Finite Group Databasebeta

← #30600757

Subgroups — #30600757

Order 15616.

478 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462548C211yesmaximal, characteristic
2#21462003C211yesmaximal, characteristic, complemented
3#21461333C211yesmaximal, characteristic, complemented
4#21461248C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic, complemented
6#21460644C211yesmaximal, characteristic, complemented
7#21456943C211yesmaximal, characteristic
8C2×#14519864—21no—
9#16840793—22no—
10#16840793—22no—
11#16840780C2211yescharacteristic
12#16840767C2211yescharacteristic
13#16840736—21no—
14#16840721C2211yescharacteristic
15#16840706C2211yescharacteristic
16#16840635—21no—
17#16840635—21no—
18#16840622—22no—
19#16840613—22no—
20#16840310C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840160C411yescharacteristic, complemented
23#16840158C411yescharacteristic, complemented
24#16840108C411yescharacteristic, complemented
25#16840079—22no—
26#16840050—22no—
27#16839813C2211yescharacteristic
28C61×#486C411yescharacteristic, complemented
29#14519864—42no—
30#14519864—42no—
31C2×#11758213—21no—
32C2×#11758217—22no—
33C2×D61⋊C8D411yescharacteristic
34C2×D61⋊C8D411yescharacteristic
35C22×#92420—22no—
36C2×C4⋊#6923—21no—
37#14519844—21no—
38#14519844—21no—
39#14519844D412yes—
40#14519834—21no—
41C22⋊#92420—24no—
42#14519832—21no—
43#14519832—21no—
44C4×#92420—21no—
45(C2×#6923)⋊C4—21no—
46(C2×#6923)⋊C4—21no—
47C2×#11758224—21no—
48C2×Q8×D61—21no—
49#14519777C2×C411yescharacteristic
50#14519777C2×C411yescharacteristic
51#14519776—21no—
52#14519776—21no—
53#14519775—21no—
54#14519764—21no—
55#14519748—21no—
56#14519748D412yes—
57D4×Dic61—21no—
58D4×Dic61D412yes—
59C4⋊C4×D61—21no—
60C4⋊C4×D61—21no—
61C42×D61C2311yescharacteristic
62#14519709—22no—
63#14519708—22no—
64#14519706—21no—
65#14519698—22no—
66#14519689—21no—
67C4×#92418C2×C411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69#127×C61C2×C411yescharacteristic
70#102×C61—22no—
71C4×C8×C61C2×C411yescharacteristic
72C22×#6923—41no—
73C2×Q8×C61—21no—
74C2×D4×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61C22⋊C412yes—
77C2×C22⋊D61—22no—
78C22×Dic61—22no—
79Q8×D61—82no—
80#11758224—41no—
81#11758224—41no—
82#11758224—41no—
83#11758224—41no—
84C2×C4×D61—41no—
85C2×C4×D61C2×D411yescharacteristic
86C2×C4×D61C2×D411yescharacteristic
87C2×C4×D61#4111yescharacteristic
88C2×Dic122—21no—
89C2×Dic122—21no—
90C2×Dic122—41no—
91#11758217—44no—
92C2×#92420—21no—
93C2×#92420—21no—
94C2×#92420—22no—
95C2×#92420—44no—
96C2×#92420—41no—
97C4⋊#6923—41no—
98C4⋊#6923—41no—
99#11758213—41no—
100#11758213—41no—
101D61⋊C8—21no—
102D61⋊C8—21no—
103D61⋊C8—21no—
104D61⋊C8SD1612yes—
105C2×C8×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C42×C61C22×C411yescharacteristic
109C22⋊Dic61—22no—
110C22⋊Dic61—22no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61C22⋊C412yes—
115#11758196—41no—
116#11758196—41no—
117C4×Dic61C2×D411yescharacteristic
118C4×Dic61C2×D411yescharacteristic
119C4×Dic61#4111yescharacteristic
120C2×#92418—21no—
121C23×C61—22no—
122C22×D61#12111yescharacteristic
123C2×#6923—81no—
124C2×#6923—81no—
125C2×#6923—81no—
126Q8×C61—81no—
127D4×C61—42no—
128D4×C61—41no—
129D4×C61—41no—
130D4×C61—81no—
131C2×C4×C61—41no—
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C22⋊D61—42no—
136C22⋊D61—42no—
137C2×Dic61—22no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—41no—
141C2×Dic61#12011yescharacteristic
142C2×Dic61#12111yescharacteristic
143C2×Dic61#12311yescharacteristic
144C4×D61—41no—
145C4×D61—41no—
146C4×D61—82no—
147C4×D61C2×SD1611yescharacteristic
148C4×D61#13511yescharacteristic
149C4×D61AGL(1,ℤ/8)11yescharacteristic
150C4×D61#13711yescharacteristic
151Dic122—41no—
152Dic122—41no—
153Dic122—82no—
154Dic122—81no—
155#92420—22no—
156#92420—42no—
157#92420—41no—
158#92420—81no—
159C8×C61—41no—
160#92418—41no—
161#9287—611nomaximal, Sylow
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#38611yescharacteristic
166C2×D61—21no—
167C2×D61—21no—
168C2×D61#46111yescharacteristic
169C2×D61#46311yescharacteristic
170#6923—161no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—81no—
174C4×C61C4×SD1611yescharacteristic
175C4×C61#44011yescharacteristic
176Dic61—21no—
177Dic61—21no—
178Dic61—42no—
179Dic61—81no—
180Dic61#45011yescharacteristic
181Dic61#45211yescharacteristic
182#3227—611no—
183#2932—611no—
184#1993—611no—
185#1548—611no—
186#1517—611no—
187#1286—611no—
188#1275—611no—
189C2×C61—82no—
190C2×C61#187711yescharacteristic
191C2×C61#190311yescharacteristic
192C2×C61#190911yescharacteristic
193D61—21no—
194D61—21no—
195C2×#131—1221no—
196#532—611no—
197C2×C4⋊Q8—611no—
198#498—611no—
199#498—611no—
200#486—611no—
201#486#692311yescharacteristic, complemented
202#481—1222no—
203#476—1222no—
204#435—1222no—
205#435—1222no—
206#431—611no—
207#431—611no—
208#423—611no—
209#408—1221no—
210C2×#97—611no—
211#382—611no—
212#340—1221no—
213#340—1221no—
214#333—1222no—
215#327—1222no—
216C61#928711yescharacteristic, complemented, Sylow
217C2×#41—1221no—
218C22×Q8—1221no—
219#131—2442no—
220#131—2442no—
221C2×M16—1221no—
222C2×M16—1222no—
223C22×C8—611no—
224C22×C8—611no—
225C22×C8—1222no—
226C4⋊Q8—611no—
227C4⋊Q8—611no—
228C4⋊Q8—1221no—
229C4⋊Q8—1221no—
230C4⋊Q8—1221no—
231C4⋊Q8C2×#692311yescharacteristic
232#127C2×#692311yescharacteristic
233#124—1221no—
234#121—612no—
235#121—1221no—
236C4×D4—612no—
237C4×D4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C42—611no—
242#106—1222no—
243#106—1222no—
244C4⋊C8—612no—
245C4⋊C8—1221no—
246C4⋊C8—1221no—
247C4⋊C8—1221no—
248#102—22no—
249#102—1222no—
250C22⋊C8—1221no—
251C22⋊C8—1224no—
252C22⋊C8—1221no—
253#97—1221no—
254#97—1221no—
255C4×C8—611no—
256C4×C8—1221no—
257C4×C8C2×#692311yescharacteristic
258#95—1221no—
259#95—1221no—
260#41—2441no—
261#41—2441no—
262#41—2441no—
263#41—2441no—
264C2×Q8—21no—
265C2×Q8—1221no—
266C2×Q8—1221no—
267C2×Q8—4882no—
268C2×Q8—2441no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—1222no—
272C2×D4C22⋊#692312yes—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—1222no—
277C22×C4—2441no—
278C22×C4—2441no—
279M16—2444no—
280M16—2441no—
281M16—2441no—
282C2×C8—21no—
283C2×C8—612no—
284C2×C8—1221no—
285C2×C8—1221no—
286C2×C8—1221no—
287C2×C8—1222no—
288C2×C8—1221no—
289C2×C8—1221no—
290C2×C8—1221no—
291C2×C8—2444no—
292C2×C8—2441no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—612no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C4⋊C4—2441no—
300C4⋊C4—2441no—
301C4⋊C4—2441no—
302C4⋊C4—2441no—
303C22⋊C4—1222no—
304C22⋊C4—1222no—
305C42—611no—
306C42—611no—
307C42—611no—
308C42C22×#692311yescharacteristic
309C23—22no—
310C23—611no—
311Q8—81no—
312Q8—4881no—
313Q8—4882no—
314Q8—2441no—
315Q8—2441no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—2442no—
321D4—2442no—
322C2×C4—41no—
323C2×C4—611no—
324C2×C4—611no—
325C2×C4—611no—
326C2×C4—611no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—611no—
330C2×C4—1222no—
331C2×C4—4882no—
332C2×C4—4881no—
333C2×C4—4881no—
334C2×C4—4881no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2442no—
338C2×C4—2442no—
339C2×C4—2441no—
340C2×C4C2×C22⋊#692311yescharacteristic
341C2×C4D4×#692311yescharacteristic
342C2×C4D4×#692311yescharacteristic
343C8—41no—
344C8—1222no—
345C8—4881no—
346C8—2441no—
347C8—2441no—
348C8—2442no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—611no—
353C22—611no—
354C22—1221no—
355C22—1221no—
356C22#1684071011yescharacteristic
357C4—21no—
358C4—21no—
359C4—81no—
360C4—611no—
361C4—611no—
362C4—1221no—
363C4—1221no—
364C4—9761no—
365C4—4881no—
366C4—2442no—
367C4SD16×#692311yescharacteristic
368C4#1684080111yescharacteristic
369C2—82no—
370C2—1221no—
371C2—1221no—
372C2#2146181211yescharacteristic
373C2#2146185011yescharacteristic
374C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#486C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#486#6923
hypocenterC61#9287
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #21462003C2semidirect
3. #21461333C2semidirect
5. #21460673C2semidirect
6. #21460644C2semidirect
22. #16840160C4semidirect
23. #16840158C4semidirect
24. #16840108C4semidirect
28. C61×#486C4semidirect
201. #486#6923semidirect
216. C61#9287semidirect

Maximal subgroups

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

subgroupindex
#214625482
#214620032
#214613332
#214612482
#214606732
#214606442
#214569432
#928761