Finite Group Databasebeta

← #30598722

Subgroups — #30598722

Order 15616.

720 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461348C212yesmaximal, complemented
2#21461300C212yesmaximal, complemented
3#21461070C211yesmaximal, characteristic, complemented
4#21461070C211yesmaximal, characteristic, complemented
5C2×#16840385C211yesmaximal, characteristic
6#16840782C2212yes—
7#16840782C2212yes—
8#16840621—24no—
9#16840618—24no—
10#16840607—21no—
11#16840607—21no—
12#16840605—21no—
13#16840605—21no—
14#16840385—24no—
15#16840385—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519772—21no—
18C2×#14519766C2211yescharacteristic
19C2×#14519763—21no—
20C2×#14519759—21no—
21C2×#14519757—21no—
22C2×#14519751C411yescharacteristic
23C2×Q8×Dic61C411yescharacteristic
24C2×#14519739C411yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27#14519844—22no—
28#14519844—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C22×Dic122—21no—
34C22×D244C2×C411yescharacteristic
35C22×C22⋊D61—21no—
36C22×C4×D61—21no—
37#14519776—22no—
38#14519776D414yes—
39#14519772—28no—
40#14519766—22no—
41#14519766D414yes—
42#14519763—28no—
43#14519759—28no—
44#14519757—28no—
45#14519751—22no—
46#14519751—22no—
47Q8×Dic61—22no—
48Q8×Dic61—22no—
49#14519739—22no—
50#14519739—22no—
51#14519736—22no—
52#14519736—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61D411yescharacteristic
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758196—21no—
58C2×C22⋊Dic61D411yescharacteristic
59C2×C22⋊Dic61D411yescharacteristic
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C22×Q8×C61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C24×C61C22⋊C411yescharacteristic
75C23×D61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—42no—
78C2×Q8×C61C22⋊C412yes—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C2×C22⋊D61—24no—
87C2×C22⋊D61—82no—
88C22×Dic61—21no—
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C22×Dic61C2×D411yescharacteristic
93C2×D244—21no—
94C2×D244—42no—
95C2×D244C22⋊C412yes—
96C2×C4×D61—24no—
97C2×C4×D61—82no—
98C2×Dic122—24no—
99C2×Dic122—82no—
100C2×#92420—42no—
101C2×#92420—42no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61C22⋊C412yes—
105C22⋊C4×C61—24no—
106C22⋊Dic61—22no—
107C22⋊Dic61—22no—
108#11758198—24no—
109#11758198—24no—
110#11758198—24no—
111#11758198—24no—
112C4⋊Dic61—24no—
113C4⋊Dic61—21no—
114C4⋊Dic61—22no—
115C4⋊Dic61—22no—
116C4⋊Dic61C22⋊C412yes—
117#11758196—24no—
118C4×Dic61—24no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61C2×D412yes—
122C4×Dic61C2×D412yes—
123C23×C61—21no—
124C23×C61—22no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61C2×C22⋊C411yescharacteristic
128C22×D61—22no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132Q8×C61—42no—
133Q8×C61—42no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C22⋊D61—84no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61—21no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61#12012yes—
155C2×Dic61#12012yes—
156D244—42no—
157D244—42no—
158C4×D61—84no—
159Dic122—84no—
160#92420—82no—
161#92420—82no—
162#8523—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—82no—
169C22×C61#37911yescharacteristic
170C22×C61#41312yes—
171C2×D61—42no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—84no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—42no—
178C4×C61—41no—
179C4×C61—42no—
180Dic61—22no—
181Dic61—22no—
182Dic61—42no—
183Dic61—41no—
184Dic61—41no—
185C2×#542—611no—
186#1556—612no—
187#1533—612no—
188#1464—611no—
189#1464—611no—
190C2×C61—22no—
191C2×C61—81no—
192C2×C61#178511yescharacteristic
193C2×C61#179511yescharacteristic
194C2×C61#179611yescharacteristic
195D61—82no—
196#542—1224no—
197#542—1224no—
198C2×C4⋊Q8—611no—
199C2×#126—1221no—
200C2×#124—611no—
201C2×#124—1221no—
202C2×#123—1221no—
203C2×C22⋊Q8—1221no—
204C2×C22⋊Q8#692311yescharacteristic
205C2×#121—611no—
206C2×C4×Q8—611no—
207C2×C4×D4—611no—
208C2×#117—611no—
209#432—612no—
210#432—612no—
211#332—1224no—
212#329—1224no—
213#324—1221no—
214#324—1221no—
215#323—1221no—
216#323—1221no—
217C61#852311yescharacteristic, complemented, Sylow
218C22×Q8—1221no—
219C22×Q8C2×#692311yescharacteristic
220C22×D4—611no—
221C22×D4—1221no—
222C23×C4—1221no—
223C23×C4C2×#692311yescharacteristic
224C4⋊Q8—614no—
225C4⋊Q8—1222no—
226#126—1228no—
227#124—614no—
228#124—1222no—
229#124—1228no—
230#123—1228no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—1228no—
234#121—1222no—
235#121—1222no—
236C4×Q8—1222no—
237C4×Q8—1222no—
238C4×D4—1222no—
239C4×D4—1222no—
240#117—1222no—
241#117—1222no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4C2×#692311yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—611no—
251C2×C22⋊C4—611no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C42—611no—
257C2×C42—611no—
258C2×C42—1221no—
259C4⋊C8—1222no—
260C4⋊C8—1222no—
261C22⋊C8—1222no—
262C22⋊C8—1222no—
263#97—1222no—
264#97—1222no—
265C24—1221no—
266C24C22⋊#692311yescharacteristic
267C2×Q8—21no—
268C2×Q8—42no—
269C2×Q8—1224no—
270C2×Q8—4882no—
271C2×Q8C22⋊#692312yes—
272C2×D4—612no—
273C2×D4—1224no—
274C2×D4—1221no—
275C2×D4—4882no—
276C2×D4—2442no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—1224no—
287C22×C4—1221no—
288C22×C4—4882no—
289C22×C4C22⋊#692311yescharacteristic
290C22×C4C22×#692311yescharacteristic
291C2×C8—2442no—
292C2×C8—2442no—
293C4⋊C4—24no—
294C4⋊C4—21no—
295C4⋊C4—612no—
296C4⋊C4—1222no—
297C4⋊C4—1222no—
298C4⋊C4—1224no—
299C4⋊C4—1224no—
300C4⋊C4—1221no—
301C4⋊C4C22⋊#692312yes—
302C22⋊C4—24no—
303C22⋊C4—1224no—
304C22⋊C4—1224no—
305C22⋊C4—1224no—
306C22⋊C4—1224no—
307C22⋊C4—1222no—
308C22⋊C4—1222no—
309C42—612no—
310C42—612no—
311C42—1224no—
312C42—1221no—
313C42—1221no—
314C23—21no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—1222no—
319C23—2442no—
320C23—2442no—
321C23—2442no—
322C23C2×C22⋊#692311yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—4884no—
326D4—4884no—
327D4—2442no—
328D4—2442no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—1221no—
341C2×C4—1222no—
342C2×C4—1221no—
343C2×C4—4884no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2442no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4C2×C22⋊#692311yescharacteristic
350C2×C4C2×C22⋊#692311yescharacteristic
351C8—4882no—
352C8—4882no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—82no—
359C22—4884no—
360C22—2442no—
361C22—2442no—
362C22—2442no—
363C22#1684070311yescharacteristic
364C22#1684074912yes—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—1222no—
371C4—1222no—
372C4—2441no—
373C4—2442no—
374C4—2441no—
375C2—22no—
376C2—81no—
377C2—4882no—
378C2#2146165611yescharacteristic
379C2#2146167011yescharacteristic
380C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61#379
prime radicalC2×#14519751C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8523
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21461348C2semidirect
2. #21461300C2semidirect
3. #21461070C2semidirect
4. #21461070C2semidirect
217. C61#8523semidirect

Maximal subgroups

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

subgroupindex
#214613482
#214613002
#214610702
C2×#168403852
#852361