Finite Group Databasebeta

← #30598299

Subgroups — #30598299

Order 15616.

788 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461725C212yesmaximal, complemented
2C4×#14519835C211yesmaximal, characteristic
3#21461218C212yesmaximal, complemented
4#21461074C211yesmaximal, characteristic
5C2×#16840447C211yesmaximal, characteristic
6C2×#14519835—21no—
7#16840733C2212yes—
8C4×C22⋊#6923C2212yes—
9#16840640—21no—
10#16840632—24no—
11#16840615—24no—
12#16840608—21no—
13#16840605—21no—
14#16840447—24no—
15#16840447—24no—
16C2×#14519772—21no—
17C2×#14519768C411yescharacteristic, complemented
18C2×#14519767—21no—
19C2×#14519760—21no—
20C2×#14519757C2211yescharacteristic
21C2×#14519756C411yescharacteristic, complemented
22C2×#14519742C411yescharacteristic, complemented
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519736—21no—
25C2×#14519735C2211yescharacteristic
26C22×C4⋊C4×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28C2×C22⋊#6923—22no—
29C2×C4×#6923—22no—
30#14519835—24no—
31#14519834—22no—
32C22⋊#92420—22no—
33(C2×#6923)⋊C4—22no—
34C22×Dic122C2×C411yescharacteristic
35C22×C22⋊D61D411yescharacteristic
36C22×C4×D61C2311yescharacteristic
37#14519772—28no—
38#14519768—22no—
39#14519768—22no—
40#14519767—28no—
41#14519760—28no—
42#14519757—22no—
43#14519757D414yes—
44#14519756—22no—
45#14519756—22no—
46#14519742—22no—
47#14519742—22no—
48C4×C22⋊D61—22no—
49C4×C22⋊D61—22no—
50#14519736—28no—
51#14519735—22no—
52#14519735D414yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196D411yescharacteristic
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2×C411yescharacteristic
68C23×C4×C61—21no—
69C23×C4×C61C2×C411yescharacteristic
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C4⋊C4×C61C2×C411yescharacteristic
75C24×C61C22⋊C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—42no—
78C22×C4×C61—24no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22×C411yescharacteristic
88C2×C22⋊D61—22no—
89C2×C22⋊D61—22no—
90C2×C22⋊D61—41no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C2×C4×D61—42no—
97C2×C4×D61C2×D412yes—
98C2×C4×D61C2×D412yes—
99C2×Dic122—21no—
100C2×Dic122—42no—
101C2×Dic122C22⋊C412yes—
102C22⋊#6923—24no—
103C22⋊#6923—42no—
104C2×#92420—42no—
105C4×#6923—44no—
106#11758213—44no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61C22⋊C412yes—
111C4⋊C4×C61C22⋊C412yes—
112C22⋊Dic61—24no—
113C22⋊Dic61—22no—
114#11758198—24no—
115#11758198—24no—
116#11758198—22no—
117C4⋊Dic61—24no—
118C4⋊Dic61—21no—
119C4⋊Dic61—24no—
120C4⋊Dic61C22⋊C412yes—
121#11758196—24no—
122#11758196—24no—
123#11758196—24no—
124#11758196—22no—
125C4×Dic61—24no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—24no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61#10412yes—
134C23×C61C2×C22⋊C411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61#12012yes—
140C2×#6923—44no—
141C2×#6923—82no—
142C2×#6923—82no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—42no—
148C2×C4×C61—48no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61C2×C22⋊C411yescharacteristic
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C22⋊D61—44no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61#12012yes—
169C4×D61—24no—
170C4×D61—42no—
171Dic122—42no—
172Dic122—42no—
173#92420—82no—
174#8363—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61#37911yescharacteristic
183C22×C61C2×#10411yescharacteristic
184C22×C61#42111yescharacteristic
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—81no—
189C2×D61—81no—
190#6923—84no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61#41012yes—
196Dic61—22no—
197Dic61—41no—
198Dic61—42no—
199Dic61—42no—
200Dic61—41no—
201C2×#552—611no—
202#1826—612no—
203C4×#99—611no—
204#1508—612no—
205#1465—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—42no—
209C2×C61#178211yescharacteristic
210C2×C61#178911yescharacteristic
211C2×C61#179611yescharacteristic
212D61—42no—
213#552—1224no—
214#552—1224no—
215C2×#126—1221no—
216C2×#125—611no—
217C2×#125—1221no—
218C2×#123—1221no—
219C2×C22⋊Q8—611no—
220C2×C22⋊Q8—611no—
221C2×C4×Q8—611no—
222C2×C4×D4—611no—
223C2×#117—611no—
224C2×#117—1221no—
225C22×C4⋊C4#692311yescharacteristic, complemented
226C2×#99—1221no—
227#407—612no—
228C4×C22⋊C4—612no—
229#342—1221no—
230#339—1224no—
231#328—1224no—
232#324—1221no—
233#323—1221no—
234C61#836311yescharacteristic, complemented, Sylow
235C22×Q8—611no—
236C22×D4—611no—
237C23×C4—21no—
238C23×C4—611no—
239C23×C4C2×#692311yescharacteristic
240C2×M16—1222no—
241#126—1228no—
242#125—1222no—
243#125—1228no—
244#125—1222no—
245#123—1228no—
246C22⋊Q8—614no—
247C22⋊Q8—1222no—
248C22⋊Q8—1222no—
249C22⋊Q8—1222no—
250C4×Q8—1222no—
251C4×Q8—1222no—
252C4×D4—1222no—
253C4×D4—1222no—
254#117—614no—
255#117—1222no—
256#117—1228no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4C2×#692311yescharacteristic
268C2×C4⋊C4C2×#692311yescharacteristic
269C2×C22⋊C4—611no—
270C2×C22⋊C4—611no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1222no—
275C2×C42—611no—
276C2×C42—1221no—
277C2×C42—1221no—
278C2×C42—1222no—
279#99—1224no—
280C22⋊C8—1222no—
281C22⋊C8—1222no—
282#95—1222no—
283C24—611no—
284C24C22⋊#692311yescharacteristic
285C2×Q8—612no—
286C2×Q8—1221no—
287C2×Q8—2442no—
288C2×D4—1222no—
289C2×D4—1222no—
290C2×D4—2441no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—611no—
300C22×C4—612no—
301C22×C4—612no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—2442no—
307C22×C4—2442no—
308C22×C4C22⋊#692311yescharacteristic
309C22×C4C22×#692311yescharacteristic
310M16—2444no—
311C2×C8—2442no—
312C4⋊C4—24no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—612no—
316C4⋊C4—1222no—
317C4⋊C4—1224no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C4⋊C4—1224no—
321C4⋊C4—1224no—
322C4⋊C4—1221no—
323C4⋊C4C22⋊#692312yes—
324C4⋊C4C22⋊#692312yes—
325C22⋊C4—1222no—
326C22⋊C4—1224no—
327C22⋊C4—1224no—
328C22⋊C4—1222no—
329C22⋊C4—1224no—
330C22⋊C4—1224no—
331C22⋊C4—2442no—
332C42—1224no—
333C42—1224no—
334C42—1221no—
335C42—1221no—
336C42—2444no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—612no—
342C23—1221no—
343C23—1221no—
344C23—2441no—
345C23—2441no—
346C23#1451984112yes—
347C23C2×C22⋊#692311yescharacteristic
348Q8—2442no—
349Q8—2442no—
350D4—2444no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—612no—
364C2×C4—1224no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—1221no—
371C2×C4—4882no—
372C2×C4—4882no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4—2444no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4C2×C22⋊#692311yescharacteristic
380C2×C4C2×C22⋊#692311yescharacteristic
381C8—4882no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—1222no—
390C22—1222no—
391C22—1222no—
392C22—4881no—
393C22—4881no—
394C22#1684070311yescharacteristic
395C22C2×#1451984111yescharacteristic
396C22#1684076011yescharacteristic
397C4—21no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—1222no—
402C4—4884no—
403C4—2442no—
404C4—2441no—
405C4—2441no—
406C4—2442no—
407C4#1684074112yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—2442no—
412C2#2146164811yescharacteristic
413C2#2146166011yescharacteristic
414C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8363
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #21461725C2semidirect
3. #21461218C2semidirect
17. C2×#14519768C4semidirect
21. C2×#14519756C4semidirect
22. C2×#14519742C4semidirect
26. C22×C4⋊C4×C61C4semidirect
225. C22×C4⋊C4#6923semidirect
234. C61#8363semidirect

Maximal subgroups

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

subgroupindex
#214617252
C4×#145198352
#214612182
#214610742
C2×#168404472
#836361