Finite Group Databasebeta

← #29581376

Subgroups — #29581376

Order 15616.

560 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 85 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#21457009C211yesmaximal, characteristic
2#21456276C211yesmaximal, characteristic
3#21444545C211yesmaximal, characteristic, complemented
4#21444421C211yesmaximal, characteristic, complemented
5#21444326C211yesmaximal, characteristic
6#21443712C211yesmaximal, characteristic, complemented
7C61×#2660C211yesmaximal, characteristic, complemented
8C2×#14519808—22no—
9#16840312C2211yescharacteristic
10#16840172—21no—
11#16840154—22no—
12#16840141—21no—
13#16840136—21no—
14#16840128—21no—
15#16840080—21no—
16#16840075—21no—
17#16840059—22no—
18#16839819C2211yescharacteristic
19#16839819C2211yescharacteristic
20C4×#11758194C2211yescharacteristic
21#16839511—22no—
22#16839504—22no—
23#16839495—22no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×#125×C61C2211yescharacteristic
26C61×#398—22no—
27C61×#395—22no—
28C61×#384—21no—
29C61×#382—21no—
30C61×#376C2211yescharacteristic
31C2×#11758222—21no—
32#14519808—48no—
33C2×#11758202—24no—
34C2×#11758201—24no—
35C2×#11758194—21no—
36#14519777D411yescharacteristic
37#14519774D411yescharacteristic
38#14519765—21no—
39#14519751—21no—
40#14519751—21no—
41C4×D244—21no—
42C4×D244—21no—
43#14519710—22no—
44#14519708—22no—
45C4⋊#92418—21no—
46C4⋊#92418D412yes—
47#14519697—22no—
48#14519697—22no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519688—21no—
52#14519688—21no—
53#14519688D412yes—
54C4×#92418—21no—
55C22×Q8×C61D412yes—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—24no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C42×C61—21no—
72C2×C42×C61C2311yescharacteristic
73#95×C61—24no—
74#95×C61—21no—
75#95×C61D412yes—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61—44no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61—21no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—41no—
87#11758222—44no—
88C2×D244—21no—
89C2×D244—21no—
90C2×D244—21no—
91C2×C4×D61—41no—
92C2×Dic122—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—84no—
106C42×C61—24no—
107C42×C61C2×D411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111#11758202—48no—
112#11758201—48no—
113#11758198—41no—
114#11758198—41no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117#11758194—24no—
118C2×#92418—22no—
119C2×#92418—22no—
120C2×#92418—41no—
121C23×C61#12011yescharacteristic
122C22×D61—41no—
123Q8×C61—44no—
124Q8×C61—82no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—24no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—44no—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12711yescharacteristic
143C22⋊D61—82no—
144C2×Dic61—41no—
145D244—42no—
146D244—42no—
147D244—81no—
148C4×D61—82no—
149Dic122—42no—
150#92418—24no—
151#92418—42no—
152#92418—81no—
153#10091—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61#39211yescharacteristic
157C22×C61#45312yes—
158C2×D61—81no—
159C2×D61—81no—
160C2×D61—81no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—44no—
167C4×C61—81no—
168C4×C61—82no—
169C4×C61—81no—
170C4×C61#45612yes—
171Dic61—81no—
172#3234—611no—
173#3152—611no—
174#2660D6111yescharacteristic, complemented
175#1558—611no—
176#1528—611no—
177#1505—611no—
178#1300—611no—
179C2×C61—22no—
180C2×C61#200211yescharacteristic
181C2×C61#200511yescharacteristic
182C2×C61#200911yescharacteristic
183D61—161no—
184C2×#137—1222no—
185#532—611no—
186C2×C4⋊Q8C2×D6111yescharacteristic
187C2×#125C2×D6111yescharacteristic
188#501—1221no—
189#497—1222no—
190#494—1221no—
191#492—1221no—
192#490—1221no—
193#481—1221no—
194#480—1221no—
195#478—1222no—
196#432—611no—
197#432—611no—
198C4×M16—611no—
199#398—22no—
200#395—22no—
201#384—21no—
202#382—21no—
203#376C2×D6111yescharacteristic
204#333—1222no—
205#330—1222no—
206#327—1222no—
207C61#1009111yescharacteristic, complemented, Sylow
208C2×#41—1221no—
209C22×Q8C22⋊D6112yes—
210#137—2448no—
211C2×Q16—1224no—
212C2×SD16—1224no—
213C2×M16—1221no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—611no—
219#127—611no—
220#125—21no—
221#125—21no—
222#125—21no—
223#125—21no—
224#124—1221no—
225#121—1221no—
226#121—1221no—
227C4×D4—1221no—
228C4×D4—1221no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4C22⋊D6111yescharacteristic
235C2×C4⋊C4C22⋊D6111yescharacteristic
236C2×C42—21no—
237C2×C42C22×D6111yescharacteristic
238#107—1222no—
239#106—1222no—
240C4⋊C8—612no—
241C4⋊C8—1221no—
242#102—1222no—
243#102—1222no—
244C22⋊C8—1221no—
245C22⋊C8—1221no—
246#97—612no—
247#97—1221no—
248#97—1221no—
249C4×C8—1221no—
250#95—24no—
251#95—21no—
252#95C22⋊D6112yes—
253#41—2444no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—44no—
257C2×Q8—1221no—
258C2×D4—1221no—
259C2×D4—1221no—
260C2×D4—1221no—
261C2×D4—2441no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—2441no—
267C22×C4C2×C22⋊D6111yescharacteristic
268C22×C4C2×C22⋊D6111yescharacteristic
269C22×C4C2×C22⋊D6111yescharacteristic
270Q16—2448no—
271SD16—2448no—
272M16—1224no—
273C2×C8—1222no—
274C2×C8—1222no—
275C2×C8—2441no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—84no—
289C4⋊C4—1221no—
290C4⋊C4—1221no—
291C22⋊C4—2441no—
292C22⋊C4—2441no—
293C42—24no—
294C42D4×D6111yescharacteristic
295C42D4×D6111yescharacteristic
296C42D4×D6111yescharacteristic
297C42#1175822411yescharacteristic
298C23—2441no—
299C23#1451974611yescharacteristic
300Q8—44no—
301Q8—82no—
302Q8—2442no—
303D4—4881no—
304D4—4882no—
305D4—2442no—
306D4—2442no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—4882no—
320C2×C4—2441no—
321C2×C4#1451974511yescharacteristic
322C2×C4#1451974511yescharacteristic
323C2×C4#1451974811yescharacteristic
324C2×C4#1451974911yescharacteristic
325C2×C4#1451974911yescharacteristic
326C2×C4#1451977311yescharacteristic
327C8—1224no—
328C8—4881no—
329C8—2442no—
330C22—21no—
331C22—21no—
332C22—4881no—
333C22—4881no—
334C22—4881no—
335C22#1683967011yescharacteristic
336C22#1683993012yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—81no—
344C4—82no—
345C4—81no—
346C4—4881no—
347C4#1683994412yes—
348C2—22no—
349C2—9761no—
350C2#2144683211yescharacteristic
351C2#2144685511yescharacteristic
352C2#2144687911yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2660C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#2660D61
hypocenterC61#10091
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2660

Complements

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

NH ≅ G/Ntype
3. #21444545C2semidirect
4. #21444421C2semidirect
6. #21443712C2semidirect
7. C61×#2660C2semidirect
174. #2660D61semidirect
207. C61#10091semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214562762
#214445452
#214444212
#214443262
#214437122
C61×#26602
#1009161