Finite Group Databasebeta

← #30601822

Subgroups — #30601822

Order 15616.

650 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 102 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#21461676C211yesmaximal, characteristic, complemented
2C2×#16840616C212yesmaximal, complemented
3C2×#16840614C212yesmaximal
4#21460995C211yesmaximal, characteristic, complemented
5#21454123C211yesmaximal, characteristic
6C2×#14519844—21no—
7C2×#14519834C2212yes—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840616—24no—
13#16840614—24no—
14C2×#14519766C411yescharacteristic, complemented
15C2×#14519753C2211yescharacteristic
16C2×#14519752C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839664C411yescharacteristic, complemented
19#16839651—21no—
20#16839651—21no—
21#16839640—21no—
22#16839640—21no—
23#16839627C411yescharacteristic, complemented
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519844—42no—
28#14519834—22no—
29#14519834—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—42no—
32(C2×#6923)⋊C4—42no—
33(C2×#6923)⋊C4D414yes—
34C22×D244—21no—
35C22×D244C2×C411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519766—42no—
40#14519753—22no—
41#14519753—22no—
42#14519752—22no—
43#14519752—22no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C42×D61—42no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63C23×D61—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×Q8×C61—22no—
73C2×Q8×C61—41no—
74C2×Q8×C61—41no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×D244—22no—
85C2×D244—41no—
86C2×D244—41no—
87C2×D244—42no—
88C2×D244—42no—
89C2×D244—41no—
90C2×D244—41no—
91C2×C4×D61—22no—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—22no—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61C2×D412yes—
101C2×C4×D61#4112yes—
102C2×#92420—42no—
103C2×#92420—42no—
104C2×#92420—42no—
105C4⋊C4×C61—22no—
106C42×C61—41no—
107#11758198—42no—
108#11758198—42no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112#11758198—41no—
113#11758198—41no—
114C4⋊Dic61—22no—
115#11758196—41no—
116#11758196—41no—
117C4×Dic61—22no—
118C4×Dic61—22no—
119C4×Dic61—41no—
120C23×C61C2×C22⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61#12112yes—
133C22×D61#13512yes—
134C22×D61AGL(1,ℤ/8)11yescharacteristic
135C22×D61#13711yescharacteristic
136C2×#6923—44no—
137C2×#6923—44no—
138C2×#6923—44no—
139C2×#6923—82no—
140C2×#6923—82no—
141C2×#6923—82no—
142C2×#6923—82no—
143Q8×C61—82no—
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—21no—
151C2×C4×C61—21no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61C4×D412yes—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61#9912yes—
170C2×Dic61#12411yescharacteristic
171C2×Dic61#12711yescharacteristic
172D244—82no—
173D244—82no—
174D244—82no—
175C4×D61—42no—
176C4×D61—42no—
177C4×D61—44no—
178C4×D61—42no—
179C4×D61—42no—
180#92420—82no—
181#9716—611nomaximal, Sylow
182C22×C61#39011yescharacteristic
183C22×C61C2×#9911yescharacteristic
184C22×C61#41011yescharacteristic
185C22×C61#41611yescharacteristic
186C22×C61#41711yescharacteristic
187C22×C61#42111yescharacteristic
188C22×C61#42111yescharacteristic
189C2×D61—22no—
190C2×D61—22no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—22no—
196C2×D61—22no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61#46312yes—
205C2×D61#46412yes—
206#6923—84no—
207C4×C61—22no—
208C4×C61—41no—
209C4×C61—41no—
210C4×C61—81no—
211C4×C61—81no—
212Dic61—21no—
213Dic61—21no—
214Dic61—22no—
215Dic61—41no—
216Dic61—81no—
217#2850—611no—
218#1797—611no—
219C2×#328—612no—
220C2×#327—612no—
221#1434—611no—
222C2×C61#151211yescharacteristic
223C2×C61#151311yescharacteristic
224C2×C61#151611yescharacteristic
225C2×C61#152111yescharacteristic
226C2×C61#196311yescharacteristic
227C2×C61#196411yescharacteristic
228C2×C61#198311yescharacteristic
229D61—22no—
230D61—22no—
231D61—161no—
232C2×#124—611no—
233C2×C22⋊Q8—611no—
234C2×#121—611no—
235C22×C42—611no—
236C2×C4⋊C8—1221no—
237C2×C22⋊C8—612no—
238#391#692311yescharacteristic, complemented
239#390—611no—
240#388—1221no—
241#388—1221no—
242#386—1221no—
243#386—1221no—
244#382—611no—
245C2×#95—612no—
246C2×#95—1221no—
247C2×#95—1221no—
248#336—1221no—
249#328—1224no—
250#327—1224no—
251C61#971611yescharacteristic, complemented, Sylow
252C22×Q8C2×#692311yescharacteristic
253C22×D4—611no—
254C22×D4—1221no—
255C23×C4—611no—
256C23×C4—611no—
257C23×C4—611no—
258C23×C4—1222no—
259C22×C8—1222no—
260#124—2442no—
261C22⋊Q8—1222no—
262C22⋊Q8—1222no—
263#121—1222no—
264#121—1222no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4C2×#692311yescharacteristic
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C42—611no—
275C2×C42—611no—
276C2×C42—611no—
277C2×C42—2442no—
278C2×C42C2×#692311yescharacteristic
279C4⋊C8—2442no—
280C22⋊C8—1222no—
281C22⋊C8—1222no—
282#95—21no—
283#95—21no—
284#95—614no—
285#95—1221no—
286#95—1221no—
287#95—1222no—
288#95—2442no—
289#95—2442no—
290C24—611no—
291C24—1221no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—1222no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4—2442no—
299C2×D4—2442no—
300C2×D4—2441no—
301C2×D4—2441no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—611no—
309C22×C4—1222no—
310C22×C4—1221no—
311C22×C4—1222no—
312C22×C4—1221no—
313C22×C4—1224no—
314C22×C4—1221no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4C22⋊#692311yescharacteristic
327C22×C4C22⋊#692311yescharacteristic
328C22×C4C22×#692311yescharacteristic
329C2×C8—2442no—
330C2×C8—2442no—
331C2×C8—2442no—
332C4⋊C4—22no—
333C4⋊C4—1222no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C22⋊C4—2442no—
337C22⋊C4—2442no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C42—41no—
344C42—1222no—
345C42—1222no—
346C42—2441no—
347C23—612no—
348C23—612no—
349C23—611no—
350C23—611no—
351C23—1221no—
352C23—1221no—
353C23—1221no—
354C23—1221no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23—2441no—
359C23—2441no—
360C23—2441no—
361C23—2441no—
362C23C2×C22⋊#692311yescharacteristic
363Q8—82no—
364D4—4882no—
365D4—4882no—
366D4—4882no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—611no—
382C2×C4—611no—
383C2×C4—612no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2444no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2444no—
401C2×C4—2444no—
402C2×C4—2444no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4D4×#692312yes—
407C8—4882no—
408C22—612no—
409C22—612no—
410C22—1222no—
411C22—1222no—
412C22—1222no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—1221no—
417C22—1222no—
418C22—4881no—
419C22—4881no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22#1684071411yescharacteristic
426C22D61⋊#9911yescharacteristic
427C22#1684074111yescharacteristic
428C22#1684075211yescharacteristic
429C22#1684075311yescharacteristic
430C22#1684076111yescharacteristic
431C22#1684076111yescharacteristic
432C4—22no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—1221no—
438C4—1221no—
439C4—1222no—
440C4—4884no—
441C4—4881no—
442C4—2441no—
443C2—1222no—
444C2—1222no—
445C2—9761no—
446C2#2146123211yescharacteristic
447C2#2146123711yescharacteristic
448C2#2146124611yescharacteristic
449C2#2146126311yescharacteristic
450C2#2146195511yescharacteristic
451C2#2146196011yescharacteristic
452C2#2146198711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C22
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9716
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #21461676C2semidirect
2. C2×#16840616C2semidirect
4. #21460995C2semidirect
14. C2×#14519766C4semidirect
18. #16839664C4semidirect
23. #16839627C4semidirect
24. C61×#391C4semidirect
238. #391#6923semidirect
251. C61#9716semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406162
C2×#168406142
#214609952
#214541232
#971661