Finite Group Databasebeta

← #29577667

Subgroups — #29577667

Order 15616.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450277C211yesmaximal, characteristic
2#21445896C211yesmaximal, characteristic, complemented
3#21444349C212yesmaximal
4#21443673C212yesmaximal, complemented
5C61×#2921C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7#16839677—21no—
8#16839677—21no—
9#16839663C411yescharacteristic, complemented
10#16839663C411yescharacteristic, complemented
11#16839653—21no—
12#16839653—21no—
13#16839631C411yescharacteristic, complemented
14#16839627C411yescharacteristic, complemented
15#16839625C2212yes—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839529—21no—
18#16839529—21no—
19#16839497—24no—
20C2×#117×C61C2211yescharacteristic
21C61×#433—24no—
22C61×#423—21no—
23C61×#407C2212yes—
24C4×M16×C61—21no—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27#14519751—22no—
28#14519751—22no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—22no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61C2×C411yescharacteristic
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—22no—
45#14519684—22no—
46C23×C4×C61C2311yescharacteristic
47C2×M16×C61—22no—
48#117×C61—22no—
49#117×C61—22no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61D411yescharacteristic
53C2×C22⋊C4×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56C4⋊C8×C61—24no—
57C22⋊C8×C61—22no—
58C22⋊C8×C61D414yes—
59#97×C61—22no—
60C4×C8×C61—22no—
61C24×C61C2×D411yescharacteristic
62C23×D61—21no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61#4112yes—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—81no—
75C2×C22⋊D61—81no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—42no—
80C2×D244—22no—
81C2×D244—41no—
82C2×D244—41no—
83M16×C61—42no—
84M16×C61—42no—
85C2×C8×C61—24no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—22no—
88C22⋊C4×C61—22no—
89C22⋊C4×C61—22no—
90C42×C61—21no—
91C42×C61—21no—
92C42×C61—21no—
93C42×C61—21no—
94C22⋊Dic61—42no—
95C22⋊Dic61—42no—
96C22⋊Dic61—42no—
97#11758198—42no—
98#11758198—42no—
99#11758198—41no—
100#11758198—41no—
101#11758198—81no—
102#11758198—81no—
103C4⋊Dic61—22no—
104#11758196—81no—
105#11758196—81no—
106C4×Dic61—41no—
107C4×Dic61—41no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C23×C61#12112yes—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C4×D412yes—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12311yescharacteristic
138C22⋊D61—84no—
139C2×Dic61—22no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—82no—
149C2×Dic61—82no—
150C2×Dic61—82no—
151D244—82no—
152C8×C61—44no—
153#9446—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61#38611yescharacteristic
162C22×C61#41011yescharacteristic
163C22×C61#41011yescharacteristic
164C2×D61—81no—
165C2×D61—81no—
166C2×D61—81no—
167C2×D61—81no—
168C2×D61—81no—
169C2×D61—81no—
170C2×D61—81no—
171C4×C61—21no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—41no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61—41no—
178Dic61—42no—
179Dic61—81no—
180Dic61—81no—
181Dic61—162no—
182#2921D6111yescharacteristic, complemented
183#2500—611no—
184#1820—611no—
185#1510—612no—
186#1284—612no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—42no—
190C2×C61#188411yescharacteristic
191C2×C61#189911yescharacteristic
192C2×C61#191011yescharacteristic
193D61—161no—
194C2×#121—611no—
195C2×#117C2×D6111yescharacteristic
196#433—24no—
197#423—21no—
198#407C2×D6112yes—
199C4×M16—21no—
200#394—1221no—
201#394—1221no—
202#390—611no—
203#390—611no—
204#388—1221no—
205#388—1221no—
206#384—611no—
207#382—611no—
208#381—612no—
209C4×C22⋊C4—611no—
210#342—1221no—
211#342—1221no—
212#327—1224no—
213C61#944611yescharacteristic, complemented, Sylow
214C22×D4—611no—
215C22×D4—1221no—
216C23×C4C22×D6111yescharacteristic
217C2×M16—22no—
218#121—1222no—
219#121—1222no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4C4×D6111yescharacteristic
226C2×C4⋊C4C4×D6111yescharacteristic
227C2×C22⋊C4—1221no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4D24411yescharacteristic
235C2×C22⋊C4D24411yescharacteristic
236C2×C42—611no—
237C2×C42—611no—
238C2×C42C22⋊D6111yescharacteristic
239C2×C42C22⋊D6111yescharacteristic
240C4⋊C8—24no—
241C22⋊C8—22no—
242C22⋊C8C22⋊D6114yes—
243#97—22no—
244C4×C8—22no—
245#95—1221no—
246#95—1221no—
247#95—1221no—
248#95—1222no—
249#95—1222no—
250C24—1221no—
251C24C2×D24411yescharacteristic
252C2×D4—1222no—
253C2×D4—4881no—
254C2×D4—4881no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2442no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—2442no—
264C22×C4#1175819811yescharacteristic
265C22×C4#1175819811yescharacteristic
266C22×C4#1175819811yescharacteristic
267C22×C4#1175819811yescharacteristic
268C22×C4C2×C4×D6111yescharacteristic
269C22×C4#1175822212yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C2×C22⋊D6112yes—
272M16—42no—
273M16—42no—
274C2×C8—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—1222no—
278C4⋊C4—4881no—
279C4⋊C4—4881no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—4881no—
283C22⋊C4—4881no—
284C22⋊C4—2442no—
285C22⋊C4—2442no—
286C22⋊C4—2442no—
287C22⋊C4—2442no—
288C22⋊C4—2442no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—2441no—
296C42—2441no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—2441no—
302C23—2441no—
303C23—2441no—
304C23—2441no—
305C23—2441no—
306C23—2441no—
307C23—2441no—
308C23C2×#1175819811yescharacteristic
309C23#1451975112yes—
310D4—4882no—
311D4—4884no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—1222no—
327C2×C4—4882no—
328C2×C4—4882no—
329C2×C4—4882no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4C4×C22⋊D6112yes—
339C2×C4#1451974611yescharacteristic
340C2×C4#1451976011yescharacteristic
341C8—44no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—4881no—
350C22—4881no—
351C22—4881no—
352C22—4881no—
353C22—4881no—
354C22—4881no—
355C22—4881no—
356C22#1683964111yescharacteristic
357C22#1683973511yescharacteristic
358C22#1683973511yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—9762no—
367C4—4881no—
368C4—4881no—
369C4—2442no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—9761no—
374C2#2144620211yescharacteristic
375C2#2144627711yescharacteristic
376C2#2144634611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2921C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#2921D61
hypocenterC61#9446
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21445896C2semidirect
4. #21443673C2semidirect
5. C61×#2921C2semidirect
9. #16839663C4semidirect
10. #16839663C4semidirect
13. #16839631C4semidirect
14. #16839627C4semidirect
182. #2921D61semidirect
213. C61#9446semidirect

Maximal subgroups

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

subgroupindex
#214502772
#214458962
#214443492
#214436732
C61×#29212
#944661