Finite Group Databasebeta

← #30601753

Subgroups — #30601753

Order 15616.

668 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 106 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×#16840615C212yesmaximal, complemented
3C2×#16840613C212yesmaximal
4#21460995C211yesmaximal, characteristic, complemented
5#21454119C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7C2×C4×#92420—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840615—24no—
13#16840613—24no—
14C2×#14519764C411yescharacteristic, complemented
15C2×#14519757C2211yescharacteristic
16C2×#14519748C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839668C411yescharacteristic, complemented
19#16839655—21no—
20#16839655—21no—
21#16839638—21no—
22#16839638—21no—
23#16839627C411yescharacteristic, complemented
24C61×#390C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519834—22no—
28#14519834D414yes—
29C4×#92420—42no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—42no—
34C22×Dic122C2×C411yescharacteristic
35C22×C22⋊D61—21no—
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519764—42no—
40#14519757—22no—
41#14519757—22no—
42#14519748—22no—
43#14519748—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C42×D61—42no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×D4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62C2×C42×C61C2×C411yescharacteristic
63C24×C61—21no—
64C23×D61C2×D411yescharacteristic
65C22×#6923—24no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—42no—
72C2×D4×C61—22no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61C2×D412yes—
97C2×C4×D61#4112yes—
98C2×Dic122—22no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C2×#92420—24no—
102C2×#92420—42no—
103C2×#92420—42no—
104C4⋊C4×C61—22no—
105C22⋊C4×C61—42no—
106C22⋊C4×C61—42no—
107C42×C61—41no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111#11758198—41no—
112#11758198—41no—
113C4⋊Dic61—22no—
114C4⋊Dic61—42no—
115C4×Dic61—22no—
116C4×Dic61—22no—
117C4×Dic61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61#12411yescharacteristic
131C22×D61#12711yescharacteristic
132C22×D61#13512yes—
133C22×D61#13512yes—
134C2×#6923—44no—
135C2×#6923—44no—
136C2×#6923—44no—
137C2×#6923—82no—
138C2×#6923—82no—
139C2×#6923—82no—
140C2×#6923—82no—
141D4×C61—82no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61#10412yes—
153C2×C4×C61C4×D412yes—
154C22⋊D61—82no—
155C22⋊D61—82no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61#9912yes—
170C2×Dic61#12112yes—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—44no—
174C4×D61—44no—
175Dic122—82no—
176#92420—44no—
177#9681—611nomaximal, Sylow
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61#39011yescharacteristic
186C22×C61C2×#9911yescharacteristic
187C22×C61#41011yescharacteristic
188C22×C61#41611yescharacteristic
189C22×C61#41611yescharacteristic
190C22×C61C2×#10411yescharacteristic
191C22×C61#42111yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—22no—
197C2×D61—22no—
198C2×D61—22no—
199C2×D61—22no—
200C2×D61#48812yes—
201C2×D61#49512yes—
202#6923—84no—
203C4×C61—22no—
204C4×C61—22no—
205C4×C61—41no—
206C4×C61—81no—
207Dic61—22no—
208Dic61—22no—
209Dic61—41no—
210Dic61—81no—
211Dic61—81no—
212#2850—611no—
213C4×C22⋊C8—611no—
214C2×#328—612no—
215C2×#327—612no—
216#1434—611no—
217C2×C61—161no—
218C2×C61#150011yescharacteristic
219C2×C61#150111yescharacteristic
220C2×C61#151611yescharacteristic
221C2×C61#152111yescharacteristic
222C2×C61#195811yescharacteristic
223C2×C61#196411yescharacteristic
224C2×C61#198011yescharacteristic
225D61—22no—
226D61—22no—
227C2×#124—611no—
228C2×C22⋊Q8—611no—
229C2×#121—611no—
230C22×C42—611no—
231C2×C22⋊C8—612no—
232C2×C4×C8—1221no—
233#391—611no—
234#390#692311yescharacteristic, complemented
235#388—1221no—
236#388—1221no—
237#386—1221no—
238#386—1221no—
239#382—611no—
240C2×#95—612no—
241C2×#95—1221no—
242C2×#95—1221no—
243#336—1221no—
244#328—1224no—
245#327—1224no—
246C61#968111yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4—1221no—
249C22×D4C2×#692311yescharacteristic
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—611no—
253C23×C4—1222no—
254C22×C8—1222no—
255#124—2442no—
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258#121—1222no—
259#121—1222no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4C2×#692311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—2442no—
273C2×C42C2×#692311yescharacteristic
274C22⋊C8—614no—
275C22⋊C8—1222no—
276C4×C8—2442no—
277#95—1221no—
278#95—1221no—
279#95—1222no—
280#95—1222no—
281#95—1224no—
282#95—1221no—
283#95—1221no—
284#95—2442no—
285C24—21no—
286C24—611no—
287C2×Q8—1222no—
288C2×Q8—2441no—
289C2×Q8—2441no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—2441no—
294C2×D4—2441no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—21no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—612no—
301C22×C4—612no—
302C22×C4—611no—
303C22×C4—1222no—
304C22×C4—1222no—
305C22×C4—1222no—
306C22×C4—1222no—
307C22×C4—1224no—
308C22×C4—1221no—
309C22×C4—1221no—
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—22no—
327C4⋊C4—1222no—
328C4⋊C4—2442no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C42—41no—
337C42—1222no—
338C42—1222no—
339C42—2441no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—611no—
348C23—611no—
349C23—612no—
350C23—612no—
351C23—1221no—
352C23—1221no—
353C23—1221no—
354C23—1221no—
355C23C2×C22⋊#692311yescharacteristic
356Q8—4882no—
357D4—82no—
358D4—4882no—
359D4—4882no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
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—1221no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4—2444no—
388C2×C4—2444no—
389C2×C4—2444no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4#1451984112yes—
397C2×C4D4×#692312yes—
398C8—2444no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—612no—
407C22—612no—
408C22—1222no—
409C22—1222no—
410C22—1221no—
411C22—1221no—
412C22—1222no—
413C22—1221no—
414C22—1221no—
415C22—1222no—
416C22#1684071411yescharacteristic
417C22D61⋊#9911yescharacteristic
418C22#1684074111yescharacteristic
419C22#1684075211yescharacteristic
420C22#1684075211yescharacteristic
421C22C2×#1451984111yescharacteristic
422C22#1684076011yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—81no—
427C4—1222no—
428C4—1222no—
429C4—4884no—
430C4—4881no—
431C4—4881no—
432C4—2441no—
433C2—161no—
434C2—1222no—
435C2—1222no—
436C2#2146119011yescharacteristic
437C2#2146119511yescharacteristic
438C2#2146124411yescharacteristic
439C2#2146126111yescharacteristic
440C2#2146194811yescharacteristic
441C2#2146196011yescharacteristic
442C2#2146198311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#9681
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C4×#14519834C2semidirect
2. C2×#16840615C2semidirect
4. #21460995C2semidirect
14. C2×#14519764C4semidirect
18. #16839668C4semidirect
23. #16839627C4semidirect
24. C61×#390C4semidirect
234. #390#6923semidirect
246. C61#9681semidirect

Maximal subgroups

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

subgroupindex
C4×#145198342
C2×#168406152
C2×#168406132
#214609952
#214541192
#968161