Finite Group Databasebeta

← #29577720

Subgroups — #29577720

Order 15616.

690 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454848C211yesmaximal, characteristic, complemented
2#21450765C211yesmaximal, characteristic
3#21444532C212yesmaximal, complemented
4#21443674C212yesmaximal, complemented
5C61×#1466C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×#14519773C412yes—
8C2×#14519751—21no—
9C2×#14519751—21no—
10#16839834—24no—
11#16839819C2212yes—
12#16839726—21no—
13C2×C22⋊#92418—21no—
14#16839671—21no—
15#16839671—21no—
16#16839663C412yes—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839505—24no—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#432C2212yes—
21C61×#323—21no—
22C61×#323—21no—
23C22×D244—21no—
24C22×D244—21no—
25C22×D244C2×C412yes—
26C22×C22⋊D61—22no—
27C22×C22⋊D61—22no—
28C2×#11758194—22no—
29C22×#92418—22no—
30#14519774—22no—
31#14519774—22no—
32#14519773—44no—
33#14519751—44no—
34#14519751—44no—
35C2×C4⋊Dic61—21no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×C4×Dic61C2×C412yes—
39C4⋊#92418D414yescomplemented
40C22⋊#92418—22no—
41C22⋊#92418—22no—
42C22⋊#92418—22no—
43#14519688—22no—
44#14519684—21no—
45C22×D4×C61C2×C411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C4×D4×C61—22no—
49C4×D4×C61—22no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61D411yescharacteristic
53C2×C42×C61C2311yescharacteristic
54C4⋊C8×C61—22no—
55C22⋊C8×C61—22no—
56#97×C61—22no—
57C24×C61C22⋊C411yescharacteristic
58C24×C61C22⋊C411yescharacteristic
59C23×D61—22no—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61—41no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—44no—
75C2×C22⋊D61—44no—
76C2×C22⋊D61—44no—
77C2×C22⋊D61—44no—
78C2×C22⋊D61—82no—
79C2×C22⋊D61—82no—
80C22×Dic61—22no—
81C2×D244—24no—
82C2×D244—42no—
83C2×D244—42no—
84C2×D244—42no—
85C2×D244—42no—
86C2×D244—41no—
87C2×D244—42no—
88C2×D244—41no—
89C2×D244—82no—
90C2×C8×C61—42no—
91C4⋊C4×C61—22no—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—22no—
94C42×C61C2×D412yes—
95C42×C61#4112yes—
96#11758198—44no—
97#11758198—44no—
98C4⋊Dic61—24no—
99C4×Dic61—42no—
100#11758194—82no—
101C2×#92418—24no—
102C2×#92418—82no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10012yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—42no—
120C22×D61—42no—
121D4×C61—44no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61C4×D412yes—
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12112yes—
137C2×C4×C61#12311yescharacteristic
138C22⋊D61—88no—
139C22⋊D61—88no—
140C2×Dic61—24no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143D244—84no—
144D244—82no—
145D244—84no—
146D244—82no—
147C8×C61—82no—
148#92418—44no—
149#9457—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—22no—
153C22×C61—42no—
154C22×C61—42no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—41no—
159C22×C61—41no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61#38611yescharacteristic
163C22×C61C2×#10011yescharacteristic
164C22×C61#41311yescharacteristic
165C2×D61—82no—
166C2×D61—82no—
167C2×D61—82no—
168C2×D61—82no—
169C2×D61—82no—
170C2×D61—82no—
171C2×D61—82no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61#45312yes—
178Dic61—44no—
179#2973—611no—
180#2541—611no—
181#1555—612no—
182#1466D6111yescharacteristic, complemented
183#1285—612no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—42no—
187C2×C61—81no—
188C2×C61#188711yescharacteristic
189C2×C61#190011yescharacteristic
190C2×C61#191011yescharacteristic
191D61—162no—
192C2×#127—612no—
193C2×#127—611no—
194C2×#121—1221no—
195C2×#121—1221no—
196C2×C4×D4C2×D6111yescharacteristic
197#435—1224no—
198#432—612no—
199#432C2×D6112yes—
200#407—1221no—
201C2×C22⋊C8—1221no—
202#392—1221no—
203#392—1221no—
204#390—612no—
205C4×C4⋊C4—611no—
206#331—1224no—
207#323—21no—
208#323—21no—
209C61#945711yescharacteristic, complemented, Sylow
210C22×D4—612no—
211C22×D4—1222no—
212C22×D4—1222no—
213C22×D4—1221no—
214C22×D4—1221no—
215C22×D4C4×D6111yescharacteristic
216C23×C4D24411yescharacteristic
217C23×C4D24411yescharacteristic
218C2×M16—1222no—
219C22×C8—1222no—
220#127—1222no—
221#127—1222no—
222#127—2444no—
223#121—2444no—
224#121—2444no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4C4×D6111yescharacteristic
229C2×C22⋊C4—1222no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4C22⋊D6111yescharacteristic
232C2×C22⋊C4C22⋊D6111yescharacteristic
233C2×C42—612no—
234C2×C42C22×D6111yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—614no—
237C22⋊C8—22no—
238C22⋊C8—1222no—
239C22⋊C8—1222no—
240C22⋊C8—1222no—
241#97—22no—
242#97—1222no—
243#95—1221no—
244C24—1222no—
245C24#1175819811yescharacteristic
246C24#1175819811yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—1224no—
251C2×D4—4882no—
252C2×D4—4882no—
253C2×D4—4882no—
254C2×D4—2442no—
255C2×D4—2442no—
256C2×D4—2442no—
257C2×D4—2442no—
258C2×D4—2444no—
259C2×D4—2444no—
260C2×D4—2444no—
261C2×D4—2444no—
262C2×D4—2441no—
263C2×D4—2442no—
264C2×D4—2441no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—1222no—
272C22×C4#1175819811yescharacteristic
273C22×C4#1175819811yescharacteristic
274C22×C4C2×C4×D6111yescharacteristic
275C22×C4C2×D24411yescharacteristic
276C22×C4C2×C22⋊D6111yescharacteristic
277M16—4882no—
278C2×C8—42no—
279C2×C8—1224no—
280C2×C8—4882no—
281C4⋊C4—22no—
282C4⋊C4—1224no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—2444no—
286C22⋊C4—2444no—
287C42—2442no—
288C42D4×D6112yes—
289C42#1175822412yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—2442no—
300C23—2442no—
301C23—2442no—
302C23—2442no—
303C23—2442no—
304C23—2442no—
305C23—2442no—
306C23#1451969312yes—
307C23C2×#1175819811yescharacteristic
308D4—44no—
309D4—4884no—
310D4—4884no—
311D4—4882no—
312D4—4888no—
313D4—4888no—
314D4—4882no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—1224no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4#1451973712yes—
331C2×C4#1451974711yescharacteristic
332C2×C4#1451974912yes—
333C2×C4#1451976111yescharacteristic
334C8—82no—
335C8—2444no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—4882no—
349C22—4882no—
350C22—4882no—
351C22—4882no—
352C22—4882no—
353C22—4882no—
354C22—4882no—
355C22#1683963911yescharacteristic
356C22C2×#1451969311yescharacteristic
357C22#1683974411yescharacteristic
358C4—21no—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—2444no—
364C4#1683993212yes—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—9762no—
370C2#2144621511yescharacteristic
371C2#2144628011yescharacteristic
372C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1466C2
socleC22×C61#386
prime radical#21450765C2
cosocleC2×C42×C61C23
hypercenter#1466D61
hypocenterC61#9457
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
1. #21454848C2semidirect
3. #21444532C2semidirect
4. #21443674C2semidirect
5. C61×#1466C2semidirect
39. C4⋊#92418D4semidirect
182. #1466D61semidirect
209. C61#9457semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214507652
#214445322
#214436742
C61×#14662
#945761