214445002
#214444762
C61×#32302
#1939361