Finite Group Databasebeta

← #30601748

Subgroups — #30601748

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×#16840614C214yesmaximal, complemented
3#21460997C211yesmaximal, characteristic, complemented
4#21454165C211yesmaximal, characteristic
5C22×C4⋊#6923—21no—
6C2×#14519834C2212yes—
7C2×C4×#92420—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840614—28no—
12C2×#14519774C411yescharacteristic, complemented
13C2×#14519752—21no—
14C2×#14519752—21no—
15C2×#14519752C2212yes—
16C2×C42×D61C2211yescharacteristic
17#16839663—21no—
18#16839663—21no—
19#16839663C412yescomplemented
20C61×#384C411yescharacteristic, 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×D244—22no—
32C22×D244—21no—
33C22×D244—21no—
34C22×D244C2×C412yes—
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519774—42no—
39#14519752—24no—
40#14519752—24no—
41#14519752—42no—
42#14519752—42no—
43#14519752—42no—
44#14519752—42no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C42×D61—42no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C412yes—
55C2×C42×C61C2×C411yescharacteristic
56C23×D61—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—
65C22×C4×C61—22no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×D244—24no—
73C2×D244—42no—
74C2×D244—42no—
75C2×D244—42no—
76C2×D244—42no—
77C2×D244—42no—
78C2×D244—42no—
79C2×D244—42no—
80C2×D244—42no—
81C2×D244—42no—
82C2×D244—42no—
83C2×D244—42no—
84C2×D244—42no—
85C2×D244—42no—
86C2×D244—42no—
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—
103C4⋊C4×C61—24no—
104C4⋊C4×C61—42no—
105C4⋊C4×C61—42no—
106C42×C61—22no—
107#11758198—44no—
108#11758198—44no—
109#11758198—44no—
110C4×Dic61—22no—
111C4×Dic61—41no—
112C4×Dic61—41no—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—42no—
119C22×D61—42no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—42no—
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—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61#9912yes—
146C2×C4×C61#10212yes—
147C2×C4×C61#10212yes—
148C2×C4×C61#10412yes—
149C2×C4×C61C4×D412yes—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61#10212yes—
157C2×Dic61#10212yes—
158C2×Dic61#12112yes—
159D244—84no—
160D244—84no—
161D244—82no—
162D244—82no—
163D244—82no—
164D244—82no—
165D244—84no—
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#39011yescharacteristic
175C22×C61C2×#9911yescharacteristic
176C22×C61#41011yescharacteristic
177C22×C61C2×#10211yescharacteristic
178C22×C61C2×#10211yescharacteristic
179C22×C61C2×#10411yescharacteristic
180C22×C61#42111yescharacteristic
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—82no—
197C2×D61—82no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61#48611yescharacteristic
201C2×D61#48611yescharacteristic
202C2×D61#49211yescharacteristic
203C2×D61#49311yescharacteristic
204#6923—84no—
205C4×C61—22no—
206C4×C61—22no—
207C4×C61—82no—
208C4×C61#32714yescomplemented
209Dic61—22no—
210Dic61—22no—
211Dic61—22no—
212#2856—611no—
213C4×C22⋊C8—611no—
214C2×#327—614no—
215C4⋊#95—611no—
216C2×C61C2×#32711yescharacteristic
217C2×C61#150211yescharacteristic
218C2×C61#151711yescharacteristic
219C2×C61#151911yescharacteristic
220C2×C61#195711yescharacteristic
221C2×C61#196411yescharacteristic
222C2×C61#198011yescharacteristic
223D61—21no—
224D61—21no—
225D61—21no—
226D61—21no—
227D61—162no—
228C2×#127—611no—
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#692311yescharacteristic, complemented
240C2×#95—612no—
241C2×#95—1221no—
242#336—1221no—
243#327—1228no—
244C61#968011yescharacteristic, complemented, Sylow
245C22×D4—612no—
246C22×D4—1222no—
247C22×D4—1221no—
248C22×D4—1221no—
249C23×C4—611no—
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—1222no—
253C22×C8—1222no—
254#127—2442no—
255#121—1224no—
256#121—1224no—
257#121—2442no—
258#121—2442no—
259#121—2442no—
260#121—2442no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—1224no—
264C2×C4⋊C4—2442no—
265C2×C4⋊C4—2442no—
266C2×C4⋊C4C2×#692312yes—
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—611no—
281C24—1222no—
282C2×D4—1224no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4—2442no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—22no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—612no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1224no—
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—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—2448no—
330C22⋊C4—2444no—
331C22⋊C4—2444no—
332C22⋊C4—2444no—
333C42—22no—
334C42—1222no—
335C42—2441no—
336C42—2441no—
337C23—611no—
338C23—611no—
339C23—611no—
340C23—611no—
341C23—611no—
342C23—611no—
343C23—1221no—
344C23—1221no—
345C23—1221no—
346C23—1221no—
347C23—2442no—
348C23—2442no—
349C23—2442no—
350C23—2442no—
351C23—2442no—
352C23—2442no—
353C23—2442no—
354C23C2×C22⋊#692311yescharacteristic
355D4—4884no—
356D4—4884no—
357D4—4882no—
358D4—4882no—
359D4—4882no—
360D4—4882no—
361D4—4884no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—612no—
370C2×C4—612no—
371C2×C4—612no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1224no—
379C2×C4—1224no—
380C2×C4—4882no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2444no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4#1451983612yes—
392C2×C4#1451983912yes—
393C2×C4#1451983912yes—
394C2×C4#1451984212yes—
395C2×C4D4×#692312yes—
396C8—2444no—
397C22—611no—
398C22—611no—
399C22—611no—
400C22—611no—
401C22—1221no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—1221no—
412C22—1221no—
413C22—4882no—
414C22—4882no—
415C22—4882no—
416C22—4882no—
417C22—4882no—
418C22—4882no—
419C22—4882no—
420C22#1684071411yescharacteristic
421C22C2×#1451983611yescharacteristic
422C22#1684074311yescharacteristic
423C22C2×#1451983911yescharacteristic
424C22C2×#1451983911yescharacteristic
425C22C2×#1451984211yescharacteristic
426C22#1684076111yescharacteristic
427C4—22no—
428C4—22no—
429C4—82no—
430C4—1222no—
431C4—1222no—
432C4—1222no—
433C4—4884no—
434C4#1684061414yescomplemented
435C2—1221no—
436C2—1221no—
437C2—1221no—
438C2—1221no—
439C2—9762no—
440C2C2×#1684061411yescharacteristic
441C2#2146120011yescharacteristic
442C2#2146125011yescharacteristic
443C2#2146125611yescharacteristic
444C2#2146194711yescharacteristic
445C2#2146195911yescharacteristic
446C2#2146198211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radical#21454165C2
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#9680
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C4×#14519834C2semidirect
2. C2×#16840614C2semidirect
3. #21460997C2semidirect
12. C2×#14519774C4semidirect
19. #16839663C4semidirect
20. C61×#384C4semidirect
27. #14519834D4semidirect
208. C4×C61#327semidirect
239. #384#6923semidirect
244. C61#9680semidirect
434. C4#16840614semidirect

Maximal subgroups

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

subgroupindex
C4×#145198342
C2×#168406142
#214609972
#214541652
#968061