Finite Group Databasebeta

← #30601813

Subgroups — #30601813

Order 15616.

742 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 110 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×#16840615C212yesmaximal, complemented
3C2×#16840613C212yesmaximal
4C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
5#21454119C211yesmaximal, characteristic
6C22×C4×#6923—21no—
7C2×C4⋊#92420—21no—
8C2×#14519834C2212yes—
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—
27C2×C4×#6923—24no—
28C2×C4×#6923—42no—
29C2×C4×#6923—42no—
30C4⋊#92420—42no—
31#14519834—22no—
32#14519834—22no—
33(C2×#6923)⋊C4—24no—
34(C2×#6923)⋊C4—22no—
35(C2×#6923)⋊C4D414yes—
36C22×Dic122C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61C2311yescharacteristic
41#14519764—42no—
42#14519757—22no—
43#14519757—22no—
44#14519748—22no—
45#14519748—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×#11758198—21no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C42×D61—42no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C22×D4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61C2×C411yescharacteristic
65C24×C61—21no—
66C23×D61C2×D411yescharacteristic
67C22×#6923—24no—
68C22×#6923—24no—
69C22×#6923—24no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C2×D4×C61—22no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C22×C4×C61—21no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61C2×D412yes—
100C2×C4×D61#4112yes—
101C2×Dic122—22no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104C2×#92420—42no—
105C2×#92420—42no—
106C2×#92420—42no—
107C4×#6923—48no—
108C4⋊C4×C61—22no—
109C22⋊C4×C61—42no—
110C22⋊C4×C61—42no—
111C42×C61—41no—
112C22⋊Dic61—42no—
113C22⋊Dic61—42no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116#11758198—42no—
117C4⋊Dic61—22no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120C4×Dic61—22no—
121C4×Dic61—22no—
122C4×Dic61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61C2×C22⋊C411yescharacteristic
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61#12112yes—
136C22×D61#13512yes—
137C22×D61AGL(1,ℤ/8)11yescharacteristic
138C22×D61#13711yescharacteristic
139C2×#6923—28no—
140C2×#6923—44no—
141C2×#6923—44no—
142C2×#6923—44no—
143C2×#6923—44no—
144C2×#6923—44no—
145C2×#6923—44no—
146D4×C61—82no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61#10412yes—
157C2×C4×C61#10412yes—
158C2×C4×C61C4×D412yes—
159C22⋊D61—82no—
160C22⋊D61—82no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61#9912yes—
175C2×Dic61#12411yescharacteristic
176C2×Dic61#12711yescharacteristic
177C4×D61—24no—
178C4×D61—24no—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61—44no—
182Dic122—82no—
183#92420—82no—
184#9713—611nomaximal, Sylow
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39011yescharacteristic
193C22×C61C2×#9911yescharacteristic
194C22×C61#41011yescharacteristic
195C22×C61#41611yescharacteristic
196C22×C61#41711yescharacteristic
197C22×C61C2×#10411yescharacteristic
198C22×C61C2×#10411yescharacteristic
199C2×D61—22no—
200C2×D61—22no—
201C2×D61—22no—
202C2×D61—22no—
203C2×D61—21no—
204C2×D61—21no—
205C2×D61—21no—
206C2×D61—21no—
207C2×D61#46312yes—
208C2×D61#46412yes—
209#6923—48no—
210C4×C61—22no—
211C4×C61—22no—
212C4×C61—22no—
213C4×C61—81no—
214Dic61—21no—
215Dic61—21no—
216Dic61—22no—
217Dic61—41no—
218Dic61—81no—
219Dic61—81no—
220#2850—611no—
221#1797—611no—
222C2×#328—612no—
223C2×#327—612no—
224C4×#95—611no—
225C2×C61—161no—
226C2×C61#150011yescharacteristic
227C2×C61#150111yescharacteristic
228C2×C61#150411yescharacteristic
229C2×C61#150911yescharacteristic
230C2×C61#196211yescharacteristic
231C2×C61#196411yescharacteristic
232C2×C61#198311yescharacteristic
233D61—22no—
234D61—22no—
235C2×#124—611no—
236C2×C22⋊Q8—611no—
237C2×#121—611no—
238C22×C42—611no—
239C22×C42—1221no—
240C2×C4⋊C8—1221no—
241C2×C22⋊C8—612no—
242#391—611no—
243#390#692311yescharacteristic, complemented
244#388—1221no—
245#388—1221no—
246#386—1221no—
247#386—1221no—
248#382—611no—
249C2×#95—612no—
250C2×#95—1221no—
251#336—1221no—
252#328—1224no—
253#327—1224no—
254C61#971311yescharacteristic, complemented, Sylow
255C22×Q8—611no—
256C22×D4—1221no—
257C22×D4C2×#692311yescharacteristic
258C23×C4—611no—
259C23×C4—611no—
260C23×C4—611no—
261C23×C4—1222no—
262C22×C8—1222no—
263#124—2442no—
264C22⋊Q8—1222no—
265C22⋊Q8—1222no—
266#121—1222no—
267#121—1222no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4C2×#692311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C42—611no—
278C2×C42—611no—
279C2×C42—611no—
280C2×C42—1224no—
281C2×C42—2442no—
282C2×C42—2442no—
283C2×C42—2442no—
284C2×C42C2×#692311yescharacteristic
285C4⋊C8—2442no—
286C22⋊C8—1222no—
287C22⋊C8—1222no—
288#95—614no—
289#95—1221no—
290#95—1221no—
291#95—1222no—
292#95—1224no—
293#95—1221no—
294#95—1221no—
295C24—21no—
296C24—611no—
297C2×Q8—1222no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—2442no—
304C2×D4—2442no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—21no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—612no—
311C22×C4—612no—
312C22×C4—611no—
313C22×C4—1222no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1222no—
317C22×C4—1222no—
318C22×C4—1224no—
319C22×C4—1224no—
320C22×C4—1224no—
321C22×C4—1222no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—2441no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2442no—
331C22×C4C22⋊#692311yescharacteristic
332C22×C4C22⋊#692311yescharacteristic
333C22×C4C22×#692311yescharacteristic
334C2×C8—2442no—
335C2×C8—2442no—
336C2×C8—2442no—
337C4⋊C4—22no—
338C4⋊C4—1222no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—2442no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2442no—
347C22⋊C4—2442no—
348C42—41no—
349C42—1222no—
350C42—1222no—
351C42—2441no—
352C42—2448no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—612no—
361C23—612no—
362C23—611no—
363C23—611no—
364C23—1221no—
365C23—1221no—
366C23—1221no—
367C23—1221no—
368C23C2×C22⋊#692311yescharacteristic
369Q8—4882no—
370D4—82no—
371D4—4882no—
372D4—4882no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—611no—
383C2×C4—611no—
384C2×C4—612no—
385C2×C4—1224no—
386C2×C4—1224no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—1228no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2444no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2444no—
405C2×C4—2444no—
406C2×C4—2444no—
407C2×C4—2444no—
408C2×C4—2444no—
409C2×C4—2444no—
410C2×C4#1451984112yes—
411C2×C4#1451984112yes—
412C2×C4D4×#692312yes—
413C8—4882no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—612no—
422C22—612no—
423C22—1222no—
424C22—1222no—
425C22—1222no—
426C22—1222no—
427C22—1221no—
428C22—1221no—
429C22—1221no—
430C22—1221no—
431C22#1684071411yescharacteristic
432C22D61⋊#9911yescharacteristic
433C22#1684074111yescharacteristic
434C22#1684075211yescharacteristic
435C22#1684075311yescharacteristic
436C22C2×#1451984111yescharacteristic
437C22C2×#1451984111yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—81no—
442C4—1221no—
443C4—1221no—
444C4—1222no—
445C4—4881no—
446C4—4881no—
447C4—2448no—
448C4—2441no—
449C2—161no—
450C2—1222no—
451C2—1222no—
452C2#2146119011yescharacteristic
453C2#2146119511yescharacteristic
454C2#2146120411yescharacteristic
455C2#2146122111yescharacteristic
456C2#2146195311yescharacteristic
457C2#2146196011yescharacteristic
458C2#2146198711yescharacteristic
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#9713
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. #21461674C2semidirect
2. C2×#16840615C2semidirect
4. C4×(C2×#6923)⋊C4C2semidirect
14. C2×#14519764C4semidirect
18. #16839668C4semidirect
23. #16839627C4semidirect
24. C61×#390C4semidirect
243. #390#6923semidirect
254. C61#9713semidirect

Maximal subgroups

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

subgroupindex
#214616742
C2×#168406152
C2×#168406132
C4×(C2×#6923)⋊C42
#214541192
#971361