Finite Group Databasebeta

← #30600786

Subgroups — #30600786

Order 15616.

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

normal subgroups only

This group has 492 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#21462314C211yesmaximal, characteristic, complemented
2#21461957C211yesmaximal, characteristic
3#21461333C211yesmaximal, characteristic, complemented
4#21461248C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic, complemented
6#21460642C211yesmaximal, characteristic, complemented
7#21456740C211yesmaximal, characteristic
8#16840780C2211yescharacteristic
9#16840767C2211yescharacteristic
10C2×#14519841—21no—
11C2×#14519841—21no—
12(C2×#6923)⋊Q8—22no—
13#16840713—21no—
14#16840711—22no—
15#16840706C2211yescharacteristic
16C4⋊C4×#6923—21no—
17#16840698C2211yescharacteristic
18#16840622—22no—
19#16840613—22no—
20#16840310C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840134C411yescharacteristic, complemented
23#16840113C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16839965—22no—
26#16839899—22no—
27#16839813C2211yescharacteristic
28C61×#492C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—22no—
32C2×C4×#6923—22no—
33C2×C4×#6923D411yescharacteristic
34C2×C4×#6923D411yescharacteristic
35#14519844—21no—
36#14519844—21no—
37#14519844—21no—
38#14519841—42no—
39#14519841—42no—
40#14519834—21no—
41C4×#92420—21no—
42(C2×#6923)⋊C4—22no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46(C2×#6923)⋊C4—21no—
47(C2×#6923)⋊C4D412yes—
48C2×#11758224—21no—
49C2×Q8×D61D412yes—
50C2×#11758200—22no—
51#14519777C2×C411yescharacteristic
52#14519777C2×C411yescharacteristic
53#14519776—21no—
54#14519776—21no—
55#14519775D412yes—
56#14519764—21no—
57#14519748—21no—
58#14519748—21no—
59D4×Dic61—21no—
60D4×Dic61—21no—
61C4⋊C4×D61—21no—
62C4⋊C4×D61—21no—
63C42×D61C2311yescharacteristic
64#14519706—21no—
65#14519702—22no—
66#14519701—22no—
67#14519689—21no—
68C4×#92418C2×C411yescharacteristic
69C2×SD16×C61—22no—
70C4⋊Q8×C61C2×C411yescharacteristic
71#127×C61C2×C411yescharacteristic
72C4×C8×C61C2×C411yescharacteristic
73C22×#6923—21no—
74C22×#6923—21no—
75C22×#6923—22no—
76C2×Q8×C61C22⋊C412yes—
77C2×D4×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×C22⋊D61—41no—
81C22×Dic61—41no—
82Q8×D61—82no—
83#11758224—41no—
84#11758224—41no—
85#11758224—41no—
86#11758224—41no—
87C2×C4×D61—22no—
88C2×C4×D61C2×D411yescharacteristic
89C2×C4×D61C2×D411yescharacteristic
90C2×C4×D61C2×D411yescharacteristic
91C2×Dic122—21no—
92C2×Dic122—22no—
93C2×Dic122C22⋊C412yes—
94C2×#92420—41no—
95C2×#92420—41no—
96C4⋊#6923—41no—
97C4⋊#6923—41no—
98C4⋊#6923—82no—
99C4×#6923—21no—
100C4×#6923—21no—
101C4×#6923—21no—
102C4×#6923—82no—
103C4×#6923SD1612yescomplemented
104#11758213—41no—
105#11758213—41no—
106SD16×C61—82no—
107C2×C8×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C42×C61C22×C411yescharacteristic
111C22⋊Dic61—41no—
112C22⋊Dic61—41no—
113#11758200—82no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118#11758196—22no—
119#11758196—22no—
120C4×Dic61C2×D411yescharacteristic
121C4×Dic61#4111yescharacteristic
122C4×Dic61#4111yescharacteristic
123C2×#92418—21no—
124C23×C61—41no—
125C22×D61#12011yescharacteristic
126C2×#6923—22no—
127C2×#6923—41no—
128C2×#6923—41no—
129C2×#6923—42no—
130C2×#6923—41no—
131C2×#6923—41no—
132C2×#6923—42no—
133C2×#6923—41no—
134C2×#6923—42no—
135Q8×C61—42no—
136D4×C61—42no—
137D4×C61—41no—
138D4×C61—41no—
139D4×C61—81no—
140C2×C4×C61—22no—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61C4×D411yescharacteristic
144C22⋊D61—81no—
145C22⋊D61—81no—
146C2×Dic61—22no—
147C2×Dic61—41no—
148C2×Dic61—81no—
149C2×Dic61—81no—
150C2×Dic61#12111yescharacteristic
151C2×Dic61#12111yescharacteristic
152C2×Dic61#12311yescharacteristic
153C4×D61—41no—
154C4×D61—41no—
155C4×D61—42no—
156C4×D61—42no—
157C4×D61C2×SD1611yescharacteristic
158C4×D61#13511yescharacteristic
159C4×D61AGL(1,ℤ/8)11yescharacteristic
160C4×D61#13711yescharacteristic
161Dic122—42no—
162Dic122—41no—
163Dic122—41no—
164Dic122—82no—
165#92420—81no—
166#92420—81no—
167C8×C61—41no—
168#92418—41no—
169#9301—611nomaximal, Sylow
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61#38611yescharacteristic
174C2×D61—21no—
175C2×D61—21no—
176C2×D61#44811yescharacteristic
177C2×D61#44911yescharacteristic
178#6923—42no—
179#6923—82no—
180#6923—81no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61C4×SD1611yescharacteristic
185C4×C61#44011yescharacteristic
186Dic61—21no—
187Dic61—21no—
188Dic61—42no—
189Dic61—81no—
190Dic61#46011yescharacteristic
191Dic61#46411yescharacteristic
192#3206—611no—
193#2331—611no—
194#1963—611no—
195#1548—611no—
196#1517—611no—
197#1286—611no—
198#1274—611no—
199C2×C61—161no—
200C2×C61#188011yescharacteristic
201C2×C61#190511yescharacteristic
202C2×C61#191111yescharacteristic
203D61—21no—
204D61—21no—
205#532—611no—
206C2×C4⋊Q8—611no—
207#492—611no—
208#492#692311yescharacteristic, complemented
209#487—611no—
210#487—611no—
211#461—1222no—
212#448—1222no—
213#431—611no—
214#431—611no—
215#423—611no—
216C2×#104—1221no—
217C2×#104—1221no—
218#391—1222no—
219#389—1221no—
220#387—1222no—
221#382—611no—
222C4×C4⋊C4—1221no—
223#376—611no—
224#333—1222no—
225#327—1222no—
226C61#930111yescharacteristic, complemented, Sylow
227C2×#41—1221no—
228C22×Q8—612no—
229C2×SD16—22no—
230C2×SD16—1222no—
231C2×M16—1221no—
232C4⋊Q8—612no—
233C4⋊Q8—611no—
234C4⋊Q8—611no—
235C4⋊Q8—1221no—
236C4⋊Q8—1221no—
237C4⋊Q8C2×#692311yescharacteristic
238#127C2×#692311yescharacteristic
239#124—1221no—
240#121—1221no—
241#121—1221no—
242C4×D4—1221no—
243C4×D4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1221no—
248C2×C42—611no—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—1222no—
252C4⋊C8—1221no—
253C4⋊C8—1221no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256#104—2442no—
257#104—2442no—
258#103—1222no—
259#103—1222no—
260C22⋊C8—1221no—
261C22⋊C8—1221no—
262C4×C8—611no—
263C4×C8—1221no—
264C4×C8C2×#692311yescharacteristic
265#95—612no—
266#95—1221no—
267#95—1221no—
268#95—1221no—
269#95—1222no—
270#95—1222no—
271#41—2441no—
272#41—2441no—
273#41—2441no—
274#41—2441no—
275C2×Q8—612no—
276C2×Q8—1222no—
277C2×Q8—1221no—
278C2×Q8—4882no—
279C2×Q8C22⋊#692312yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—2441no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—1222no—
290C22×C4—1221no—
291C22×C4—2441no—
292SD16—82no—
293SD16—4882no—
294M16—2441no—
295M16—2441no—
296C2×C8—21no—
297C2×C8—1221no—
298C2×C8—2441no—
299C2×C8—2441no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1222no—
303C4⋊C4—1222no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C4⋊C4—4882no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C22⋊C4—2441no—
312C22⋊C4—2441no—
313C42—611no—
314C42—611no—
315C42—611no—
316C42—612no—
317C42—1221no—
318C42—1221no—
319C42—1221no—
320C42—4882no—
321C42C22×#692311yescharacteristic
322C23—41no—
323C23—611no—
324Q8—42no—
325Q8—4882no—
326Q8—2442no—
327Q8—2441no—
328Q8—2441no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—4881no—
334D4—4881no—
335C2×C4—22no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—611no—
339C2×C4—611no—
340C2×C4—611no—
341C2×C4—611no—
342C2×C4—611no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—4881no—
346C2×C4—4881no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2442no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4C2×C22⋊#692311yescharacteristic
361C2×C4D4×#692311yescharacteristic
362C2×C4D4×#692311yescharacteristic
363C8—41no—
364C8—4881no—
365C8—4881no—
366C8—2441no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—611no—
371C22—611no—
372C22—1221no—
373C22—1221no—
374C22#1684071011yescharacteristic
375C4—21no—
376C4—21no—
377C4—42no—
378C4—611no—
379C4—611no—
380C4—1221no—
381C4—1221no—
382C4—4881no—
383C4—4882no—
384C4—4881no—
385C4—2442no—
386C4—2442no—
387C4SD16×#692311yescharacteristic
388C4#1684080111yescharacteristic
389C2—161no—
390C2—1221no—
391C2—1221no—
392C2#2146181511yescharacteristic
393C2#2146185511yescharacteristic
394C2#2146186511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#492C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#492#6923
hypocenterC61#9301
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #21462314C2semidirect
3. #21461333C2semidirect
5. #21460673C2semidirect
6. #21460642C2semidirect
22. #16840134C4semidirect
23. #16840113C4semidirect
24. #16840112C4semidirect
28. C61×#492C4semidirect
103. C4×#6923SD16semidirect
208. #492#6923semidirect
226. C61#9301semidirect

Maximal subgroups

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

subgroupindex
#214623142
#214619572
#214613332
#214612482
#214606732
#214606422
#214567402
#930161