Finite Group Databasebeta

← #30601820

Subgroups — #30601820

Order 15616.

764 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 114 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#21461676C211yesmaximal, characteristic, complemented
2C2×#16840613C214yesmaximal, complemented
3#21460995C211yesmaximal, characteristic, complemented
4#21454161C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×#14519834C2212yes—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840613—28no—
12C2×#14519748—21no—
13C2×#14519748—21no—
14C2×#14519748C2212yes—
15C2×C42×D61C2211yescharacteristic
16#16839665—21no—
17#16839665—21no—
18#16839665C412yescomplemented
19#16839632C411yescharacteristic, complemented
20C2×#127×C61C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23#14519844—42no—
24#14519834—24no—
25(C2×#6923)⋊C4—22no—
26(C2×#6923)⋊C4—42no—
27(C2×#6923)⋊C4—42no—
28(C2×#6923)⋊C4D414yescomplemented
29C22×C22⋊D61—22no—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61C2311yescharacteristic
33#14519748—24no—
34#14519748—24no—
35#14519748—44no—
36#14519748—42no—
37#14519748—42no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C412yes—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C42×D61—42no—
48C22×D4×C61—21no—
49C22×D4×C61—21no—
50C22×D4×C61C2×C412yes—
51#127×C61—42no—
52C2×C42×C61C2×C411yescharacteristic
53C24×C61—22no—
54C23×D61C2×D411yescharacteristic
55C22×#6923—24no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×#6923—42no—
61C22×#6923—42no—
62C2×D4×C61—24no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C2×D4×C61—42no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C2×C22⋊D61—44no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C2×C22⋊D61—44no—
78C22×Dic61—22no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—41no—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61D812yes—
90C2×C4×D61SD1612yes—
91C2×C4×D61C2×D412yes—
92C2×C4×D61#4112yes—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C42×C61—22no—
97C22⋊Dic61—44no—
98C22⋊Dic61—44no—
99C22⋊Dic61—42no—
100C22⋊Dic61—42no—
101C4⋊Dic61—24no—
102C4⋊Dic61—42no—
103C4⋊Dic61—42no—
104C4×Dic61—22no—
105C4×Dic61—41no—
106C4×Dic61—41no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61#12112yes—
120C22×D61C2×D811yescharacteristic
121C22×D61C2×SD1611yescharacteristic
122C22×D61AGL(1,ℤ/8)12yes—
123C2×#6923—44no—
124C2×#6923—44no—
125C2×#6923—44no—
126C2×#6923—82no—
127C2×#6923—82no—
128C2×#6923—82no—
129C2×#6923—82no—
130D4×C61—84no—
131D4×C61—84no—
132D4×C61—84no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61#9912yes—
140C2×C4×C61#10212yes—
141C2×C4×C61C4×D412yes—
142C22⋊D61—84no—
143C22⋊D61—84no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154C2×Dic61#10212yes—
155C2×Dic61#12411yescharacteristic
156C2×Dic61#12711yescharacteristic
157C4×D61—42no—
158C4×D61—42no—
159C4×D61—42no—
160C4×D61—42no—
161C4×D61—44no—
162#92420—82no—
163#9715—611nomaximal, Sylow
164C22×C61—82no—
165C22×C61—82no—
166C22×C61—82no—
167C22×C61—82no—
168C22×C61—82no—
169C22×C61—82no—
170C22×C61—82no—
171C22×C61#39011yescharacteristic
172C22×C61C2×#9911yescharacteristic
173C22×C61#41011yescharacteristic
174C22×C61C2×#10211yescharacteristic
175C22×C61#41811yescharacteristic
176C22×C61#42111yescharacteristic
177C22×C61#42111yescharacteristic
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61#45912yes—
187C2×D61#46012yes—
188#6923—84no—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—22no—
192Dic61—21no—
193Dic61—21no—
194Dic61—22no—
195Dic61—41no—
196Dic61—82no—
197#2856—611no—
198#1797—611no—
199C2×#327—614no—
200#1434—611no—
201C2×C61—162no—
202C2×C61#151011yescharacteristic
203C2×C61#151411yescharacteristic
204C2×C61#151711yescharacteristic
205C2×C61#151911yescharacteristic
206C2×C61#196311yescharacteristic
207C2×C61#196411yescharacteristic
208C2×C61#198211yescharacteristic
209D61—22no—
210D61—22no—
211C2×#127#692311yescharacteristic, complemented
212C2×#121—612no—
213C2×#121—1221no—
214C2×#121—1221no—
215C22×C42—611no—
216C2×C4⋊C8—1221no—
217C2×C22⋊C8—612no—
218#390—612no—
219#390—1221no—
220#390—1221no—
221#384—611no—
222C2×#95—612no—
223C2×#95—1221no—
224C2×#95—1221no—
225#336—1221no—
226#327—1228no—
227C61#971511yescharacteristic, complemented, Sylow
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—1222no—
231C22×D4C2×#692312yes—
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—611no—
235C23×C4—1222no—
236C22×C8—1222no—
237#127—42no—
238#121—1224no—
239#121—1224no—
240#121—2442no—
241#121—2442no—
242#121—2444no—
243C2×C4⋊C4—612no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—1222no—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—611no—
252C2×C42—2442no—
253C2×C42C2×#692311yescharacteristic
254C4⋊C8—2442no—
255C22⋊C8—1224no—
256#95—614no—
257#95—1222no—
258#95—2442no—
259#95—2442no—
260C24—22no—
261C24—611no—
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—2444no—
272C2×D4—2442no—
273C2×D4—2442no—
274C2×D4—2444no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—612no—
278C22×C4—612no—
279C22×C4—612no—
280C22×C4—612no—
281C22×C4—611no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1224no—
285C22×C4—1222no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4—2442no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4C22⋊#692311yescharacteristic
298C22×C4C22⋊#692311yescharacteristic
299C22×C4C22×#692311yescharacteristic
300C2×C8—2442no—
301C2×C8—2442no—
302C2×C8—2442no—
303C4⋊C4—1224no—
304C4⋊C4—2442no—
305C4⋊C4—2442no—
306C22⋊C4—2442no—
307C22⋊C4—2442no—
308C22⋊C4—2444no—
309C22⋊C4—2444no—
310C42—22no—
311C42—1222no—
312C42—2441no—
313C42—2441no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—612no—
322C23—611no—
323C23—611no—
324C23—612no—
325C23—1221no—
326C23—1221no—
327C23—1221no—
328C23—1221no—
329C23C2×C22⋊#692311yescharacteristic
330D4—84no—
331D4—84no—
332D4—84no—
333D4—4884no—
334D4—4884no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—611no—
342C2×C4—611no—
343C2×C4—612no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—4882no—
352C2×C4—4882no—
353C2×C4—4882no—
354C2×C4—4882no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2444no—
360C2×C4—2444no—
361C2×C4—2444no—
362C2×C4—2444no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4#1451983512yes—
367C2×C4#1451983912yes—
368C2×C4D4×#692312yes—
369C8—4882no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—612no—
378C22—612no—
379C22—1221no—
380C22—1221no—
381C22—1221no—
382C22—1221no—
383C22—1222no—
384C22—1222no—
385C22—1222no—
386C22—1222no—
387C22#1684071411yescharacteristic
388C22C2×#1451983511yescharacteristic
389C22#1684074211yescharacteristic
390C22C2×#1451983911yescharacteristic
391C22#1684075411yescharacteristic
392C22#1684076011yescharacteristic
393C22#1684076011yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—1222no—
398C4—1221no—
399C4—1221no—
400C4—4884no—
401C4—4882no—
402C4—2441no—
403C2—162no—
404C2—1222no—
405C2—1222no—
406C2#2146122411yescharacteristic
407C2#2146123911yescharacteristic
408C2#2146124811yescharacteristic
409C2#2146125511yescharacteristic
410C2#2146195711yescharacteristic
411C2#2146195811yescharacteristic
412C2#2146198611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454161C2
cosocleC22×C4×D61C23
hypercenterC2×#127#6923
hypocenterC61#9715
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461676C2semidirect
2. C2×#16840613C2semidirect
3. #21460995C2semidirect
18. #16839665C4semidirect
19. #16839632C4semidirect
20. C2×#127×C61C4semidirect
28. (C2×#6923)⋊C4D4semidirect
211. C2×#127#6923semidirect
227. C61#9715semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406132
#214609952
#214541612
#971561