Finite Group Databasebeta

← #30600418

Subgroups — #30600418

Order 15616.

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

normal subgroups only

This group has 756 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
1C2×#16840698C211yesmaximal, characteristic, complemented
2#21461711C211yesmaximal, characteristic, complemented
3C2×#16840613C212yesmaximal
4#21460692C212yesmaximal, complemented
5#21454149C211yesmaximal, characteristic
6C22×C4×#6923—21no—
7C2×#14519834C2212yes—
8#16840698—24no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840628—21no—
13#16840613—24no—
14C2×#14519759—21no—
15C2×#14519748C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839665C411yescharacteristic, complemented
18#16839661C411yescharacteristic, complemented
19#16839660C411yescharacteristic, complemented
20#16839656—21no—
21#16839655—21no—
22#16839638—21no—
23#16839619C2211yescharacteristic
24C61×#390C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4×#6923—24no—
28C2×C4×#6923—24no—
29C2×C4×#6923—24no—
30#14519834—22no—
31#14519834—22no—
32(C2×#6923)⋊C4—24no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4D414yes—
35C22×C22⋊D61—21no—
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519759—44no—
40#14519748—22no—
41#14519748—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
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—22no—
54C42×D61—22no—
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—24no—
67C22×#6923—24no—
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—21no—
89C2×C4×D61—21no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61C2×D412yes—
99C2×C4×D61#4112yes—
100C2×#92420—42no—
101C2×#92420—42no—
102C2×#92420—42no—
103C4×#6923—216no—
104C4⋊C4×C61—41no—
105C22⋊C4×C61—42no—
106C22⋊C4×C61—42no—
107C42×C61—22no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111#11758198—42no—
112C4⋊Dic61—22no—
113C4⋊Dic61—42no—
114C4⋊Dic61—41no—
115#11758196—42no—
116C4×Dic61—22no—
117C4×Dic61—22no—
118C4×Dic61—22no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61C2×C22⋊C411yescharacteristic
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—21no—
131C22×D61#12011yescharacteristic
132C22×D61#12311yescharacteristic
133C22×D61#13512yes—
134C22×D61AGL(1,ℤ/8)12yes—
135C2×#6923—28no—
136C2×#6923—44no—
137C2×#6923—44no—
138C2×#6923—44no—
139C2×#6923—44no—
140C2×#6923—44no—
141C2×#6923—44no—
142D4×C61—82no—
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—21no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61#10412yes—
154C2×C4×C61C4×D412yes—
155C22⋊D61—82no—
156C22⋊D61—82no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61#10412yes—
171C2×Dic61#12112yes—
172C4×D61—24no—
173C4×D61—24no—
174C4×D61—42no—
175C4×D61—42no—
176C4×D61—42no—
177C4×D61—42no—
178#92420—82no—
179#9149—611nomaximal, Sylow
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#38611yescharacteristic
188C22×C61#41011yescharacteristic
189C22×C61#41011yescharacteristic
190C22×C61#41611yescharacteristic
191C22×C61#41811yescharacteristic
192C22×C61C2×#10411yescharacteristic
193C22×C61C2×#10411yescharacteristic
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—22no—
199C2×D61—22no—
200C2×D61—22no—
201C2×D61—22no—
202C2×D61#44912yes—
203C2×D61#48212yes—
204#6923—48no—
205C4×C61—22no—
206C4×C61—22no—
207C4×C61—41no—
208C4×C61—81no—
209Dic61—22no—
210Dic61—22no—
211Dic61—41no—
212Dic61—81no—
213Dic61—81no—
214#2854—611no—
215C2×#376—611no—
216#1817—611no—
217C2×#327—612no—
218#1294—612no—
219C2×C61—161no—
220C2×C61#150011yescharacteristic
221C2×C61#150011yescharacteristic
222C2×C61#150511yescharacteristic
223C2×C61#150511yescharacteristic
224C2×C61#187511yescharacteristic
225C2×C61#189611yescharacteristic
226C2×C61#189911yescharacteristic
227D61—22no—
228D61—22no—
229C2×#123—1221no—
230C2×#121—611no—
231C22×C42—611no—
232C22×C42—1221no—
233C2×C22⋊C8—612no—
234#390—611no—
235#390#692311yescharacteristic, complemented
236#389—611no—
237#389—611no—
238#388—1221no—
239#388—1221no—
240#386—1221no—
241#380—611no—
242#376—1224no—
243C2×#95—612no—
244C2×#95—1221no—
245#336—1221no—
246#336—1221no—
247#327—1224no—
248C61#914911yescharacteristic, complemented, Sylow
249C22×D4—1221no—
250C22×D4C2×#692311yescharacteristic
251C23×C4—611no—
252C23×C4—611no—
253C23×C4—611no—
254C23×C4—1222no—
255C22×C8—1222no—
256#123—2444no—
257#121—1222no—
258#121—1222no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4C2×#692311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—611no—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42—1224no—
276C2×C42—1224no—
277C2×C42—1224no—
278C2×C42C2×#692311yescharacteristic
279C22⋊C8—1222no—
280C22⋊C8—1222no—
281#95—614no—
282#95—1221no—
283#95—1221no—
284#95—1222no—
285#95—1224no—
286#95—1221no—
287C24—21no—
288C24—611no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2442no—
295C2×D4—2442no—
296C22×C4—21no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—612no—
300C22×C4—612no—
301C22×C4—611no—
302C22×C4—1222no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1222no—
310C22×C4—1224no—
311C22×C4—1224no—
312C22×C4—1224no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4C22⋊#692311yescharacteristic
322C22×C4C22⋊#692311yescharacteristic
323C22×C4C22×#692311yescharacteristic
324C2×C8—2442no—
325C2×C8—2442no—
326C2×C8—2442no—
327C4⋊C4—41no—
328C4⋊C4—1222no—
329C4⋊C4—2441no—
330C4⋊C4—2442no—
331C4⋊C4—2442no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C22⋊C4—2442no—
338C42—22no—
339C42—1222no—
340C42—1222no—
341C42—12216no—
342C42—1222no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—611no—
351C23—611no—
352C23—612no—
353C23—612no—
354C23—1221no—
355C23—1221no—
356C23—1221no—
357C23—1221no—
358C23C2×C22⋊#692311yescharacteristic
359D4—82no—
360D4—4882no—
361D4—4882no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—612no—
373C2×C4—612no—
374C2×C4—1224no—
375C2×C4—1224no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1228no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2444no—
395C2×C4—2444no—
396C2×C4—2444no—
397C2×C4—2444no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4#1451984112yes—
401C2×C4D4×#692312yes—
402C8—4882no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—612no—
411C22—612no—
412C22—1221no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—1222no—
417C22—1222no—
418C22—1222no—
419C22—1222no—
420C22#1684071011yescharacteristic
421C22#1684074111yescharacteristic
422C22#1684074211yescharacteristic
423C22#1684075211yescharacteristic
424C22#1684075411yescharacteristic
425C22C2×#1451984111yescharacteristic
426C22D61⋊#10411yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—81no—
431C4—1222no—
432C4—1222no—
433C4—4881no—
434C4—4881no—
435C4—2448no—
436C4—2441no—
437C2—161no—
438C2—1222no—
439C2—1222no—
440C2#2146119011yescharacteristic
441C2#2146119111yescharacteristic
442C2#2146120811yescharacteristic
443C2#2146120911yescharacteristic
444C2#2146180711yescharacteristic
445C2#2146183711yescharacteristic
446C2#2146184211yescharacteristic
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#9149
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. C2×#16840698C2semidirect
2. #21461711C2semidirect
4. #21460692C2semidirect
17. #16839665C4semidirect
18. #16839661C4semidirect
19. #16839660C4semidirect
24. C61×#390C4semidirect
235. #390#6923semidirect
248. C61#9149semidirect

Maximal subgroups

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

subgroupindex
C2×#168406982
#214617112
C2×#168406132
#214606922
#214541492
#914961