Finite Group Databasebeta

← #29576580

Subgroups — #29576580

Order 15616.

522 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 107 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#21456594C211yesmaximal, characteristic
2#21448899C211yesmaximal, characteristic
3#21446803C211yesmaximal, characteristic
4#21444390C211yesmaximal, characteristic
5C2×#16839498C211yesmaximal, characteristic
6#21443665C211yesmaximal, characteristic
7C61×#1294C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C2×#14519756C2211yescharacteristic
10#16840098—21no—
11#16840098C412yescomplemented
12#16840003—22no—
13#16839920—21no—
14#16839920C412yescomplemented
15C2×C22⋊#92418C2211yescharacteristic
16#16839666—21no—
17#16839647—21no—
18#16839644—21no—
19#16839635—21no—
20#16839613—21no—
21C4×C22⋊Dic61—21no—
22C4×C22⋊Dic61C2211yescharacteristic
23#16839528—21no—
24#16839528—21no—
25#16839498—22no—
26#16839498—21no—
27#16839498—21no—
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C2×#95×C61C2211yescharacteristic
31C22×Dic122C2×C411yescharacteristic
32C22×Dic122D411yescharacteristic
33C2×Dic244—22no—
34C22×#92418—21no—
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756C2×C412yes—
42#14519756D412yes—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61D411yescharacteristic
52C2×C22⋊Dic61D411yescharacteristic
53C2×C4×Dic61—21no—
54C2×C4×Dic61—21no—
55C2×C4×Dic61—21no—
56#14519711—22no—
57#14519702—22no—
58#14519702—22no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66C23×C4×C61—21no—
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61C2×C412yes—
73#95×C61—21no—
74#95×C61—21no—
75#95×C61D411yescharacteristic
76#95×C61D411yescharacteristic
77C24×C61C2×D411yescharacteristic
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—42no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—81no—
85C22×C4×C61—81no—
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—81no—
102C2×Dic122—81no—
103C2×Dic122C22⋊C412yes—
104C2×C8×C61—22no—
105C2×C8×C61—81no—
106C4⋊C4×C61—81no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊Dic61—22no—
110C22⋊Dic61—22no—
111C22⋊Dic61—21no—
112C22⋊Dic61—21no—
113C22⋊Dic61—21no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116C22⋊Dic61Q1612yescomplemented
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C22⋊C412yes—
121#11758196—22no—
122#11758196—22no—
123#11758196—42no—
124#11758196—41no—
125#11758196—41no—
126#11758196—81no—
127C4×Dic61—42no—
128C4×Dic61—81no—
129C4×Dic61—81no—
130C2×#92418—41no—
131C2×#92418—81no—
132Dic244—82no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61C4×D411yescharacteristic
136C23×C61#12011yescharacteristic
137C23×C61#12111yescharacteristic
138C23×C61C2×Q1611yescharacteristic
139C23×C61#13511yescharacteristic
140C23×C61#13711yescharacteristic
141C23×C61#13711yescharacteristic
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—82no—
155C2×C4×C61—82no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12311yescharacteristic
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—22no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—81no—
178C2×Dic61—81no—
179Dic122—42no—
180Dic122—42no—
181Dic122—81no—
182Dic122—81no—
183C8×C61—42no—
184#92418—81no—
185#9276—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61#38611yescharacteristic
197C22×C61C4×Q1611yescharacteristic
198C22×C61#44111yescharacteristic
199C22×C61#45111yescharacteristic
200C22×C61#45211yescharacteristic
201C22×C61#46211yescharacteristic
202C22×C61#46411yescharacteristic
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—81no—
209C4×C61—161no—
210Dic61—42no—
211Dic61—42no—
212Dic61—42no—
213Dic61—81no—
214Dic61—81no—
215Dic61—81no—
216#3189—611no—
217#2368—611no—
218#1996—611no—
219#1520—611no—
220C2×#328—611no—
221#1294D6111yescharacteristic
222#1282—611no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61#187811yescharacteristic
230C2×C61#190611yescharacteristic
231C2×C61#191311yescharacteristic
232C2×C22⋊Q8—611no—
233C2×C22⋊Q8—611no—
234#484—612no—
235#484—1221no—
236#467—1222no—
237#451—612no—
238#451—1221no—
239C2×C22⋊C8—611no—
240C2×C22⋊C8C2×D6111yescharacteristic
241#391—1221no—
242#387—1221no—
243#387—1221no—
244#385—1221no—
245#380C2×D6111yescharacteristic
246C4×C4⋊C4—1221no—
247C4×C22⋊C4—611no—
248C4×C22⋊C4—1221no—
249C2×#95C2×D6111yescharacteristic
250#342—1221no—
251#342—1221no—
252#328—1222no—
253#328—1221no—
254#328—1221no—
255C61#927611yescharacteristic, complemented, Sylow
256C22×Q8—611no—
257C22×Q8—611no—
258C23×C4—21no—
259C23×C4C22×D6111yescharacteristic
260C2×Q16—1222no—
261C22×C8—21no—
262C22×C8—1221no—
263C22⋊Q8—612no—
264C22⋊Q8—612no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C22⋊Q8—1221no—
269C22⋊Q8—1221no—
270C22⋊Q8—1221no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—611no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C42—1221no—
284C2×C42—1221no—
285C2×C42—1221no—
286#107—1222no—
287#103—1222no—
288#103—1222no—
289C22⋊C8—21no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C22⋊C8C4×D6112yes—
293#95—21no—
294#95—21no—
295#95—1221no—
296#95—1221no—
297#95—1221no—
298#95—1221no—
299#95—1221no—
300#95D24411yescharacteristic
301#95D24411yescharacteristic
302C24D4×D6111yescharacteristic
303C2×Q8—612no—
304C2×Q8—1221no—
305C2×Q8—1221no—
306C2×Q8—1221no—
307C2×Q8—4881no—
308C2×Q8—4881no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4C2×C4×D6111yescharacteristic
324C22×C4C2×D24411yescharacteristic
325C22×C4#1175822211yescharacteristic
326C22×C4D4×D6111yescharacteristic
327C22×C4D4×D6111yescharacteristic
328C22×C4#1175822611yescharacteristic
329Q16—4882no—
330C2×C8—22no—
331C2×C8—81no—
332C2×C8—4881no—
333C2×C8—2441no—
334C4⋊C4—81no—
335C4⋊C4—612no—
336C4⋊C4—1222no—
337C4⋊C4—1222no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—4881no—
342C4⋊C4—2442no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—612no—
348C22⋊C4—1222no—
349C22⋊C4—1222no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C42—4881no—
356C42—4881no—
357C42—2442no—
358C23—41no—
359C23—41no—
360C23#1451973911yescharacteristic
361C23#1451974711yescharacteristic
362C23#1451975011yescharacteristic
363C23Q16×D6111yescharacteristic
364C23#1451979611yescharacteristic
365C23#1451980611yescharacteristic
366C23#1451980611yescharacteristic
367Q8—4881no—
368Q8—4881no—
369Q8—2442no—
370Q8—2442no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—1222no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4C22⋊C4×D6111yescharacteristic
405C2×C4C4×D24411yescharacteristic
406C2×C4#1451975211yescharacteristic
407C2×C4#1451976211yescharacteristic
408C8—42no—
409C8—4881no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22#1683964011yescharacteristic
421C22#1683985611yescharacteristic
422C22#1683986811yescharacteristic
423C22#1683991911yescharacteristic
424C22#1683992511yescharacteristic
425C22#1683997111yescharacteristic
426C22#1683998311yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—81no—
433C4—161no—
434C4—4881no—
435C4—4881no—
436C4—4881no—
437C4—2442no—
438C4—2442no—
439C4—2442no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#2144616111yescharacteristic
447C2#2144631911yescharacteristic
448C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1294C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1294D61
hypocenterC61#9276
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
11. #16840098C4semidirect
14. #16839920C4semidirect
116. C22⋊Dic61Q16semidirect
255. C61#9276semidirect

Maximal subgroups

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

subgroupindex
#214565942
#214488992
#214468032
#214443902
C2×#168394982
#214436652
C61×#12942
#927661