Finite Group Databasebeta

← #29581439

Subgroups — #29581439

Order 15616.

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

normal subgroups only

This group has 512 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#21456822C211yesmaximal, characteristic
2#21456361C211yesmaximal, characteristic, complemented
3#21444536C211yesmaximal, characteristic, complemented
4#21444417C211yesmaximal, characteristic
5#21444330C211yesmaximal, characteristic, complemented
6#21443712C211yesmaximal, characteristic, complemented
7C61×#2640C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840168—22no—
10#16840165—21no—
11#16840158—21no—
12#16840157—21no—
13#16840121—21no—
14#16839981—22no—
15#16839918—21no—
16#16839913—21no—
17#16839819C2211yescharacteristic
18#16839819C2211yescharacteristic
19#16839802—22no—
20C4×#11758194C2211yescharacteristic
21#16839508—22no—
22#16839502—22no—
23#16839498—22no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×#124×C61C2211yescharacteristic
26C61×#397—22no—
27C61×#393—22no—
28C61×#391—21no—
29C61×#390—21no—
30C61×#376C2211yescharacteristic
31C2×#11758202—22no—
32C2×#11758200—22no—
33C2×#11758194—21no—
34#14519777D411yescharacteristic
35#14519777D411yescharacteristic
36#14519769—21no—
37#14519756—21no—
38#14519756—21no—
39C4×Dic122—21no—
40C4×Dic122—21no—
41#14519736—21no—
42#14519710—24no—
43#14519708—24no—
44C4⋊#92418—21no—
45C4⋊#92418D412yes—
46#14519701—22no—
47#14519698—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519688—21no—
51#14519688—21no—
52#14519688D412yes—
53C4×#92418—21no—
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×D4×C61D411yescharacteristic
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61#124×C61—21no—
62#124×C61—21no—
63#124×C61—21no—
64#124×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61D412yes—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—24no—
70C2×C42×C61—21no—
71C2×C42×C61C2311yescharacteristic
72#95×C61—24no—
73#95×C61—21no—
74#95×C61D412yes—
75C24×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—21no—
78C2×Q8×C61—42no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61—82no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—42no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C2×Dic122—21no—
92C2×Dic122—21no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—84no—
102C42×C61—42no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107C22⋊Dic61—41no—
108#11758202—82no—
109#11758200—82no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114#11758196—41no—
115#11758196—41no—
116C4×Dic61—41no—
117#11758194—24no—
118C2×#92418—22no—
119C2×#92418—22no—
120C2×#92418—41no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—41no—
124C23×C61—42no—
125C23×C61#12011yescharacteristic
126Q8×C61—42no—
127Q8×C61—82no—
128Q8×C61—82no—
129Q8×C61—81no—
130D4×C61—42no—
131D4×C61—81no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—44no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12711yescharacteristic
148C2×Dic61—41no—
149C2×Dic61—41no—
150Dic122—42no—
151Dic122—81no—
152#92418—24no—
153#92418—42no—
154#92418—81no—
155#10100—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—84no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61#39211yescharacteristic
163C22×C61#45612yes—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—82no—
170C4×C61—81no—
171C4×C61—82no—
172C4×C61#45512yes—
173Dic61—81no—
174Dic61—81no—
175#3215—611no—
176#3165—611no—
177#2640D6111yescharacteristic
178#1556—611no—
179#1527—611no—
180#1506—611no—
181#1300—611no—
182C2×C61—22no—
183C2×C61—161no—
184C2×C61#200411yescharacteristic
185C2×C61#200511yescharacteristic
186C2×C61#200811yescharacteristic
187#533—611no—
188C2×C4⋊Q8C2×D6111yescharacteristic
189C2×#124C2×D6111yescharacteristic
190#501—1222no—
191#500—1221no—
192#498—1221no—
193#497—1221no—
194#489—1221no—
195#464—1222no—
196#451—1221no—
197#450—1221no—
198#432—611no—
199#432—611no—
200#428—1222no—
201C4×M16—611no—
202#397—22no—
203#393—22no—
204#391—21no—
205#390—21no—
206#376C2×D6111yescharacteristic
207#332—1222no—
208#329—1222no—
209#328—1222no—
210C61#1010011yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C22×Q8C22⋊D6111yescharacteristic
213C22×D4C22⋊D6111yescharacteristic
214C2×Q16—1222no—
215C2×SD16—1222no—
216C2×M16—1221no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—611no—
222C4⋊Q8—611no—
223#125—1221no—
224#124—21no—
225#124—21no—
226#124—21no—
227#124—21no—
228C22⋊Q8—1221no—
229C22⋊Q8—1221no—
230C4×Q8—1221no—
231C4×Q8—1221no—
232#117—1221no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D6112yes—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—24no—
238C2×C42—21no—
239C2×C42C22×D6111yescharacteristic
240#107—1224no—
241#106—1224no—
242C4⋊C8—612no—
243C4⋊C8—1221no—
244#103—1222no—
245#102—1222no—
246C22⋊C8—1221no—
247C22⋊C8—1221no—
248#97—612no—
249#97—1221no—
250#97—1221no—
251C4×C8—1221no—
252#95—24no—
253#95—21no—
254#95C22⋊D6112yes—
255C24—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—82no—
262C2×Q8—1221no—
263C2×Q8—1221no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—42no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C2×C22⋊D6111yescharacteristic
272C22×C4C2×C22⋊D6111yescharacteristic
273Q16—4882no—
274SD16—4882no—
275M16—1224no—
276C2×C8—1222no—
277C2×C8—1222no—
278C2×C8—2441no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—1221no—
284C4⋊C4—1221no—
285C4⋊C4—1221no—
286C4⋊C4—1221no—
287C4⋊C4—2441no—
288C4⋊C4—2441no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—84no—
294C22⋊C4—2441no—
295C42—42no—
296C42—2441no—
297C42D4×D6111yescharacteristic
298C42D4×D6111yescharacteristic
299C42D4×D6111yescharacteristic
300C42#1175822411yescharacteristic
301C23—42no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23#1451974611yescharacteristic
306Q8—42no—
307Q8—81no—
308Q8—82no—
309Q8—82no—
310Q8—4881no—
311Q8—2442no—
312D4—42no—
313D4—81no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4#1451974511yescharacteristic
327C2×C4#1451974511yescharacteristic
328C2×C4#1451974811yescharacteristic
329C2×C4#1451974911yescharacteristic
330C2×C4#1451974911yescharacteristic
331C2×C4#1451977311yescharacteristic
332C8—1224no—
333C8—4881no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22—84no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1683967011yescharacteristic
342C22#1683994512yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—81no—
350C4—82no—
351C4—4881no—
352C4—4881no—
353C4#1683993912yes—
354C2—22no—
355C2—161no—
356C2#2144684811yescharacteristic
357C2#2144685311yescharacteristic
358C2#2144687211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2640C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#2640D61
hypocenterC61#10100
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2640

Complements

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

NH ≅ G/Ntype
2. #21456361C2semidirect
3. #21444536C2semidirect
5. #21444330C2semidirect
6. #21443712C2semidirect
210. C61#10100semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214563612
#214445362
#214444172
#214443302
#214437122
C61×#26402
#1010061