Finite Group Databasebeta

← #30600423

Subgroups — #30600423

Order 15616.

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

normal subgroups only

This group has 630 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
1C2×#16840706C211yesmaximal, characteristic
2#21461711C211yesmaximal, characteristic
3C2×#16840616C212yesmaximal
4#21460692C212yesmaximal
5#21454160C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×#14519834C2212yes—
8#16840706—24no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840628—21no—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839669C411yescharacteristic, complemented
17#16839661C411yescharacteristic, complemented
18#16839660C411yescharacteristic, complemented
19#16839645—21no—
20#16839644—21no—
21#16839636—21no—
22#16839624—21no—
23#16839619C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4⋊#6923—24no—
28C2×C4⋊#6923—42no—
29C2×C4⋊#6923—42no—
30#14519834—22no—
31#14519834—22no—
32(C2×#6923)⋊C4—24no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4D414yes—
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519753—22no—
39#14519753—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C42×D61—22no—
51C42×D61—22no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C2×Q8×C61—22no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22×C411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61SD1612yes—
94C2×C4×D61Q1612yes—
95C2×C4×D61C2×D412yes—
96C2×C4×D61#4112yes—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C4⋊#6923—48no—
101C4⋊C4×C61—41no—
102C42×C61—22no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106#11758198—41no—
107C4⋊Dic61—22no—
108C4⋊Dic61—42no—
109C4⋊Dic61—41no—
110#11758196—42no—
111#11758196—42no—
112C4×Dic61—22no—
113C4×Dic61—22no—
114C4×Dic61—22no—
115C23×C61C2×C22⋊C411yescharacteristic
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61#12011yescharacteristic
121C22×D61#12311yescharacteristic
122C22×D61C2×SD1611yescharacteristic
123C22×D61C2×Q1611yescharacteristic
124C22×D61AGL(1,ℤ/8)11yescharacteristic
125C22×D61#13711yescharacteristic
126C2×#6923—44no—
127C2×#6923—44no—
128C2×#6923—44no—
129C2×#6923—82no—
130C2×#6923—82no—
131C2×#6923—82no—
132C2×#6923—82no—
133Q8×C61—82no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61#10312yes—
148C2×C4×C61C4×D412yes—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61#10312yes—
163C2×Dic61#12112yes—
164C4×D61—22no—
165C4×D61—22no—
166C4×D61—22no—
167C4×D61—22no—
168C4×D61—42no—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61—42no—
172#92420—82no—
173#9152—611nomaximal, Sylow
174C22×C61#38611yescharacteristic
175C22×C61#41011yescharacteristic
176C22×C61#41011yescharacteristic
177C22×C61C2×#10311yescharacteristic
178C22×C61#41711yescharacteristic
179C22×C61#42111yescharacteristic
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#45011yescharacteristic
194C2×D61#45111yescharacteristic
195C2×D61#48311yescharacteristic
196C2×D61#48411yescharacteristic
197#6923—84no—
198C4×C61—22no—
199C4×C61—22no—
200C4×C61—41no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—22no—
204Dic61—22no—
205Dic61—41no—
206Dic61—81no—
207Dic61—81no—
208#2855—611no—
209C2×#382—611no—
210#1817—611no—
211C2×#328—612no—
212#1294—612no—
213C2×C61#151211yescharacteristic
214C2×C61#151211yescharacteristic
215C2×C61#151811yescharacteristic
216C2×C61#151811yescharacteristic
217C2×C61#187211yescharacteristic
218C2×C61#189711yescharacteristic
219C2×C61#189911yescharacteristic
220D61—21no—
221D61—21no—
222D61—21no—
223D61—21no—
224C2×C22⋊Q8—611no—
225C22×C4⋊C4—1221no—
226C22×C42—611no—
227C2×C22⋊C8—612no—
228#391—611no—
229#391#692311yescharacteristic, complemented
230#389—611no—
231#389—611no—
232#387—1221no—
233#387—1221no—
234#385—1221no—
235#382—1224no—
236#381—1221no—
237#380—611no—
238C2×#95—612no—
239C2×#95—1221no—
240#336—1221no—
241#336—1221no—
242#328—1224no—
243C61#915211yescharacteristic, complemented, Sylow
244C22×Q8C2×#692311yescharacteristic
245C23×C4—611no—
246C23×C4—611no—
247C23×C4—611no—
248C23×C4—1222no—
249C22×C8—1222no—
250C22⋊Q8—1222no—
251C22⋊Q8—1222no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1224no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—2442no—
259C2×C4⋊C4—2442no—
260C2×C4⋊C4C2×#692311yescharacteristic
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C42—611no—
264C2×C42—611no—
265C2×C42—611no—
266C2×C42—1222no—
267C2×C42—1222no—
268C2×C42C2×#692311yescharacteristic
269C22⋊C8—1222no—
270C22⋊C8—1222no—
271#95—21no—
272#95—21no—
273#95—614no—
274#95—1221no—
275#95—1221no—
276#95—1221no—
277#95—1221no—
278#95—1222no—
279#95—1224no—
280#95—1221no—
281C24—611no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—612no—
293C22×C4—611no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1224no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—2441no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4—2441no—
307C22×C4—2442no—
308C22×C4—2442no—
309C22×C4—2442no—
310C22×C4—2442no—
311C22×C4—2442no—
312C22×C4—2442no—
313C22×C4C22⋊#692311yescharacteristic
314C22×C4C22⋊#692311yescharacteristic
315C22×C4C22×#692311yescharacteristic
316C2×C8—2442no—
317C2×C8—2442no—
318C2×C8—2442no—
319C4⋊C4—41no—
320C4⋊C4—1222no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4—2442no—
324C4⋊C4—2442no—
325C4⋊C4—2448no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C42—22no—
331C42—1222no—
332C42—1222no—
333C42—1222no—
334C23—611no—
335C23—611no—
336C23—611no—
337C23—611no—
338C23—611no—
339C23—611no—
340C23—1221no—
341C23—1221no—
342C23—1221no—
343C23—1221no—
344C23C2×C22⋊#692311yescharacteristic
345Q8—82no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—612no—
360C2×C4—612no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—4882no—
373C2×C4—4882no—
374C2×C4—4882no—
375C2×C4—4882no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2444no—
381C2×C4—2444no—
382C2×C4—2444no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4#1451984012yes—
390C2×C4D4×#692312yes—
391C8—4882no—
392C22—611no—
393C22—611no—
394C22—611no—
395C22—611no—
396C22—1221no—
397C22—1221no—
398C22—1221no—
399C22—1221no—
400C22—1221no—
401C22—1221no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22#1684071011yescharacteristic
409C22#1684074111yescharacteristic
410C22#1684074211yescharacteristic
411C22C2×#1451984011yescharacteristic
412C22#1684075311yescharacteristic
413C22#1684075911yescharacteristic
414C22#1684076111yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—1222no—
421C4—1222no—
422C4—4884no—
423C4—4881no—
424C4—4881no—
425C4—2441no—
426C2—1221no—
427C2—1221no—
428C2—1221no—
429C2—1221no—
430C2#2146123111yescharacteristic
431C2#2146123211yescharacteristic
432C2#2146125211yescharacteristic
433C2#2146125311yescharacteristic
434C2#2146180411yescharacteristic
435C2#2146183811yescharacteristic
436C2#2146184211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9152
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
16. #16839669C4semidirect
17. #16839661C4semidirect
18. #16839660C4semidirect
24. C61×#391C4semidirect
229. #391#6923semidirect
243. C61#9152semidirect

Maximal subgroups

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

subgroupindex
C2×#168407062
#214617112
C2×#168406162
#214606922
#214541602
#915261