Finite Group Databasebeta

← #30601749

Subgroups — #30601749

Order 15616.

826 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×#14519834C211yesmaximal, characteristic, complemented
2C2×#16840613C214yesmaximal, complemented
3#21460997C211yesmaximal, characteristic, complemented
4#21454161C211yesmaximal, characteristic
5C22×C4⋊#6923—21no—
6C2×#14519834C2212yes—
7C2×C4×#92420—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—
23C2×C4⋊#6923—24no—
24C2×C4⋊#6923—42no—
25C2×C4⋊#6923—42no—
26#14519834—22no—
27#14519834D414yescomplemented
28C4×#92420—42no—
29(C2×#6923)⋊C4—24no—
30(C2×#6923)⋊C4—42no—
31C22×C22⋊D61—22no—
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519748—24no—
36#14519748—24no—
37#14519748—42no—
38#14519748—42no—
39#14519748—42no—
40#14519748—42no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C412yes—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—22no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C42×D61—42no—
51C22×D4×C61—21no—
52C22×D4×C61—21no—
53C22×D4×C61C2×C412yes—
54#127×C61—42no—
55C2×C42×C61C2×C411yescharacteristic
56C24×C61—22no—
57C23×D61C2×D411yescharacteristic
58C22×#6923—24no—
59C22×#6923—42no—
60C22×#6923—42no—
61C22×#6923—42no—
62C22×#6923—42no—
63C22×#6923—42no—
64C22×#6923—42no—
65C2×D4×C61—24no—
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—
73C2×D4×C61—42no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C22×Dic61—22no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C2×D411yescharacteristic
87C2×C4×D61—22no—
88C2×C4×D61—22no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61D812yes—
94C2×C4×D61D812yes—
95C2×C4×D61SD1612yes—
96C2×C4×D61SD1612yes—
97C2×C4×D61C2×D412yes—
98C2×C4×D61#4112yes—
99C2×#92420—24no—
100C2×#92420—42no—
101C2×#92420—42no—
102C4⋊#6923—48no—
103C42×C61—22no—
104C22⋊Dic61—44no—
105C22⋊Dic61—44no—
106C22⋊Dic61—44no—
107C4⋊Dic61—24no—
108C4⋊Dic61—42no—
109C4⋊Dic61—42no—
110C4×Dic61—22no—
111C4×Dic61—41no—
112C4×Dic61—41no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61C2×C22⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61#12411yescharacteristic
126C22×D61#12711yescharacteristic
127C22×D61C2×D811yescharacteristic
128C22×D61C2×D811yescharacteristic
129C22×D61C2×SD1611yescharacteristic
130C22×D61C2×SD1611yescharacteristic
131C2×#6923—44no—
132C2×#6923—44no—
133C2×#6923—44no—
134C2×#6923—82no—
135C2×#6923—82no—
136C2×#6923—82no—
137C2×#6923—82no—
138D4×C61—84no—
139D4×C61—84no—
140D4×C61—84no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61#9912yes—
146C2×C4×C61#10212yes—
147C2×C4×C61#10212yes—
148C2×C4×C61#10412yes—
149C2×C4×C61C4×D412yes—
150C22⋊D61—82no—
151C22⋊D61—82no—
152C22⋊D61—82no—
153C22⋊D61—82no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61#10212yes—
164C2×Dic61#10212yes—
165C2×Dic61#12112yes—
166C4×D61—24no—
167C4×D61—24no—
168C4×D61—42no—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61—42no—
172#92420—44no—
173#9680—611nomaximal, Sylow
174C22×C61—82no—
175C22×C61—82no—
176C22×C61—82no—
177C22×C61—82no—
178C22×C61—82no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61#39011yescharacteristic
182C22×C61C2×#9911yescharacteristic
183C22×C61#41011yescharacteristic
184C22×C61C2×#10211yescharacteristic
185C22×C61C2×#10211yescharacteristic
186C22×C61C2×#10411yescharacteristic
187C22×C61#42111yescharacteristic
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—21no—
200C2×D61#48611yescharacteristic
201C2×D61#48611yescharacteristic
202C2×D61#49211yescharacteristic
203C2×D61#49311yescharacteristic
204#6923—84no—
205C4×C61—22no—
206C4×C61—22no—
207C4×C61#32714yescomplemented
208Dic61—22no—
209Dic61—22no—
210Dic61—22no—
211Dic61—82no—
212#2856—611no—
213C4×C22⋊C8—611no—
214C2×#327—614no—
215C4⋊#95—611no—
216C2×C61—162no—
217C2×C61C2×#32711yescharacteristic
218C2×C61#150211yescharacteristic
219C2×C61#151711yescharacteristic
220C2×C61#151911yescharacteristic
221C2×C61#195711yescharacteristic
222C2×C61#196411yescharacteristic
223C2×C61#198011yescharacteristic
224D61—21no—
225D61—21no—
226D61—21no—
227D61—21no—
228C2×#127#692311yescharacteristic, complemented
229C2×#121—612no—
230C2×#121—1221no—
231C2×#121—1221no—
232C22×C4⋊C4—1221no—
233C22×C42—611no—
234C2×C22⋊C8—612no—
235C2×C4×C8—1221no—
236#390—612no—
237#390—1221no—
238#390—1221no—
239#384—611no—
240C2×#95—612no—
241C2×#95—1221no—
242#336—1221no—
243#327—1228no—
244C61#968011yescharacteristic, complemented, Sylow
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—1222no—
248C22×D4C2×#692312yes—
249C23×C4—611no—
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—1222no—
253C22×C8—1222no—
254#127—42no—
255#121—1224no—
256#121—1224no—
257#121—2442no—
258#121—2442no—
259#121—2442no—
260#121—2442no—
261C2×C4⋊C4—612no—
262C2×C4⋊C4—1224no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—2442no—
266C2×C4⋊C4—2442no—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1222no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—611no—
273C2×C42—2442no—
274C2×C42C2×#692311yescharacteristic
275C22⋊C8—614no—
276C22⋊C8—1222no—
277C4×C8—2442no—
278#95—1224no—
279#95—2442no—
280C24—22no—
281C24—611no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—2442no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—612no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—1222no—
307C22×C4—1222no—
308C22×C4—1224no—
309C22×C4—1222no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4C22⋊#692311yescharacteristic
321C22×C4C22⋊#692311yescharacteristic
322C22×C4C22×#692311yescharacteristic
323C2×C8—1224no—
324C2×C8—2442no—
325C2×C8—2442no—
326C4⋊C4—1224no—
327C4⋊C4—2448no—
328C4⋊C4—2442no—
329C4⋊C4—2442no—
330C22⋊C4—2444no—
331C22⋊C4—2444no—
332C22⋊C4—2444no—
333C42—22no—
334C42—1222no—
335C42—2441no—
336C42—2441no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—611no—
345C23—611no—
346C23—611no—
347C23—611no—
348C23—611no—
349C23—611no—
350C23—1221no—
351C23—1221no—
352C23—1221no—
353C23—1221no—
354C23C2×C22⋊#692311yescharacteristic
355D4—84no—
356D4—84no—
357D4—84no—
358D4—4882no—
359D4—4882no—
360D4—4882no—
361D4—4882no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—612no—
367C2×C4—612no—
368C2×C4—612no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1224no—
376C2×C4—1224no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4—2444no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4#1451983512yes—
392C2×C4#1451983912yes—
393C2×C4#1451983912yes—
394C2×C4#1451984112yes—
395C2×C4D4×#692312yes—
396C8—2444no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—611no—
405C22—611no—
406C22—611no—
407C22—611no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—1221no—
412C22—1221no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—1221no—
417C22—1221no—
418C22—1221no—
419C22—1221no—
420C22#1684071411yescharacteristic
421C22C2×#1451983511yescharacteristic
422C22#1684074211yescharacteristic
423C22C2×#1451983911yescharacteristic
424C22C2×#1451983911yescharacteristic
425C22C2×#1451984111yescharacteristic
426C22#1684076011yescharacteristic
427C4—22no—
428C4—22no—
429C4—1222no—
430C4—1222no—
431C4—1222no—
432C4—4884no—
433C4—4882no—
434C4#1684061314yescomplemented
435C2—162no—
436C2—1221no—
437C2—1221no—
438C2—1221no—
439C2—1221no—
440C2C2×#1684061311yescharacteristic
441C2#2146119911yescharacteristic
442C2#2146124811yescharacteristic
443C2#2146125511yescharacteristic
444C2#2146194711yescharacteristic
445C2#2146195811yescharacteristic
446C2#2146198311yescharacteristic
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#9680
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. C4×#14519834C2semidirect
2. C2×#16840613C2semidirect
3. #21460997C2semidirect
18. #16839665C4semidirect
19. #16839632C4semidirect
20. C2×#127×C61C4semidirect
27. #14519834D4semidirect
207. C4×C61#327semidirect
228. C2×#127#6923semidirect
244. C61#9680semidirect
434. C4#16840613semidirect

Maximal subgroups

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

subgroupindex
C4×#145198342
C2×#168406132
#214609972
#214541612
#968061