Finite Group Databasebeta

← #30606543

Subgroups — #30606543

Order 15616.

564 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462309C211yesmaximal, characteristic
2D61⋊#354C211yesmaximal, characteristic
3#99×#6923C212yesmaximal
4#21460912C212yesmaximal
5#354×D61C211yesmaximal, characteristic
6C2×Q8×#6923C411yescharacteristic, complemented
7C2×D4×#6923C2211yescharacteristic
8D61⋊#99C2212yes—
9(C2×#6923)⋊Q8C411yescharacteristic, complemented
10#16840714C2211yescharacteristic
11#16840712—22no—
12#16840709—22no—
13#16840708C411yescharacteristic, complemented
14C22⋊C4×#6923—22no—
15#16840698C411yescharacteristic, complemented
16#16840669C412yescomplemented
17#16840668C412yescomplemented
18#16840639—22no—
19#124×D61C2211yescharacteristic
20#99×D61C2212yes—
21#16839557C411yescharacteristic, complemented
22#16839556C412yescomplemented
23C61×#354C411yescharacteristic, complemented
24C23×#6923D412yes—
25Q8×#6923—42no—
26D4×#6923—42no—
27C2×C4⋊#6923—21no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923C2×C411yescharacteristic
30C2×C4⋊#6923C2×C411yescharacteristic
31C2×C22⋊#6923—22no—
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923D412yes—
34C2×C4×#6923—21no—
35C2×C4×#6923—42no—
36C2×C4×#6923C2×C411yescharacteristic
37C2×C4×#6923C2×C411yescharacteristic
38#14519836C2×C414yes—
39#14519835—22no—
40(C2×#6923)⋊C4—21no—
41(C2×#6923)⋊C4—22no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—42no—
44C2×Q8×D61C2×C411yescharacteristic
45C2×D4×D61C2311yescharacteristic
46#14519766C2×C411yescharacteristic
47#14519765C2×C411yescharacteristic
48#14519764C2×C411yescharacteristic
49#14519763—22no—
50C22⋊C4×D61—22no—
51C22⋊C4×D61—22no—
52C42×D61C2×C411yescharacteristic
53#14519692C2×C412yes—
54#14519691—22no—
55#124×C61C2×C411yescharacteristic
56#99×C61C2×C412yes—
57C23×D61C2×D412yes—
58C22×#6923—22no—
59C22×#6923—22no—
60C22×#6923—21no—
61C22×#6923—21no—
62C22×#6923—42no—
63C22×#6923—82no—
64C22×#6923C22⋊C411yescharacteristic
65C22×#6923C22⋊C412yes—
66C22×#6923C22⋊C411yescharacteristic
67C2×Q8×C61C4211yescharacteristic, complemented
68C2×D4×C61C22×C411yescharacteristic
69C2×C22⋊D61#4112yes—
70Q8×D61—41no—
71Q8×D61—41no—
72D4×D61—41no—
73D4×D61—41no—
74C2×D244C22×C411yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—42no—
78C2×C4×D61C22×C411yescharacteristic
79C2×Dic122—21no—
80C2×Dic122C4211yescharacteristic, complemented
81C22⋊#6923—24no—
82C22⋊#6923—42no—
83C22⋊#6923—42no—
84C22⋊#6923—82no—
85C4⋊#6923—41no—
86C4⋊#6923—41no—
87C4⋊#6923—42no—
88C4⋊#6923—82no—
89C4×#6923—22no—
90C4×#6923—41no—
91C4×#6923—42no—
92C4×#6923—84no—
93C22⋊C4×C61—22no—
94C22⋊C4×C61—22no—
95C42×C61C4211yescharacteristic, complemented
96C22⋊Dic61—22no—
97C22⋊Dic61—22no—
98#11758198—22no—
99#11758198—22no—
100#11758198—42no—
101C4×Dic61—21no—
102C4×Dic61C4211yescharacteristic, complemented
103C23×C61C4×D412yes—
104C22×D61—22no—
105C22×D61—22no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61C2×C22⋊C411yescharacteristic
109C22×D61C4×D412yes—
110C2×#6923—21no—
111C2×#6923—22no—
112C2×#6923—41no—
113C2×#6923—44no—
114C2×#6923—42no—
115C2×#6923—41no—
116C2×#6923—42no—
117C2×#6923—44no—
118C2×#6923—41no—
119C2×#6923—41no—
120C2×#6923—41no—
121C2×#6923—41no—
122C2×#6923—82no—
123C2×#6923—82no—
124C2×#6923—82no—
125C2×#6923#9912yes—
126Q8×C61—41no—
127D4×C61—41no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—42no—
131C2×C4×C61C2×C4211yescharacteristic
132C22⋊D61—42no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—42no—
136C2×Dic61#11711yescharacteristic
137D244—41no—
138C4×D61—21no—
139C4×D61—21no—
140C4×D61—41no—
141C4×D61—41no—
142C4×D61—41no—
143C4×D61—41no—
144C4×D61—82no—
145C4×D61—82no—
146Dic122—42no—
147Dic122—41no—
148#12627—611nomaximal, Sylow
149C22×C61—22no—
150C22×C61—42no—
151C22×C61C4×C22⋊C411yescharacteristic
152C2×D61—21no—
153C2×D61—21no—
154C2×D61—22no—
155C2×D61—42no—
156C2×D61—42no—
157C2×D61—42no—
158C2×D61—82no—
159C2×D61C2×#9911yescharacteristic
160C2×D61#41011yescharacteristic
161#6923—22no—
162#6923—42no—
163#6923—41no—
164#6923—81no—
165#6923—84no—
166#6923—81no—
167C4×C61—21no—
168C4×C61—41no—
169C4×C61—41no—
170C4×C61—82no—
171Dic61—21no—
172Dic61—41no—
173Dic61—41no—
174Dic61—82no—
175#2326—611no—
176C2×#354—611no—
177C2×#354—611no—
178C4×#99—612no—
179#1393—612no—
180C2×C61—21no—
181C2×C61—42no—
182C2×C61C4×#9911yescharacteristic
183D61—21no—
184D61—42no—
185D61#212711yescharacteristic
186D61#212911yescharacteristic
187C2×#124—611no—
188C2×C4×Q8—611no—
189C2×C4×D4—611no—
190C2×#99—612no—
191C2×#99—612no—
192#391—611no—
193#390—611no—
194#388—1222no—
195#385—1222no—
196#384—611no—
197C4×C22⋊C4—1222no—
198#376—611no—
199#354—612no—
200#354—612no—
201#354—611no—
202#354—612no—
203#354#692311yescharacteristic, complemented
204#342—1222no—
205C61#1262711yescharacteristic, complemented, Sylow
206C22×Q8—611no—
207C22×D4—611no—
208C23×C4—612no—
209#124—611no—
210#124—611no—
211#124—611no—
212#124—1222no—
213#124C2×#692311yescharacteristic
214C4×Q8—2442no—
215C4×D4—2442no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—611no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1221no—
220C2×C22⋊C4—612no—
221C2×C22⋊C4—1222no—
222C2×C22⋊C4—1222no—
223C2×C22⋊C4—1222no—
224C2×C22⋊C4—1222no—
225C2×C42—611no—
226C2×C42—611no—
227C2×C42—611no—
228C2×C42—1221no—
229C2×C42—2442no—
230#99—614no—
231#99—612no—
232#99—1222no—
233#99—1222no—
234#99C2×#692312yes—
235#95—1221no—
236#95—1221no—
237#95—1222no—
238#95—2442no—
239C24—612no—
240C2×Q8—611no—
241C2×Q8—1221no—
242C2×Q8—2441no—
243C2×Q8—2441no—
244C2×Q8C4×#692311yescharacteristic, complemented
245C2×D4—612no—
246C2×D4—611no—
247C2×D4—2441no—
248C2×D4—2441no—
249C2×D4C22×#692311yescharacteristic
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—612no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1222no—
257C22×C4—1222no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—4882no—
261C22×C4—2442no—
262C22×C4—2442no—
263C4⋊C4—4882no—
264C4⋊C4—2442no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—1222no—
270C22⋊C4—1222no—
271C22⋊C4—1222no—
272C22⋊C4—1224no—
273C22⋊C4—1222no—
274C22⋊C4—4882no—
275C22⋊C4—2442no—
276C22⋊C4—2442no—
277C22⋊C4—2442no—
278C42—611no—
279C42—1221no—
280C42—1222no—
281C42—4884no—
282C42—2441no—
283C42—2442no—
284C42C4×#692311yescharacteristic, complemented
285C23—611no—
286C23—612no—
287C23—1222no—
288C23—1222no—
289C23—2442no—
290C23—2442no—
291C23D4×#692312yes—
292Q8—41no—
293Q8—2441no—
294Q8—2442no—
295D4—41no—
296D4—2441no—
297D4—2442no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—612no—
302C2×C4—611no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1222no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—4882no—
310C2×C4—4882no—
311C2×C4—4882no—
312C2×C4—4882no—
313C2×C4—4882no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2444no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2444no—
324C2×C4—2441no—
325C2×C4—2442no—
326C2×C4—2441no—
327C2×C4—2442no—
328C2×C4—2441no—
329C2×C4C2×C4×#692311yescharacteristic
330C22—22no—
331C22—42no—
332C22—611no—
333C22—611no—
334C22—1221no—
335C22—1221no—
336C22—1222no—
337C22—4882no—
338C22—2442no—
339C22—2442no—
340C22—2442no—
341C22C22⋊C4×#692311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—82no—
346C4—1222no—
347C4—1221no—
348C4—4882no—
349C4—4881no—
350C4—4881no—
351C4—4884no—
352C4—2441no—
353C4—2442no—
354C4—2441no—
355C4—2441no—
356C2—21no—
357C2—42no—
358C2—611no—
359C2—611no—
360C2—1221no—
361C2—2442no—
362C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#354C4
socleC2×C61C4×#99
prime radical, cosocleC2×D4×D61C23
hypercenter#354#6923
hypocenterC61#12627
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×#6923C4semidirect
9. (C2×#6923)⋊Q8C4semidirect
13. #16840708C4semidirect
15. #16840698C4semidirect
16. #16840669C4semidirect
17. #16840668C4semidirect
21. #16839557C4semidirect
22. #16839556C4semidirect
23. C61×#354C4semidirect
67. C2×Q8×C61C42semidirect
80. C2×Dic122C42semidirect
95. C42×C61C42semidirect
102. C4×Dic61C42semidirect
203. #354#6923semidirect
205. C61#12627semidirect
244. C2×Q8C4×#6923semidirect
284. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623092
D61⋊#3542
#99×#69232
#214609122
#354×D612
#1262761