Finite Group Databasebeta

← #29595174

Subgroups — #29595174

Order 15616.

720 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 115 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#21448383C211yesmaximal, characteristic
2C2×#16839554C212yesmaximal
3#99×Dic61C212yesmaximal, complemented
4#21443944C211yesmaximal, characteristic, complemented
5C61×#1393C211yesmaximal, characteristic, complemented
6C2×#14519773C2211yescharacteristic
7C2×D4×Dic61C2211yescharacteristic
8C2×#14519739C411yescharacteristic, complemented
9C2×#14519737—22no—
10C2×#14519692C2212yes—
11#16839664C411yescharacteristic, complemented
12#16839662—22no—
13#16839631C411yescharacteristic, complemented
14C22⋊C4×Dic61—22no—
15C42×Dic61C411yescharacteristic, complemented
16#16839554C418yescomplemented
17#16839528—22no—
18C2×#99×C61C2212yes—
19C61×#390C2211yescharacteristic
20C22×D244C2×C411yescharacteristic
21C22×C22⋊D61—22no—
22C23×Dic61D412yes—
23C22×C4×D61—21no—
24#14519773—21no—
25#14519773—21no—
26#14519773C2×C414yes—
27D4×Dic61—41no—
28D4×Dic61—41no—
29#14519739—41no—
30#14519739—41no—
31#14519737—48no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758196—22no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61D412yes—
36C2×#11758198—22no—
37C2×#11758198—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61—42no—
41C2×C4×Dic61C2×C411yescharacteristic
42C2×C4×Dic61C2×C411yescharacteristic
43#14519692—22no—
44#14519692C2×C414yescomplemented
45#14519684—42no—
46C22×D4×C61C2311yescharacteristic
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—22no—
50C2×C42×C61C2×C411yescharacteristic
51#99×C61—22no—
52#99×C61C2×C414yescomplemented
53C24×C61C2×D412yes—
54C23×D61—21no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61C22×C412yes—
58C2×D4×C61#4111yescharacteristic
59C2×D4×C61#4111yescharacteristic
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—42no—
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—24no—
65C2×C22⋊D61—24no—
66C2×C22⋊D61—44no—
67C2×C22⋊D61—44no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—44no—
72C22×Dic61—42no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C412yes—
75C22×Dic61C22⋊C411yescharacteristic
76C2×D244—21no—
77C2×D244—41no—
78C2×D244—41no—
79C2×D244C4212yescomplemented
80C2×C4×D61—44no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—44no—
86C22⋊C4×C61—42no—
87C42×C61—41no—
88C22⋊Dic61—24no—
89C22⋊Dic61—42no—
90C22⋊Dic61—42no—
91#11758198—42no—
92#11758198—44no—
93C4⋊Dic61—41no—
94#11758196—44no—
95C4×Dic61—24no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—42no—
100C4×Dic61—84no—
101C4×Dic61C4212yescomplemented
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61C4×D412yes—
108C23×C61C4×D412yes—
109C22×D61—22no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—41no—
113C22×D61—41no—
114D4×C61—42no—
115D4×C61—41no—
116D4×C61—41no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—42no—
123C2×C4×C61—44no—
124C2×C4×C61—41no—
125C2×C4×C61—82no—
126C2×C4×C61—82no—
127C2×C4×C61C2×C4211yescharacteristic
128C2×C4×C61#11711yescharacteristic
129C22⋊D61—48no—
130C22⋊D61—48no—
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—22no—
135C2×Dic61—21no—
136C2×Dic61—22no—
137C2×Dic61—44no—
138C2×Dic61—41no—
139C2×Dic61—42no—
140C2×Dic61—44no—
141C2×Dic61—82no—
142C2×Dic61—82no—
143C2×Dic61—82no—
144C2×Dic61#9911yescharacteristic
145C2×Dic61#9911yescharacteristic
146D244—42no—
147D244—42no—
148C4×D61—82no—
149C4×D61—82no—
150C4×#353—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—22no—
154C22×C61—22no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—82no—
160C22×C61C4×C22⋊C411yescharacteristic
161C22×C61C2×#9911yescharacteristic
162C22×C61#41011yescharacteristic
163C2×D61—44no—
164C2×D61—42no—
165C2×D61—82no—
166C2×D61—81no—
167C2×D61—81no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—42no—
171C4×C61—84no—
172C4×C61—81no—
173Dic61—24no—
174Dic61—21no—
175Dic61—42no—
176Dic61—41no—
177Dic61—84no—
178Dic61#35312yescomplemented
179C4×#127—611no—
180C2×#353—612no—
181C4×#99—612no—
182#1393—611no—
183#1393D6111yescharacteristic, complemented
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—42no—
187C2×C61—42no—
188C2×C61C4×#9911yescharacteristic
189C2×C61C2×#35311yescharacteristic
190C2×C61#212611yescharacteristic
191D61—82no—
192C2×#127—611no—
193C2×C4×D4—611no—
194C2×C4×D4—611no—
195C2×C4×D4—1222no—
196C2×#99—612no—
197C2×#99C2×D6112yes—
198#390—611no—
199#390—1222no—
200#390C2×D6111yescharacteristic
201#384—611no—
202C4×C22⋊C4—1222no—
203C43—611no—
204#353—618no—
205#342—1222no—
206C61C4×#35311yescharacteristic, complemented, Sylow
207C22×D4—611no—
208C22×D4—1222no—
209C22×D4C22×D6111yescharacteristic
210C23×C4—612no—
211C23×C4—1221no—
212#127—614no—
213#127—1221no—
214#127—1221no—
215C4×D4—2448no—
216C4×D4—2441no—
217C4×D4—2441no—
218C4×D4—2441no—
219C4×D4—2441no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4C4×D6111yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—612no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4—1221no—
228C2×C22⋊C4—1222no—
229C2×C42—611no—
230C2×C42—611no—
231C2×C42—1221no—
232C2×C42—1221no—
233C2×C42—2442no—
234C2×C42C4×D6111yescharacteristic
235#99—22no—
236#99—614no—
237#99—1222no—
238#99C4×D6114yescomplemented
239#95—2442no—
240C24—1221no—
241C24D4×D6112yes—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—612no—
245C2×D4—1224no—
246C2×D4—1224no—
247C2×D4—1221no—
248C2×D4—2441no—
249C2×D4—2441no—
250C2×D4—2444no—
251C2×D4—2444no—
252C2×D4C2×C4×D6112yes—
253C2×D4#1175822411yescharacteristic
254C2×D4#1175822611yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—612no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—1221no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4—2444no—
267C22×C4—2444no—
268C22×C4—2442no—
269C22×C4C2×C4×D6111yescharacteristic
270C4⋊C4—41no—
271C4⋊C4—2441no—
272C4⋊C4—2444no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—1224no—
277C22⋊C4—2444no—
278C22⋊C4—2442no—
279C22⋊C4—2442no—
280C22⋊C4—2442no—
281C42—41no—
282C42—612no—
283C42—1221no—
284C42—1221no—
285C42—1224no—
286C42—1221no—
287C42—4884no—
288C42—2442no—
289C23—22no—
290C23—22no—
291C23—42no—
292C23—42no—
293C23—1222no—
294C23—2442no—
295C23—2442no—
296C23—2441no—
297C23—2441no—
298C23C22⋊C4×D6111yescharacteristic
299C23#1451973712yes—
300C23#1451973912yes—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—2442no—
305D4—2442no—
306D4—2448no—
307D4—2448no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—611no—
319C2×C4—611no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—1221no—
325C2×C4—1222no—
326C2×C4—4882no—
327C2×C4—4882no—
328C2×C4—4882no—
329C2×C4—4882no—
330C2×C4—4882no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4—2441no—
334C2×C4—2442no—
335C2×C4C42×D6111yescharacteristic
336C2×C4#1451973511yescharacteristic
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22—4882no—
347C22—4881no—
348C22—4881no—
349C22—2444no—
350C22—2442no—
351C22#1683960911yescharacteristic
352C22#99×D6111yescharacteristic
353C22#1683973411yescharacteristic
354C4—21no—
355C4—21no—
356C4—42no—
357C4—84no—
358C4—81no—
359C4—612no—
360C4—1224no—
361C4—1221no—
362C4—4884no—
363C4—2442no—
364C4—2441no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—42no—
369C2—4882no—
370C2#2144560611yescharacteristic
371C2#353×D6111yescharacteristic
372C2#2144748311yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C2×C42
FrattiniC22×D4C22×D61
FittingC61×#1393C2
socleC22×C61C2×#99
prime radicalC2×#14519773C22
cosocleC22×D4×C61C23
hypercenter#1393D61
hypocenterC61C4×#353
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
3. #99×Dic61C2semidirect
4. #21443944C2semidirect
5. C61×#1393C2semidirect
8. C2×#14519739C4semidirect
11. #16839664C4semidirect
13. #16839631C4semidirect
15. C42×Dic61C4semidirect
16. #16839554C4semidirect
44. #14519692C2×C4semidirect
52. #99×C61C2×C4semidirect
79. C2×D244C42semidirect
101. C4×Dic61C42semidirect
178. Dic61#353semidirect
183. #1393D61semidirect
206. C61C4×#353semidirect
238. #99C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214483832
C2×#168395542
#99×Dic612
#214439442
C61×#13932
C4×#35361