Finite Group Databasebeta

← #29577997

Subgroups — #29577997

Order 15616.

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

normal subgroups only

This group has 558 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#21450089C211yesmaximal, characteristic
2#21445902C211yesmaximal, characteristic, complemented
3#21444145C212yesmaximal
4#21443728C212yesmaximal, complemented
5C61×#2921C211yesmaximal, characteristic, complemented
6C22×#11758198C2211yescharacteristic
7#16839726C2212yes—
8#16839697—21no—
9#16839688—21no—
10#16839653—21no—
11#16839653—21no—
12#16839634C411yescharacteristic
13#16839634C411yescharacteristic
14#16839625C2211yescharacteristic
15#16839617C411yescharacteristic
16#16839617C411yescharacteristic
17#16839531—21no—
18#16839531—21no—
19#16839485—24no—
20C2×#117×C61C2211yescharacteristic
21C61×#433—24no—
22C61×#423—21no—
23C61×#407C2212yes—
24C4×M16×C61—21no—
25C24×D61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C2×#11758194—22no—
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×C22⋊Dic61C2×C411yescharacteristic
31C2×C22⋊Dic61C2×C411yescharacteristic
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—22no—
36C2×#11758198—21no—
37C2×#11758198—22no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C22⋊#92418—22no—
41C22⋊#92418—22no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C23×C4×C61C2311yescharacteristic
47C2×M16×C61—22no—
48#117×C61—22no—
49#117×C61—22no—
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61C2×C411yescharacteristic
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×C61C22×C411yescharacteristic
62C23×D61—21no—
63C23×D61—22no—
64C23×D61—42no—
65C23×D61—41no—
66C23×D61—41no—
67C23×D61—81no—
68C22×C4×C61—41no—
69C22×C4×C61—41no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61#4112yes—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—22no—
82C22×Dic61—81no—
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—41no—
95C22⋊Dic61—41no—
96#11758198—42no—
97#11758198—42no—
98#11758198—41no—
99#11758198—42no—
100#11758198—41no—
101#11758198—81no—
102#11758198—81no—
103#11758196—81no—
104#11758196—81no—
105#11758194—42no—
106#11758194—42no—
107C2×#92418—42no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C23×C61C4×D412yes—
114C22×D61—22no—
115C22×D61—24no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—42no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—84no—
124C22×D61—84no—
125C22×D61—82no—
126C22×D61—82no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61#12112yes—
143C2×C4×C61#12311yescharacteristic
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—82no—
154C8×C61—44no—
155#92418—82no—
156#9512—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—22no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61#38611yescharacteristic
165C22×C61#41011yescharacteristic
166C22×C61#41011yescharacteristic
167C2×D61—42no—
168C2×D61—42no—
169C2×D61—42no—
170C2×D61—81no—
171C2×D61—81no—
172C2×D61—84no—
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—84no—
176C4×C61—22no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—41no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—41no—
183Dic61—81no—
184Dic61—81no—
185Dic61—81no—
186#2921D6111yescharacteristic, complemented
187#2482—611no—
188#1822—611no—
189#1462—612no—
190#1306—612no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—42no—
194C2×C61#189111yescharacteristic
195C2×C61#189911yescharacteristic
196C2×C61#191411yescharacteristic
197D61—82no—
198C2×#117C2×D6111yescharacteristic
199C22×C22⋊C4—611no—
200#433—24no—
201#423—21no—
202#407—612no—
203#407C2×D6112yes—
204C4×M16—21no—
205#399—1221no—
206#397—1221no—
207#388—1221no—
208#388—1221no—
209#385—611no—
210#385—611no—
211#381—611no—
212#379—611no—
213#379—611no—
214#343—1221no—
215#343—1221no—
216#323—1224no—
217C61#951211yescharacteristic, complemented, Sylow
218C25—611no—
219C23×C4—611no—
220C23×C4C22×D6111yescharacteristic
221C2×M16—22no—
222C2×M16—1222no—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4D24411yescharacteristic
228C2×C4⋊C4D24411yescharacteristic
229C2×C22⋊C4—611no—
230C2×C22⋊C4—611no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1222no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4C4×D6111yescharacteristic
240C2×C22⋊C4C4×D6111yescharacteristic
241C2×C42C22⋊D6111yescharacteristic
242C2×C42C22⋊D6111yescharacteristic
243C4⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—1222no—
246C22⋊C8—1222no—
247C22⋊C8C22⋊D6114yes—
248#97—22no—
249C4×C8—22no—
250#95—1221no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254C24—1222no—
255C24—1221no—
256C24—4881no—
257C24—2441no—
258C24—2441no—
259C24—2442no—
260C24C2×C4×D6111yescharacteristic
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—1222no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—4881no—
268C22×C4#1175819811yescharacteristic
269C22×C4#1175819811yescharacteristic
270C22×C4#1175819811yescharacteristic
271C22×C4#1175819811yescharacteristic
272C22×C4C2×D24411yescharacteristic
273C22×C4#1175822212yes—
274C22×C4C2×C22⋊D6111yescharacteristic
275C22×C4C2×C22⋊D6112yes—
276M16—42no—
277M16—42no—
278M16—2442no—
279M16—2442no—
280C2×C8—24no—
281C2×C8—2442no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—4881no—
285C4⋊C4—4881no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—4881no—
289C22⋊C4—4881no—
290C22⋊C4—2442no—
291C22⋊C4—2442no—
292C22⋊C4—2441no—
293C22⋊C4—2442no—
294C22⋊C4—2441no—
295C22⋊C4—2441no—
296C22⋊C4—2441no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—1224no—
306C23—1222no—
307C23—4882no—
308C23—4882no—
309C23—4884no—
310C23—4884no—
311C23—2441no—
312C23—2441no—
313C23—2441no—
314C23—2442no—
315C23—2441no—
316C23—2441no—
317C23—2441no—
318C23C2×#1175819811yescharacteristic
319C23C4×C22⋊D6112yes—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—4882no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4#1451974611yescharacteristic
345C2×C4#1451975112yes—
346C2×C4#1451976011yescharacteristic
347C8—44no—
348C8—4882no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—42no—
355C22—42no—
356C22—4881no—
357C22—4881no—
358C22—4884no—
359C22—4884no—
360C22—4881no—
361C22—4881no—
362C22—2442no—
363C22—2442no—
364C22—2442no—
365C22#1683964111yescharacteristic
366C22#1683973511yescharacteristic
367C22#1683973511yescharacteristic
368C4—21no—
369C4—22no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—4881no—
376C4—4881no—
377C4—4881no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—4882no—
382C2#2144623711yescharacteristic
383C2#2144627711yescharacteristic
384C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2921C2
socleC22×C61#386
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#2921D61
hypocenterC61#9512
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21445902C2semidirect
4. #21443728C2semidirect
5. C61×#2921C2semidirect
186. #2921D61semidirect
217. C61#9512semidirect

Maximal subgroups

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

subgroupindex
#214500892
#214459022
#214441452
#214437282
C61×#29212
#951261