Finite Group Databasebeta

← #29581401

Subgroups — #29581401

Order 15616.

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

normal subgroups only

This group has 480 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#21457005C211yesmaximal, characteristic
2#21451769C211yesmaximal, characteristic, complemented
3#21444536C211yesmaximal, characteristic, complemented
4#21444420C211yesmaximal, characteristic, complemented
5#21444327C211yesmaximal, characteristic
6#21443713C211yesmaximal, characteristic, complemented
7C61×#3141C211yesmaximal, characteristic
8C2×#14519769C2211yescharacteristic
9#16840168—21no—
10#16840126—21no—
11#16840078—21no—
12#16840073—21no—
13#16840062—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16#16839700—22no—
17#16839700—22no—
18#16839669—21no—
19#16839669—21no—
20#16839608C2211yescharacteristic
21#16839508—22no—
22#16839503—22no—
23#16839496—22no—
24C61×#532C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#489—22no—
27C61×#487—21no—
28C61×#486—21no—
29C61×#417—22no—
30C4×M16×C61C2211yescharacteristic
31#14519769—21no—
32#14519769—21no—
33#14519769—21no—
34#14519769—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×C4×Dic61—21no—
40#14519710—22no—
41#14519708—22no—
42C4⋊#92418—21no—
43C4⋊#92418D412yes—
44#14519698—22no—
45#14519698—22no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48#14519688—21no—
49#14519688—21no—
50#14519684—24no—
51#14519684—24no—
52#14519684—21no—
53#14519684D412yes—
54C2×#41×C61—21no—
55C22×Q8×C61—21no—
56C2×M16×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61D411yescharacteristic
62#127×C61D411yescharacteristic
63#124×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D412yes—
70C2×C42×C61C2311yescharacteristic
71#103×C61—24no—
72#102×C61—24no—
73#97×C61D412yes—
74C4×C8×C61—21no—
75#41×C61—44no—
76C2×Q8×C61—21no—
77C2×Q8×C61—44no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—41no—
84C22×C4×C61—22no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×Dic61—22no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92M16×C61—42no—
93C2×C8×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C42×C61C2×D411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—41no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—42no—
115C2×#92418—22no—
116C2×#92418—41no—
117C23×C61—41no—
118C23×C61#12011yescharacteristic
119Q8×C61—42no—
120Q8×C61—82no—
121Q8×C61—82no—
122D4×C61—42no—
123D4×C61—42no—
124D4×C61—81no—
125D4×C61—82no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—41no—
129C2×C4×C61—44no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—82no—
133C2×C4×C61#12011yescharacteristic
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12711yescharacteristic
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C8×C61—42no—
147#92418—42no—
148#92418—81no—
149#10095—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#39211yescharacteristic
156C22×C61#45412yes—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—82no—
163C4×C61—81no—
164C4×C61#45412yes—
165Dic61—82no—
166Dic61—81no—
167Dic61—81no—
168#3234—611no—
169#3141D6111yescharacteristic
170#2642—611no—
171#1556—611no—
172#1528—611no—
173#1505—611no—
174#1300—611no—
175C2×C61—22no—
176C2×C61—161no—
177C2×C61#200311yescharacteristic
178C2×C61#200311yescharacteristic
179C2×C61#200911yescharacteristic
180#532C2×D6111yescharacteristic
181C2×C4⋊Q8C2×D6111yescharacteristic
182C2×#125—611no—
183#501—1221no—
184#490—1221no—
185#489—22no—
186#487—21no—
187#486—21no—
188#481—1221no—
189#480—1221no—
190#478—1222no—
191#432—611no—
192#432—611no—
193#417—22no—
194C4×M16C2×D6111yescharacteristic
195#400—1222no—
196#400—1222no—
197#391—1221no—
198#391—1221no—
199#376—611no—
200#332—1222no—
201#330—1222no—
202#327—1222no—
203C61#1009511yescharacteristic, complemented, Sylow
204C2×#41—21no—
205C22×Q8—21no—
206C2×M16—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C22⋊D6111yescharacteristic
212#127C22⋊D6111yescharacteristic
213#125—1221no—
214#125—1221no—
215#125—1221no—
216#125—1221no—
217#124—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4C22⋊D6112yes—
228C2×C42—1221no—
229C2×C42C22×D6111yescharacteristic
230#107—1222no—
231#106—1222no—
232C4⋊C8—612no—
233C4⋊C8—1221no—
234#103—24no—
235#102—24no—
236#102—1222no—
237#102—1222no—
238C22⋊C8—1221no—
239C22⋊C8—1221no—
240#97—1221no—
241#97—1221no—
242#97C22⋊D6112yes—
243C4×C8—21no—
244#95—612no—
245#95—1224no—
246#95—1224no—
247#95—1221no—
248#41—44no—
249C2×Q8—21no—
250C2×Q8—44no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—1222no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4D4×D6111yescharacteristic
263C22×C4D4×D6111yescharacteristic
264C22×C4C2×C22⋊D6111yescharacteristic
265M16—42no—
266C2×C8—22no—
267C2×C8—1222no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—2442no—
286C42D4×D6111yescharacteristic
287C42#1175822411yescharacteristic
288C42C2×C22⋊D6111yescharacteristic
289C42C2×C22⋊D6111yescharacteristic
290C23—41no—
291C23#1451974511yescharacteristic
292Q8—42no—
293Q8—82no—
294Q8—82no—
295D4—42no—
296D4—42no—
297D4—81no—
298D4—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4#1451974511yescharacteristic
314C2×C4#1451974611yescharacteristic
315C2×C4#1451974811yescharacteristic
316C2×C4#1451974911yescharacteristic
317C2×C4#1451974911yescharacteristic
318C2×C4#1451977311yescharacteristic
319C8—42no—
320C8—4881no—
321C8—2442no—
322C22—21no—
323C22—21no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#1683967011yescharacteristic
328C22#1683993412yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—4882no—
337C4—4881no—
338C4—4881no—
339C4#1683993312yes—
340C2—22no—
341C2—161no—
342C2#2144683811yescharacteristic
343C2#2144684111yescharacteristic
344C2#2144687711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3141C2
socleC22×C61#392
prime radicalC61×#532C22
hypercenter#3141D61
hypocenterC61#10095
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. #21451769C2semidirect
3. #21444536C2semidirect
4. #21444420C2semidirect
6. #21443713C2semidirect
203. C61#10095semidirect

Maximal subgroups

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

subgroupindex
#214570052
#214517692
#214445362
#214444202
#214443272
#214437132
C61×#31412
#1009561