Finite Group Databasebeta

← #29580775

Subgroups — #29580775

Order 15616.

568 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455620C211yesmaximal, characteristic, complemented
2#21450728C211yesmaximal, characteristic
3#21444383C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic
5#21444215C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3101C211yesmaximal, characteristic
8C2×#14519756—21no—
9C2×C4×Dic122C2211yescharacteristic
10#16840066—22no—
11#16840054—21no—
12#16840044—21no—
13#16839815C2211yescharacteristic
14#16839805—21no—
15#16839786C2211yescharacteristic
16#16839770—21no—
17#16839693—21no—
18#16839676—21no—
19#16839650—22no—
20#16839628C2211yescharacteristic
21#16839622—21no—
22#16839499—22no—
23#16839494—22no—
24D4⋊#92418—22no—
25C61×#533C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#483—21no—
28C61×#476—21no—
29C61×#461—21no—
30C61×#450—21no—
31C4×SD16×C61—22no—
32C61×#431C2211yescharacteristic
33C22×Dic122D411yescharacteristic
34C2×#11758194—21no—
35#14519756—42no—
36#14519756—42no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—22no—
44C2×#11758196—21no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—22no—
47C2×C4×Dic61—21no—
48#14519710—22no—
49#14519708—22no—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53#14519700—22no—
54#14519700—22no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57#14519684—22no—
58#14519684—21no—
59#14519684D412yes—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C2×SD16×C61—22no—
63C4⋊Q8×C61D411yescharacteristic
64C4⋊Q8×C61D411yescharacteristic
65#125×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74#117×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#106×C61—22no—
79C4⋊C8×C61—21no—
80#103×C61—22no—
81#102×C61—22no—
82C22⋊C8×C61—21no—
83C4×C8×C61D412yes—
84C24×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—42no—
90C22×C4×C61—21no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C2×Dic122—21no—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×Dic122—21no—
107C2×Dic122—21no—
108C2×Dic122SD1612yescomplemented
109SD16×C61—44no—
110C2×C8×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C22⋊Dic61—44no—
128C22⋊Dic61—44no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133#11758196—44no—
134#11758196—41no—
135#11758196—41no—
136C4×Dic61—41no—
137C4×Dic61—41no—
138#11758194—41no—
139#11758194—41no—
140C2×#92418—22no—
141C2×#92418—41no—
142C23×C61—22no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61#12011yescharacteristic
150Q8×C61—42no—
151Q8×C61—81no—
152D4×C61—22no—
153D4×C61—22no—
154D4×C61—41no—
155D4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12711yescharacteristic
173C2×C4×C61C2×SD1611yescharacteristic
174C2×C4×C61#13511yescharacteristic
175C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C61#13711yescharacteristic
177C2×Dic61—22no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—42no—
185Dic122—22no—
186Dic122—22no—
187Dic122—41no—
188Dic122—41no—
189C8×C61—42no—
190#92418—42no—
191#92418—81no—
192#10001—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—42no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61#39211yescharacteristic
203C22×C61#45011yescharacteristic
204C22×C61#45211yescharacteristic
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—81no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61#44811yescharacteristic
213C4×C61#44911yescharacteristic
214C4×C61#46511yescharacteristic
215C4×C61#46811yescharacteristic
216Dic61—42no—
217Dic61—82no—
218Dic61—81no—
219#3101D6111yescharacteristic
220#3078—611no—
221#2536—611no—
222#1518—611no—
223#1486—611no—
224#1479—611no—
225#1286—611no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—82no—
229C2×C61#200011yescharacteristic
230C2×C61#200911yescharacteristic
231C2×C61#201011yescharacteristic
232#533C2×D6111yescharacteristic
233C2×C22⋊Q8—1221no—
234C2×C4×Q8—611no—
235C2×C4×D4C2×D6111yescharacteristic
236#483—21no—
237#479—1222no—
238#477—1221no—
239#476—21no—
240#475—1221no—
241#461—21no—
242#450—21no—
243C4×SD16—22no—
244#431—611no—
245#431C2×D6111yescharacteristic
246#428—1221no—
247#423—611no—
248#419—1221no—
249#398—1221no—
250#394—1221no—
251#388—1222no—
252#382—611no—
253#380—1221no—
254#328—1222no—
255Q8⋊C8—1222no—
256D4⋊C8—1222no—
257C61#1000111yescharacteristic, complemented, Sylow
258C22×Q8—611no—
259C22×D4C22⋊D6111yescharacteristic
260C23×C4—21no—
261C2×SD16—22no—
262C2×M16—1221no—
263C4⋊Q8C22⋊D6111yescharacteristic
264C4⋊Q8C22⋊D6111yescharacteristic
265#125—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—2442no—
269C22⋊Q8—2442no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—1221no—
273C4×Q8—1221no—
274C4×Q8—1221no—
275C4×Q8—1221no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280#117—21no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1222no—
289C2×C42—1221no—
290C2×C42C22×D6111yescharacteristic
291#107—1222no—
292#106—22no—
293#106—1222no—
294C4⋊C8—21no—
295C4⋊C8—612no—
296C4⋊C8—1221no—
297C4⋊C8—1221no—
298#103—22no—
299#103—1222no—
300#103—1222no—
301#102—22no—
302C22⋊C8—21no—
303C22⋊C8—1221no—
304C4×C8—1221no—
305C4×C8C22⋊D6112yes—
306#95—612no—
307#95—1222no—
308#95—1221no—
309C24—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—612no—
313C2×Q8—1221no—
314C2×Q8—1221no—
315C2×Q8—1221no—
316C2×Q8—1221no—
317C2×Q8—1221no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—1221no—
329C22×C4—1222no—
330C22×C4—1221no—
331C22×C4D4×D6111yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—44no—
335M16—2441no—
336M16—2441no—
337C2×C8—22no—
338C2×C8—1222no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—2444no—
351C4⋊C4—2441no—
352C4⋊C4—2441no—
353C4⋊C4—2441no—
354C4⋊C4—2441no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—2444no—
359C22⋊C4—2444no—
360C42—41no—
361C42—2441no—
362C42—2441no—
363C42D4×D6111yescharacteristic
364C42#1175822411yescharacteristic
365C42C2×C22⋊D6111yescharacteristic
366C42C2×C22⋊D6111yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23#1451974511yescharacteristic
375Q8—42no—
376Q8—81no—
377Q8—1222no—
378Q8—1222no—
379Q8—2441no—
380Q8—2441no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—1222no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4#1451974511yescharacteristic
405C2×C4#1451974611yescharacteristic
406C2×C4#1451974811yescharacteristic
407C2×C4#1451974911yescharacteristic
408C2×C4#1451974911yescharacteristic
409C2×C4#1451977311yescharacteristic
410C2×C4SD16×D6111yescharacteristic
411C2×C4#1451979811yescharacteristic
412C2×C4#1451980411yescharacteristic
413C2×C4#1451980611yescharacteristic
414C8—42no—
415C8—4881no—
416C8—2442no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#1683967011yescharacteristic
427C22#1683991111yescharacteristic
428C22#1683992111yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—4882no—
437C4—4881no—
438C4—2442no—
439C4#1683989911yescharacteristic
440C4#1683990611yescharacteristic
441C4#1683998811yescharacteristic
442C4#1684000511yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2#2144681911yescharacteristic
447C2#2144687611yescharacteristic
448C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3101C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#3101D61
hypocenterC61#10001
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21455620C2semidirect
3. #21444383C2semidirect
5. #21444215C2semidirect
6. #21443678C2semidirect
108. C2×Dic122SD16semidirect
257. C61#10001semidirect

Maximal subgroups

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

subgroupindex
#214556202
#214507282
#214443832
#214442442
#214442152
#214436782
C61×#31012
#1000161