Finite Group Databasebeta

← #29596281

Subgroups — #29596281

Order 15616.

642 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 87 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#21448308C211yesmaximal, characteristic
2#21445746C212yesmaximal
3#21443946C211yesmaximal, characteristic, complemented
4#21443945C212yesmaximal, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C22×#11758226C2211yescharacteristic
7C2×#14519735C411yescharacteristic
8C2×C42×D61C411yescharacteristic
9C2×#14519696C2212yes—
10C2×#14519693—22no—
11#16839725—22no—
12#16839669C2211yescharacteristic
13#16839664C411yescharacteristic
14#16839663C411yescharacteristic
15#16839640—22no—
16#16839609—22no—
17C2×#101×C61C2212yes—
18C61×#391C2211yescharacteristic
19C2×#11758226—22no—
20C2×#11758226—21no—
21C2×#11758226—21no—
22C2×#11758226—42no—
23C2×#11758226—42no—
24C22×D244C2×C411yescharacteristic
25C22×D244D412yes—
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C2×#11758194—22no—
30#14519735—42no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758198—21no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C42×D61—42no—
39#14519696—22no—
40#14519696—22no—
41#14519693—44no—
42#14519689—44no—
43#14519684—22no—
44C22×Q8×C61C2311yescharacteristic
45C2×M16×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C42×C61C2×C411yescharacteristic
48#101×C61—22no—
49#101×C61—22no—
50#95×C61—22no—
51C23×D61C22⋊C412yes—
52C23×D61C22⋊C411yescharacteristic
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61#4112yes—
56C2×Q8×C61#4111yescharacteristic
57C2×Q8×C61#4111yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61C22⋊C411yescharacteristic
65#11758226—44no—
66#11758226—48no—
67#11758226—44no—
68C2×D244—24no—
69C2×D244—21no—
70C2×D244—21no—
71C2×D244—22no—
72C2×D244—22no—
73C2×D244—41no—
74C2×D244—41no—
75C2×D244—42no—
76C2×C4×D61—22no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—24no—
83C2×C4×D61—42no—
84C2×C4×D61—42no—
85C2×C4×D61—42no—
86C2×C4×D61—42no—
87M16×C61—42no—
88M16×C61—42no—
89C2×C8×C61—42no—
90C4⋊C4×C61—41no—
91C42×C61—41no—
92#11758198—44no—
93#11758198—44no—
94#11758198—42no—
95C4⋊Dic61—41no—
96C4×Dic61—42no—
97C4×Dic61—42no—
98C4×Dic61—41no—
99#11758194—42no—
100#11758194—42no—
101C2×#92418—42no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—22no—
106C22×D61—22no—
107C22×D61—22no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61#9912yes—
113Q8×C61—41no—
114Q8×C61—42no—
115Q8×C61—41no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61#11711yescharacteristic
126C2×C4×C61#11711yescharacteristic
127C2×C4×C61#12312yes—
128C2×C4×C61#12312yes—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—22no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61#9912yes—
137D244—44no—
138D244—44no—
139D244—42no—
140D244—42no—
141C4×D61—24no—
142C4×D61—24no—
143C4×D61—44no—
144C4×D61—44no—
145C4×D61—48no—
146C8×C61—82no—
147#92418—82no—
148#12861—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61#38111yescharacteristic
152C22×C61C2×#9911yescharacteristic
153C22×C61C2×#9911yescharacteristic
154C2×D61—22no—
155C2×D61—42no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—42no—
159C2×D61—44no—
160C2×D61—42no—
161C2×D61—42no—
162C2×D61—42no—
163C2×D61—82no—
164C4×C61—22no—
165C4×C61—22no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—42no—
169C4×C61—81no—
170Dic61—22no—
171Dic61—42no—
172Dic61—81no—
173#2297—611no—
174#1785—612no—
175#1394—612no—
176#1394—611no—
177#1394D6111yescharacteristic, complemented
178C2×C61—21no—
179C2×C61—21no—
180C2×C61#181911yescharacteristic
181C2×C61#213211yescharacteristic
182C2×C61#213211yescharacteristic
183D61—42no—
184D61—82no—
185C22×#41—611no—
186C2×#117—611no—
187C22×C42—611no—
188C2×#101—612no—
189C2×#101C2×D6112yes—
190C2×#100—1222no—
191#407—1222no—
192#391—611no—
193#391C2×D6111yescharacteristic
194#390—611no—
195#390—611no—
196#386—1222no—
197C4×C22⋊C4—1222no—
198C61#1286111yescharacteristic, complemented, Sylow
199C2×#41—1222no—
200C2×#41—1221no—
201C2×#41—1221no—
202C2×#41—2442no—
203C2×#41—2442no—
204C22×Q8C22×D6111yescharacteristic
205C22×D4—611no—
206C22×D4—612no—
207C23×C4—611no—
208C23×C4—612no—
209C23×C4—1221no—
210C2×M16—22no—
211C2×M16—1222no—
212#117—2442no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4C4×D6111yescharacteristic
215C2×C22⋊C4—1221no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1222no—
218C2×C42—611no—
219C2×C42—1221no—
220C2×C42—1221no—
221C2×C42—2442no—
222C2×C42C4×D6111yescharacteristic
223#101—22no—
224#101—22no—
225#101—1222no—
226#101—1222no—
227#100—2444no—
228C22⋊C8—2444no—
229#95—22no—
230#95—1222no—
231C24—611no—
232C24—612no—
233#41—2448no—
234#41—2444no—
235#41—2444no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8#1175822212yes—
239C2×Q8#1175822411yescharacteristic
240C2×Q8#1175822611yescharacteristic
241C2×D4—1224no—
242C2×D4—1221no—
243C2×D4—1221no—
244C2×D4—1222no—
245C2×D4—1222no—
246C2×D4—2441no—
247C2×D4—2441no—
248C2×D4—2442no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—611no—
252C22×C4—1222no—
253C22×C4—1224no—
254C22×C4—1222no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1222no—
259C22×C4—1222no—
260C22×C4—1221no—
261C22×C4—2442no—
262C22×C4—2442no—
263C22×C4—2442no—
264C22×C4—2442no—
265C22×C4C2×C4×D6111yescharacteristic
266C22×C4D4×D6112yes—
267M16—42no—
268M16—42no—
269M16—2442no—
270M16—2442no—
271C2×C8—42no—
272C2×C8—2442no—
273C4⋊C4—41no—
274C4⋊C4—2441no—
275C22⋊C4—2444no—
276C22⋊C4—2444no—
277C22⋊C4—2442no—
278C42—41no—
279C42—2441no—
280C42—2442no—
281C42—2442no—
282C23—612no—
283C23—1221no—
284C23—1221no—
285C23—1222no—
286C23—1222no—
287C23—1222no—
288C23—2442no—
289C23—2442no—
290C23—2441no—
291C23—2441no—
292C23C22⋊C4×D6111yescharacteristic
293Q8—42no—
294Q8—41no—
295Q8—41no—
296D4—2444no—
297D4—2444no—
298D4—2442no—
299D4—2442no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—612no—
310C2×C4—1224no—
311C2×C4—1224no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1222no—
315C2×C4—2444no—
316C2×C4—2444no—
317C2×C4—2448no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4#1451973411yescharacteristic
323C2×C4#1451973511yescharacteristic
324C2×C4#1451975912yes—
325C2×C4#1451976212yes—
326C8—82no—
327C8—4882no—
328C22—21no—
329C22—21no—
330C22—1222no—
331C22—4882no—
332C22—2444no—
333C22—2442no—
334C22—2442no—
335C22—2442no—
336C22—2442no—
337C22—2442no—
338C22—2441no—
339C22—2441no—
340C22#1683962411yescharacteristic
341C22#99×D6111yescharacteristic
342C22#99×D6111yescharacteristic
343C4—22no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—81no—
349C4—1222no—
350C4—4881no—
351C4—2442no—
352C2—21no—
353C2—21no—
354C2—4882no—
355C2—2442no—
356C2#2144589211yescharacteristic
357C2#2144750111yescharacteristic
358C2#2144750311yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12861
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #21443946C2semidirect
4. #21443945C2semidirect
5. C61×#1394C2semidirect
177. #1394D61semidirect
198. C61#12861semidirect

Maximal subgroups

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

subgroupindex
#214483082
#214457462
#214439462
#214439452
C61×#13942
#1286161