Finite Group Databasebeta

← #30601809

Subgroups — #30601809

Order 15616.

856 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 130 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#21461674C211yesmaximal, characteristic, complemented
2C2×#16840613C214yesmaximal, complemented
3C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
4#21454161C211yesmaximal, characteristic
5C22×C4×#6923—21no—
6C2×C4⋊#92420—21no—
7C2×#14519834C2212yes—
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—
23C2×C4×#6923—24no—
24C2×C4×#6923—42no—
25C2×C4×#6923—42no—
26C4⋊#92420—42no—
27#14519834—24no—
28(C2×#6923)⋊C4—24no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4D414yescomplemented
31C22×C22⋊D61—22no—
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519748—24no—
36#14519748—24no—
37#14519748—44no—
38#14519748—42no—
39#14519748—42no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C412yes—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—22no—
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C42×D61—42no—
50C22×D4×C61—21no—
51C22×D4×C61—21no—
52C22×D4×C61C2×C412yes—
53#127×C61—42no—
54C2×C42×C61C2×C411yescharacteristic
55C24×C61—22no—
56C23×D61C2×D411yescharacteristic
57C22×#6923—24no—
58C22×#6923—24no—
59C22×#6923—24no—
60C22×#6923—42no—
61C22×#6923—42no—
62C22×#6923—42no—
63C22×#6923—42no—
64C2×D4×C61—24no—
65C2×D4×C61—42no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C2×D4×C61—42no—
71C2×D4×C61—42no—
72C2×D4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C2×C22⋊D61—44no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—44no—
80C22×Dic61—22no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—22no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61D812yes—
92C2×C4×D61SD1612yes—
93C2×C4×D61C2×D412yes—
94C2×C4×D61#4112yes—
95C2×#92420—42no—
96C2×#92420—42no—
97C2×#92420—42no—
98C4×#6923—48no—
99C42×C61—22no—
100C22⋊Dic61—44no—
101C22⋊Dic61—44no—
102C22⋊Dic61—42no—
103C22⋊Dic61—42no—
104C4⋊Dic61—24no—
105C4⋊Dic61—42no—
106C4⋊Dic61—42no—
107C4×Dic61—22no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61C2×C22⋊C411yescharacteristic
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61#12112yes—
123C22×D61C2×D811yescharacteristic
124C22×D61C2×SD1611yescharacteristic
125C22×D61AGL(1,ℤ/8)12yes—
126C2×#6923—28no—
127C2×#6923—44no—
128C2×#6923—44no—
129C2×#6923—44no—
130C2×#6923—44no—
131C2×#6923—44no—
132C2×#6923—44no—
133D4×C61—84no—
134D4×C61—84no—
135D4×C61—84no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61#9912yes—
141C2×C4×C61#10212yes—
142C2×C4×C61#10412yes—
143C2×C4×C61#10412yes—
144C2×C4×C61C4×D412yes—
145C22⋊D61—84no—
146C22⋊D61—84no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—42no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61#10212yes—
158C2×Dic61#12411yescharacteristic
159C2×Dic61#12711yescharacteristic
160C4×D61—24no—
161C4×D61—24no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—44no—
165#92420—82no—
166#9712—611nomaximal, Sylow
167C22×C61—82no—
168C22×C61—82no—
169C22×C61—82no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61—82no—
173C22×C61—82no—
174C22×C61#39011yescharacteristic
175C22×C61C2×#9911yescharacteristic
176C22×C61#41011yescharacteristic
177C22×C61C2×#10211yescharacteristic
178C22×C61#41811yescharacteristic
179C22×C61C2×#10411yescharacteristic
180C22×C61C2×#10411yescharacteristic
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—22no—
188C2×D61—22no—
189C2×D61#45912yes—
190C2×D61#46012yes—
191#6923—48no—
192C4×C61—22no—
193C4×C61—22no—
194C4×C61#32714yescomplemented
195Dic61—21no—
196Dic61—21no—
197Dic61—22no—
198Dic61—41no—
199Dic61—82no—
200#2856—611no—
201#1797—611no—
202C2×#327—614no—
203C4×#95—611no—
204C2×C61—162no—
205C2×C61C2×#32711yescharacteristic
206C2×C61#150211yescharacteristic
207C2×C61#150511yescharacteristic
208C2×C61#150711yescharacteristic
209C2×C61#196211yescharacteristic
210C2×C61#196411yescharacteristic
211C2×C61#198211yescharacteristic
212D61—22no—
213D61—22no—
214C2×#127#692311yescharacteristic, complemented
215C2×#121—612no—
216C2×#121—1221no—
217C2×#121—1221no—
218C22×C42—611no—
219C22×C42—1221no—
220C2×C4⋊C8—1221no—
221C2×C22⋊C8—612no—
222#390—612no—
223#390—1221no—
224#390—1221no—
225#384—611no—
226C2×#95—612no—
227C2×#95—1221no—
228#336—1221no—
229#327—1228no—
230C61#971211yescharacteristic, complemented, Sylow
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—1222no—
234C22×D4C2×#692312yes—
235C23×C4—611no—
236C23×C4—611no—
237C23×C4—611no—
238C23×C4—1222no—
239C22×C8—1222no—
240#127—42no—
241#121—1224no—
242#121—1224no—
243#121—2442no—
244#121—2442no—
245#121—2444no—
246C2×C4⋊C4—612no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C42—611no—
253C2×C42—611no—
254C2×C42—611no—
255C2×C42—1224no—
256C2×C42—2442no—
257C2×C42—2442no—
258C2×C42—2442no—
259C2×C42C2×#692311yescharacteristic
260C4⋊C8—2442no—
261C22⋊C8—1224no—
262#95—614no—
263#95—1222no—
264#95—1224no—
265C24—22no—
266C24—611no—
267C2×D4—24no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—2444no—
277C2×D4—2442no—
278C2×D4—2442no—
279C2×D4—2444no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—612no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—612no—
286C22×C4—611no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—1224no—
291C22×C4—1224no—
292C22×C4—1224no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2441no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4—2442no—
301C22×C4—2442no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—2442no—
306C2×C8—2442no—
307C2×C8—2442no—
308C4⋊C4—1224no—
309C4⋊C4—2442no—
310C4⋊C4—2442no—
311C22⋊C4—2442no—
312C22⋊C4—2442no—
313C22⋊C4—2444no—
314C22⋊C4—2444no—
315C42—22no—
316C42—1222no—
317C42—2441no—
318C42—2441no—
319C42—2448no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—612no—
328C23—611no—
329C23—611no—
330C23—612no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23—1221no—
335C23C2×C22⋊#692311yescharacteristic
336D4—84no—
337D4—84no—
338D4—84no—
339D4—4884no—
340D4—4884no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—612no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1224no—
353C2×C4—1224no—
354C2×C4—1228no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2444no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2444no—
365C2×C4—2444no—
366C2×C4—2444no—
367C2×C4—2444no—
368C2×C4—2444no—
369C2×C4—2444no—
370C2×C4#1451983512yes—
371C2×C4#1451983912yes—
372C2×C4#1451984112yes—
373C2×C4#1451984112yes—
374C2×C4D4×#692312yes—
375C8—4882no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—612no—
384C22—612no—
385C22—1221no—
386C22—1221no—
387C22—1222no—
388C22—1222no—
389C22—1221no—
390C22—1221no—
391C22—1222no—
392C22—1222no—
393C22#1684071411yescharacteristic
394C22C2×#1451983511yescharacteristic
395C22#1684074211yescharacteristic
396C22C2×#1451983911yescharacteristic
397C22#1684075411yescharacteristic
398C22C2×#1451984111yescharacteristic
399C22C2×#1451984111yescharacteristic
400C4—22no—
401C4—22no—
402C4—1222no—
403C4—1221no—
404C4—1221no—
405C4—4882no—
406C4—2448no—
407C4—2441no—
408C4#1684061314yescomplemented
409C2—162no—
410C2—1222no—
411C2—1222no—
412C2C2×#1684061311yescharacteristic
413C2#2146119911yescharacteristic
414C2#2146120811yescharacteristic
415C2#2146121511yescharacteristic
416C2#2146195311yescharacteristic
417C2#2146195811yescharacteristic
418C2#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#9712
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. #21461674C2semidirect
2. C2×#16840613C2semidirect
3. C4×(C2×#6923)⋊C4C2semidirect
18. #16839665C4semidirect
19. #16839632C4semidirect
20. C2×#127×C61C4semidirect
30. (C2×#6923)⋊C4D4semidirect
194. C4×C61#327semidirect
214. C2×#127#6923semidirect
230. C61#9712semidirect
408. C4#16840613semidirect

Maximal subgroups

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

subgroupindex
#214616742
C2×#168406132
C4×(C2×#6923)⋊C42
#214541612
#971261