Finite Group Databasebeta

← #30600124

Subgroups — #30600124

Order 15616.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 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#21462565C211yesmaximal, characteristic
2#21461995C211yesmaximal, characteristic, complemented
3#21461200C211yesmaximal, characteristic
4#21461128C211yesmaximal, characteristic, complemented
5#21460654C211yesmaximal, characteristic, complemented
6#21460651C211yesmaximal, characteristic, complemented
7#21456380C211yesmaximal, characteristic
8C2×#14519865—21no—
9#16840796—21no—
10#16840792—22no—
11#16840790—21no—
12#16840781C2211yescharacteristic
13D61⋊(C4⋊C8)C2211yescharacteristic
14#16840737—21no—
15C4×#11758213C2211yescharacteristic
16C4×C4⋊#6923C2211yescharacteristic
17#16840632—21no—
18#16840632—21no—
19#16840614—22no—
20Q8⋊#92420—21no—
21Q8⋊#92420—21no—
22#16840317C2211yescharacteristic
23#16840313C2211yescharacteristic
24#16840135C411yescharacteristic, complemented
25#16840114C411yescharacteristic, complemented
26#16840110C411yescharacteristic, complemented
27#16839992—22no—
28#16839849—22no—
29C4×C8×D61C2211yescharacteristic
30C61×#498C411yescharacteristic, complemented
31#14519865—42no—
32#14519865—42no—
33C2×#11758213—22no—
34C2×#11758213D411yescharacteristic
35C2×#11758213D411yescharacteristic
36C2×D61⋊C8—22no—
37C2×D61⋊C8—21no—
38C2×C4⋊#6923—21no—
39C2×C4×#6923—21no—
40#14519844—21no—
41C4⋊#92420—21no—
42C4⋊#92420D411yescharacteristic
43C4⋊#92420D411yescharacteristic
44#14519834—22no—
45#14519834—22no—
46#14519834—21no—
47#14519832—21no—
48C4×#92420—21no—
49C4×#92420—21no—
50(C2×#6923)⋊C4—21no—
51C2×#11758226—21no—
52C2×#11758190—22no—
53C2×C8×D61—21no—
54#14519777C2×C411yescharacteristic
55#14519774C2×C411yescharacteristic
56#14519768—21no—
57#14519766—21no—
58#14519757—21no—
59#14519753—21no—
60#14519752—21no—
61#14519752D412yes—
62Q8×Dic61—21no—
63#14519742—21no—
64#14519739—21no—
65#14519739D412yes—
66#14519735—21no—
67C42×D61C2311yescharacteristic
68#14519700—22no—
69#14519697—22no—
70C4×#92418C2×C411yescharacteristic
71C8×Dic61—21no—
72C4⋊Q8×C61C2×C411yescharacteristic
73C4⋊Q8×C61C2×C411yescharacteristic
74#106×C61—22no—
75C4×C8×C61C2×C411yescharacteristic
76C22×#6923—41no—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79#11758226—41no—
80#11758226—41no—
81#11758226—41no—
82#11758226—41no—
83C2×D244—22no—
84C2×D244—21no—
85C2×D244—21no—
86C2×D244C22⋊C412yes—
87C2×C4×D61—22no—
88C2×C4×D61C2×D411yescharacteristic
89C2×C4×D61C2×D411yescharacteristic
90C2×C4×D61#4111yescharacteristic
91C2×Dic122—21no—
92C2×#92420—21no—
93C2×#92420—21no—
94C2×#92420—21no—
95C2×#92420—21no—
96C2×#92420—41no—
97C4⋊#6923—22no—
98C4⋊#6923—41no—
99C4×#6923—41no—
100C4×#6923—41no—
101#11758213—21no—
102#11758213—21no—
103#11758213—21no—
104#11758213—44no—
105#11758213SD1612yescomplemented
106D61⋊C8—44no—
107D61⋊C8—41no—
108D61⋊C8—41no—
109C2×C8×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C22⋊C412yes—
114C42×C61C22×C411yescharacteristic
115#11758198—22no—
116#11758198—22no—
117#11758198—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—41no—
121#11758196—41no—
122C4×Dic61—41no—
123C4×Dic61C2×D411yescharacteristic
124C4×Dic61C2×D411yescharacteristic
125C4×Dic61#4111yescharacteristic
126C2×#92418—21no—
127#11758190—44no—
128C8×D61—42no—
129C22×D61—22no—
130C22×D61#12111yescharacteristic
131C2×#6923—42no—
132C2×#6923—81no—
133C2×#6923—81no—
134Q8×C61—41no—
135Q8×C61—41no—
136Q8×C61—41no—
137Q8×C61—41no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×Dic61—41no—
144C2×Dic61#12011yescharacteristic
145C2×Dic61#12111yescharacteristic
146C2×Dic61#12311yescharacteristic
147D244—42no—
148D244—42no—
149D244—42no—
150D244—41no—
151D244—41no—
152D244—41no—
153D244—41no—
154C4×D61—21no—
155C4×D61—21no—
156C4×D61—41no—
157C4×D61—42no—
158C4×D61—42no—
159C4×D61C2×SD1611yescharacteristic
160C4×D61#13511yescharacteristic
161C4×D61AGL(1,ℤ/8)11yescharacteristic
162C4×D61#13711yescharacteristic
163Dic122—42no—
164#92420—22no—
165#92420—41no—
166#92420—41no—
167#92420—41no—
168#92420—81no—
169C8×C61—22no—
170#92418—41no—
171#9055—611nomaximal, Sylow
172C22×C61#38611yescharacteristic
173C2×D61—21no—
174C2×D61—21no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61#46111yescharacteristic
179C2×D61#46311yescharacteristic
180#6923—82no—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—81no—
184C4×C61C4×SD1611yescharacteristic
185C4×C61#44011yescharacteristic
186C4×C61#44311yescharacteristic
187C4×C61#44411yescharacteristic
188Dic61—21no—
189Dic61—21no—
190Dic61—81no—
191Dic61#45011yescharacteristic
192Dic61#45211yescharacteristic
193#3167—611no—
194#2935—611no—
195#1986—611no—
196#1502—611no—
197#1481—611no—
198#1279—611no—
199#1278—611no—
200C2×C61#187611yescharacteristic
201C2×C61#187711yescharacteristic
202C2×C61#188411yescharacteristic
203D61—21no—
204D61—21no—
205D61—82no—
206C2×#131—1221no—
207#533—611no—
208#532—611no—
209#498#692311yescharacteristic, complemented
210#492—611no—
211#487—611no—
212#486—611no—
213#465—1222no—
214C4×SD16—1222no—
215#436—1221no—
216#435—1222no—
217C8×D4—1221no—
218#431—611no—
219C2×C4⋊C8—611no—
220#408—1221no—
221C4×M16—611no—
222C2×C4×C8—611no—
223C4×C4⋊C4—611no—
224#339—1221no—
225#339—1221no—
226#327—1222no—
227Q8⋊C8—1221no—
228Q8⋊C8—1221no—
229C61#905511yescharacteristic, complemented, Sylow
230C2×#41—1221no—
231C2×SD16—1222no—
232#131—2442no—
233#131—2442no—
234C2×M16—611no—
235C2×M16—611no—
236C2×M16—1222no—
237C22×C8—1221no—
238C22×C8—1221no—
239C22×C8—1222no—
240C4⋊Q8—611no—
241C4⋊Q8C2×#692311yescharacteristic
242C4⋊Q8C2×#692311yescharacteristic
243#127—611no—
244#125—1221no—
245#124—1221no—
246C22⋊Q8—1221no—
247C22⋊Q8—1221no—
248#121—612no—
249#121—1221no—
250C4×Q8—1221no—
251C4×Q8—1221no—
252C4×D4—612no—
253C4×D4—1221no—
254#117—1221no—
255C2×C4⋊C4—1221no—
256C2×C42—611no—
257C2×C42—1221no—
258#106—22no—
259C4⋊C8—611no—
260C4⋊C8—611no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263#103—1222no—
264#102—1222no—
265C22⋊C8—1222no—
266C22⋊C8—1222no—
267C22⋊C8—1221no—
268#97—1221no—
269C4×C8—611no—
270C4×C8—1221no—
271C4×C8—1221no—
272C4×C8—1221no—
273C4×C8C2×#692311yescharacteristic
274#95—1221no—
275#41—2441no—
276#41—2441no—
277#41—2441no—
278#41—2441no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—1221no—
282C2×D4—612no—
283C2×D4—1222no—
284C2×D4—1221no—
285C2×D4—1221no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—1222no—
290C22×C4—2441no—
291SD16—2444no—
292M16—612no—
293M16—1221no—
294M16—1221no—
295M16—1221no—
296M16—2444no—
297C2×C8—21no—
298C2×C8—1221no—
299C2×C8—1221no—
300C2×C8—1221no—
301C2×C8—1221no—
302C2×C8—1221no—
303C2×C8—2442no—
304C2×C8—2444no—
305C2×C8—2441no—
306C2×C8—2441no—
307C2×C8—2441no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4C22⋊#692312yes—
318C22⋊C4—1222no—
319C22⋊C4—1222no—
320C22⋊C4—2441no—
321C42—611no—
322C42—611no—
323C42—611no—
324C42—2441no—
325C42—2441no—
326C42—2441no—
327C42C22×#692311yescharacteristic
328C23—611no—
329C23—1222no—
330Q8—41no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—2442no—
335D4—2442no—
336D4—2442no—
337D4—2442no—
338D4—2441no—
339D4—2441no—
340D4—2441no—
341D4—2441no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—611no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—611no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—4881no—
354C2×C4—4881no—
355C2×C4—2441no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2441no—
360C2×C4C2×C22⋊#692311yescharacteristic
361C2×C4D4×#692311yescharacteristic
362C2×C4D4×#692311yescharacteristic
363C8—22no—
364C8—1222no—
365C8—4881no—
366C8—2441no—
367C8—2441no—
368C8—2441no—
369C8—2441no—
370C22—611no—
371C22—611no—
372C22—1221no—
373C22—1221no—
374C22—2442no—
375C22—2442no—
376C22—2442no—
377C22#1684071011yescharacteristic
378C4—21no—
379C4—42no—
380C4—81no—
381C4—611no—
382C4—611no—
383C4—1221no—
384C4—1221no—
385C4—4882no—
386C4—4881no—
387C4SD16×#692311yescharacteristic
388C4#1684080111yescharacteristic
389C4#1684080511yescharacteristic
390C4#1684080711yescharacteristic
391C2—1221no—
392C2—1221no—
393C2—4882no—
394C2#2146181111yescharacteristic
395C2#2146181211yescharacteristic
396C2#2146182111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#498C4
socleC22×C61#386
prime radical#16840313C22
cosocleC42×D61C23
hypercenter#498#6923
hypocenterC61#9055
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #21461995C2semidirect
4. #21461128C2semidirect
5. #21460654C2semidirect
6. #21460651C2semidirect
24. #16840135C4semidirect
25. #16840114C4semidirect
26. #16840110C4semidirect
30. C61×#498C4semidirect
105. #11758213SD16semidirect
209. #498#6923semidirect
229. C61#9055semidirect

Maximal subgroups

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

subgroupindex
#214625652
#214619952
#214612002
#214611282
#214606542
#214606512
#214563802
#905561