Finite Group Databasebeta

← #30600761

Subgroups — #30600761

Order 15616.

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

normal subgroups only

This group has 490 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#21461335C211yesmaximal, characteristic, complemented
4#21461248C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic, complemented
6#21460644C211yesmaximal, characteristic, complemented
7#21456740C211yesmaximal, 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#16840621—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
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—
49C2×#11758200—22no—
50#14519777C2×C411yescharacteristic
51#14519777C2×C411yescharacteristic
52#14519776—21no—
53#14519776—21no—
54#14519775—21no—
55#14519764—21no—
56#14519748—21no—
57#14519748D412yes—
58D4×Dic61—21no—
59D4×Dic61D412yes—
60C4⋊C4×D61—21no—
61C4⋊C4×D61—21no—
62C42×D61C2311yescharacteristic
63#14519706—21no—
64#14519702—22no—
65#14519701—22no—
66#14519689—21no—
67C4×#92418C2×C411yescharacteristic
68C2×SD16×C61—22no—
69C4⋊Q8×C61C2×C411yescharacteristic
70#127×C61C2×C411yescharacteristic
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⋊C8SD1612yescomplemented
105SD16×C61—82no—
106C2×C8×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C42×C61C22×C411yescharacteristic
110C22⋊Dic61—22no—
111C22⋊Dic61—22no—
112#11758200—82no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61C22⋊C412yes—
117#11758196—41no—
118#11758196—41no—
119C4×Dic61C2×D411yescharacteristic
120C4×Dic61C2×D411yescharacteristic
121C4×Dic61#4111yescharacteristic
122C2×#92418—21no—
123C23×C61—22no—
124C22×D61#12111yescharacteristic
125C2×#6923—81no—
126C2×#6923—81no—
127C2×#6923—81no—
128Q8×C61—42no—
129D4×C61—42no—
130D4×C61—41no—
131D4×C61—41no—
132D4×C61—81no—
133C2×C4×C61—41no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C22⋊D61—42no—
138C22⋊D61—42no—
139C2×Dic61—22no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61#12011yescharacteristic
144C2×Dic61#12111yescharacteristic
145C2×Dic61#12311yescharacteristic
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—82no—
149C4×D61C2×SD1611yescharacteristic
150C4×D61#13511yescharacteristic
151C4×D61AGL(1,ℤ/8)11yescharacteristic
152C4×D61#13711yescharacteristic
153Dic122—41no—
154Dic122—41no—
155Dic122—42no—
156Dic122—82no—
157#92420—22no—
158#92420—42no—
159#92420—41no—
160#92420—81no—
161C8×C61—41no—
162#92418—41no—
163#9289—611nomaximal, Sylow
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61#38611yescharacteristic
168C2×D61—21no—
169C2×D61—21no—
170C2×D61#46111yescharacteristic
171C2×D61#46311yescharacteristic
172#6923—161no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—81no—
176C4×C61C4×SD1611yescharacteristic
177C4×C61#44011yescharacteristic
178Dic61—21no—
179Dic61—21no—
180Dic61—42no—
181Dic61—81no—
182Dic61#45011yescharacteristic
183Dic61#45211yescharacteristic
184#3206—611no—
185#2932—611no—
186#1993—611no—
187#1549—611no—
188#1517—611no—
189#1286—611no—
190#1275—611no—
191C2×C61—82no—
192C2×C61#187711yescharacteristic
193C2×C61#190211yescharacteristic
194C2×C61#191011yescharacteristic
195D61—21no—
196D61—21no—
197C2×#131—1221no—
198#532—611no—
199C2×C4⋊Q8—611no—
200#492—611no—
201#492#692311yescharacteristic, complemented
202#487—611no—
203#487—611no—
204#461—1222no—
205#448—1222no—
206#435—1222no—
207#435—1222no—
208#431—611no—
209#431—611no—
210#423—611no—
211#408—1221no—
212C2×#97—611no—
213#382—611no—
214#340—1221no—
215#340—1221no—
216#332—1222no—
217#327—1222no—
218C61#928911yescharacteristic, complemented, Sylow
219C2×#41—1221no—
220C22×Q8—1221no—
221C2×SD16—22no—
222C2×SD16—1222no—
223#131—2442no—
224#131—2442no—
225C2×M16—1221no—
226C2×M16—1222no—
227C22×C8—611no—
228C22×C8—611no—
229C22×C8—1222no—
230C4⋊Q8—611no—
231C4⋊Q8—611no—
232C4⋊Q8—1221no—
233C4⋊Q8—1221no—
234C4⋊Q8—1221no—
235C4⋊Q8C2×#692311yescharacteristic
236#127C2×#692311yescharacteristic
237#124—1221no—
238#121—612no—
239#121—1221no—
240C4×D4—612no—
241C4×D4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C42—611no—
246C4⋊C8—612no—
247C4⋊C8—1221no—
248C4⋊C8—1221no—
249C4⋊C8—1221no—
250#103—1222no—
251#103—1222no—
252C22⋊C8—1221no—
253C22⋊C8—1224no—
254C22⋊C8—1221no—
255#97—1221no—
256#97—1221no—
257C4×C8—611no—
258C4×C8—1221no—
259C4×C8C2×#692311yescharacteristic
260#95—1221no—
261#95—1221no—
262#41—2441no—
263#41—2441no—
264#41—2441no—
265#41—2441no—
266C2×Q8—21no—
267C2×Q8—1221no—
268C2×Q8—1221no—
269C2×Q8—4882no—
270C2×Q8—2441no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—1222no—
274C2×D4C22⋊#692312yes—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—1222no—
279C22×C4—2441no—
280C22×C4—2441no—
281SD16—82no—
282SD16—4882no—
283M16—2444no—
284M16—2441no—
285M16—2441no—
286C2×C8—21no—
287C2×C8—612no—
288C2×C8—1221no—
289C2×C8—1221no—
290C2×C8—1221no—
291C2×C8—1222no—
292C2×C8—1221no—
293C2×C8—1221no—
294C2×C8—1221no—
295C2×C8—2444no—
296C2×C8—2441no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—612no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C22⋊C4—1222no—
308C22⋊C4—1222no—
309C42—611no—
310C42—611no—
311C42—611no—
312C42C22×#692311yescharacteristic
313C23—22no—
314C23—611no—
315Q8—42no—
316Q8—4882no—
317Q8—2442no—
318Q8—2441no—
319Q8—2441no—
320D4—42no—
321D4—41no—
322D4—41no—
323D4—81no—
324D4—2442no—
325D4—2442no—
326C2×C4—41no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—611no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—1222no—
335C2×C4—4882no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2441no—
344C2×C4C2×C22⋊#692311yescharacteristic
345C2×C4D4×#692311yescharacteristic
346C2×C4D4×#692311yescharacteristic
347C8—41no—
348C8—1222no—
349C8—4881no—
350C8—2441no—
351C8—2441no—
352C8—2442no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—611no—
357C22—611no—
358C22—1221no—
359C22—1221no—
360C22#1684071011yescharacteristic
361C4—21no—
362C4—21no—
363C4—81no—
364C4—611no—
365C4—611no—
366C4—1221no—
367C4—1221no—
368C4—9761no—
369C4—4881no—
370C4—2442no—
371C4SD16×#692311yescharacteristic
372C4#1684080111yescharacteristic
373C2—82no—
374C2—1221no—
375C2—1221no—
376C2#2146181211yescharacteristic
377C2#2146184811yescharacteristic
378C2#2146186211yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#492C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#492#6923
hypocenterC61#9289
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #21462003C2semidirect
3. #21461335C2semidirect
5. #21460673C2semidirect
6. #21460644C2semidirect
22. #16840134C4semidirect
23. #16840113C4semidirect
24. #16840112C4semidirect
28. C61×#492C4semidirect
104. D61⋊C8SD16semidirect
201. #492#6923semidirect
218. C61#9289semidirect

Maximal subgroups

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

subgroupindex
#214625482
#214620032
#214613352
#214612482
#214606732
#214606442
#214567402
#928961