Finite Group Databasebeta

← #29595819

Subgroups — #29595819

Order 15616.

876 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16839555C211yesmaximal, characteristic
2C2×#16839552C212yesmaximal, complemented
3#21446233C212yesmaximal, complemented
4#21446218C211yesmaximal, characteristic
5C2×C61×#559C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839767—21no—
8C2×#14519694C2211yescharacteristic
9C2×#14519692C2212yes—
10#16839625—22no—
11#16839555—22no—
12#16839552—24no—
13#16839528—22no—
14#16839522—21no—
15C61×#559—22no—
16C61×#559—24no—
17C61×#559C414yescomplemented
18C2×#127×C61C2211yescharacteristic
19C2×#124×C61—22no—
20C2×#123×C61—21no—
21C2×#123×C61C2212yes—
22C2×C4×Q8×C61—21no—
23C2×C4×D4×C61—22no—
24C2×#11758194—21no—
25D4×Dic61—42no—
26#14519736—21no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29#14519698—42no—
30#14519694—22no—
31#14519692—22no—
32#14519692—22no—
33#14519684—44no—
34C22×Q8×C61—21no—
35C22×D4×C61—22no—
36C22×D4×C61D411yescharacteristic
37C22×D4×C61C2311yescharacteristic
38C23×C4×C61—21no—
39C23×C4×C61D412yes—
40#127×C61—22no—
41#127×C61C2×C412yes—
42#127×C61D412yes—
43#124×C61—28no—
44#124×C61—28no—
45#123×C61—24no—
46#123×C61—24no—
47#123×C61—22no—
48#123×C61—22no—
49#123×C61C2×C414yes—
50#123×C61D414yes—
51C4×Q8×C61—24no—
52C4×Q8×C61—24no—
53C4×D4×C61—28no—
54C4×D4×C61—28no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—22no—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61D412yes—
61C2×C42×C61—22no—
62C2×C42×C61D411yescharacteristic
63C24×C61—21no—
64C24×C61C2×D412yes—
65C2×Q8×C61—24no—
66C2×Q8×C61—44no—
67C2×D4×C61—24no—
68C2×D4×C61—24no—
69C2×D4×C61—21no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—48no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61C22⋊C412yes—
76C2×D4×C61C22×C411yescharacteristic
77C2×D4×C61C2×D411yescharacteristic
78C2×D4×C61C2×D412yes—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—24no—
83C22×C4×C61—22no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61—81no—
87C22×C4×C61—81no—
88C22×C4×C61C22⋊C414yes—
89C22×C4×C61C2×D411yescharacteristic
90C22×Dic61—24no—
91C22×Dic61—42no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—24no—
95C22⋊C4×C61—28no—
96C22⋊C4×C61—24no—
97C22⋊C4×C61—24no—
98C22⋊C4×C61—24no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61C22⋊C414yes—
101C42×C61—28no—
102C42×C61—21no—
103C42×C61C22⋊C412yes—
104C22⋊Dic61—42no—
105C22⋊Dic61—41no—
106C22⋊Dic61—42no—
107C22⋊Dic61—42no—
108C4⋊Dic61—22no—
109C4×Dic61—41no—
110#11758194—41no—
111#11758194—41no—
112C2×#92418—41no—
113C23×C61—22no—
114C23×C61—22no—
115C23×C61—22no—
116C23×C61—44no—
117C23×C61—44no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61C2×C22⋊C412yes—
121C23×C61#12011yescharacteristic
122C23×C61#12012yes—
123Q8×C61—48no—
124D4×C61—41no—
125D4×C61—42no—
126D4×C61—41no—
127D4×C61—42no—
128D4×C61—48no—
129D4×C61—48no—
130D4×C61—81no—
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—24no—
136C2×C4×C61—24no—
137C2×C4×C61—44no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—44no—
141C2×C4×C61—82no—
142C2×C4×C61—84no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61#12011yescharacteristic
145C2×Dic61—42no—
146C2×Dic61—84no—
147C2×Dic61—82no—
148C2×Dic61—84no—
149#92418—81no—
150#12760—611nomaximal, Sylow
151C22×C61—22no—
152C22×C61—22no—
153C22×C61—22no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61—44no—
157C22×C61—44no—
158C22×C61—42no—
159C22×C61—82no—
160C22×C61—82no—
161C22×C61—82no—
162C22×C61#37911yescharacteristic
163C22×C61B(4,𝔽2)11yescharacteristic
164C22×C61#45811yescharacteristic
165C4×C61—24no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—44no—
169C4×C61—44no—
170Dic61—82no—
171Dic61—162no—
172C2×#559D6111yescharacteristic
173C2×#353—611no—
174C2×#352—612no—
175#1890—612no—
176#1887—611no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61—42no—
180C2×C61—42no—
181C2×C61—82no—
182C2×C61#179011yescharacteristic
183C2×C61#219812yes—
184#559—22no—
185#559—24no—
186#559Dic6114yescomplemented
187C2×#127C2×D6111yescharacteristic
188C2×#124—22no—
189C2×#123—21no—
190C2×#123C2×D6112yes—
191#518—611no—
192C2×C4×Q8—21no—
193C2×C4×D4—22no—
194#418—1221no—
195C2×#100—611no—
196C2×#99—612no—
197#381—1222no—
198#353—1222no—
199#352—1224no—
200#342—1222no—
201#339—1221no—
202C61#1276011yescharacteristic, complemented, Sylow
203C22×Q8—21no—
204C22×D4—22no—
205C22×D4C22⋊D6111yescharacteristic
206C22×D4C22×D6111yescharacteristic
207C23×C4—21no—
208C23×C4C22⋊D6112yes—
209C2×M16—1221no—
210#127—22no—
211#127C2×Dic6112yes—
212#127C22⋊D6112yes—
213#124—28no—
214#124—28no—
215#123—24no—
216#123—24no—
217#123—22no—
218#123—22no—
219#123C2×Dic6114yes—
220#123C22⋊D6114yes—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—2442no—
226#117—1221no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1222no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4C22⋊D6112yes—
235C2×C42—22no—
236C2×C42C22⋊D6111yescharacteristic
237#102—2442no—
238#100—1222no—
239#99—1222no—
240#99—1222no—
241#95—2444no—
242C24—21no—
243C24C2×C22⋊D6112yes—
244C2×Q8—24no—
245C2×Q8—44no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊Dic6112yes—
255C2×D4C22×Dic6111yescharacteristic
256C2×D4C2×C22⋊D6111yescharacteristic
257C2×D4C2×C22⋊D6112yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—1224no—
268C22×C4—2442no—
269C22×C4C22⋊Dic6114yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271M16—2441no—
272M16—2441no—
273C2×C8—2441no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—1222no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—2442no—
284C22⋊C4—2442no—
285C22⋊C4—2442no—
286C22⋊C4—2441no—
287C22⋊C4C22⋊Dic6114yes—
288C42—28no—
289C42—21no—
290C42—2441no—
291C42C22⋊Dic6112yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—44no—
296C23—44no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic6112yes—
300C23#1451974612yes—
301C23#1451974611yescharacteristic
302Q8—48no—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—81no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—4882no—
323C2×C4—4884no—
324C2×C4—4884no—
325C2×C4—2442no—
326C2×C4C2×C22⋊Dic6111yescharacteristic
327C2×C4#1451974611yescharacteristic
328C8—4881no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22#1683961811yescharacteristic
341C22#1683994911yescharacteristic
342C22#1683995211yescharacteristic
343C4—24no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—9762no—
349C4—4882no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2#2144576711yescharacteristic
356C2#2144781512yes—
subgroupG/H
centerC22#16839949
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×#559C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#127×C61C22
cosocleC22×D4×C61C23
hypercenterC2×#559D61
hypocenterC61#12760
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#16839552C2semidirect
3. #21446233C2semidirect
17. C61×#559C4semidirect
186. #559Dic61semidirect
202. C61#12760semidirect

Maximal subgroups

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

subgroupindex
C2×#168395552
C2×#168395522
#214462332
#214462182
C2×C61×#5592
#1276061