Finite Group Databasebeta

← #29577475

Subgroups — #29577475

Order 15616.

720 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 93 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#21450751C211yesmaximal, characteristic
2#21448414C211yesmaximal, characteristic, complemented
3#21444335C211yesmaximal, characteristic, complemented
4#21444145C211yesmaximal, characteristic, complemented
5#21443705C211yesmaximal, characteristic, complemented
6#21443704C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8C2×#14519758—21no—
9C2×#14519745—21no—
10C22×#11758198C2211yescharacteristic
11#16839726C2211yescharacteristic
12#16839697C412yescomplemented
13#16839670C412yescomplemented
14#16839662—22no—
15#16839653—22no—
16#16839651—22no—
17#16839620—21no—
18#16839617—21no—
19C4×C22⋊Dic61C2211yescharacteristic
20C4×C22⋊Dic61C2211yescharacteristic
21#16839496—22no—
22#16839485—22no—
23C2×#121×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C4×C22⋊C4×C61C2211yescharacteristic
26C61×#327—22no—
27C24×D61D411yescharacteristic
28C22×C22⋊D61—22no—
29C23×Dic61D411yescharacteristic
30C2×#11758194—21no—
31#14519758—44no—
32#14519745—44no—
33C2×#11758196—22no—
34C2×#11758196—22no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—22no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198C2×C412yes—
46C2×#11758198D412yes—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—22no—
52#14519684C2×C412yes—
53#14519684D412yescomplemented
54C22×D4×C61—21no—
55C22×D4×C61C2×C411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×M16×C61—21no—
58#121×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C42×C61—21no—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#95×C61D412yescomplemented
69C24×C61—21no—
70C24×C61C2×D411yescharacteristic
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—42no—
74C23×D61—42no—
75C23×D61—81no—
76C23×D61C22⋊C412yes—
77C2×D4×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—42no—
80C2×D4×C61—42no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—81no—
84C22×C4×C61—21no—
85C22×C4×C61—22no—
86C22×C4×C61—81no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C2×C22⋊D61—44no—
94C2×C22⋊D61—44no—
95C2×C22⋊D61—82no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—22no—
100C22×Dic61—42no—
101C22×Dic61C22⋊C412yes—
102M16×C61—81no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C42×C61—81no—
110C22⋊Dic61—24no—
111C22⋊Dic61—44no—
112C22⋊Dic61—42no—
113#11758198—22no—
114#11758198—22no—
115#11758198—22no—
116#11758198—22no—
117#11758198—44no—
118#11758198—82no—
119#11758198—82no—
120#11758196—44no—
121#11758196—82no—
122C4×Dic61—81no—
123C4×Dic61—81no—
124#11758194—81no—
125C2×#92418—41no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—42no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61#12011yescharacteristic
136C23×C61#12311yescharacteristic
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—22no—
140C22×D61—42no—
141C22×D61—42no—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—81no—
147C22×D61—81no—
148C22×D61—88no—
149D4×C61—81no—
150D4×C61—81no—
151D4×C61—82no—
152D4×C61—82no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×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—42no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C22⋊D61—88no—
170C2×Dic61—24no—
171C2×Dic61—22no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—81no—
180C8×C61—81no—
181#92418—81no—
182#9413—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—44no—
191C22×C61—82no—
192C22×C61—82no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61#38611yescharacteristic
197C22×C61#42112yes—
198C22×C61#45312yes—
199C2×D61—44no—
200C2×D61—42no—
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—84no—
204C2×D61—84no—
205C2×D61—82no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—82no—
211C4×C61—81no—
212Dic61—44no—
213Dic61—42no—
214Dic61—82no—
215#2539—611no—
216#2312—611no—
217#1507—611no—
218#1507D6111yescharacteristic, complemented
219#1462—611no—
220#1297—611no—
221#1297—611no—
222C2×C61—22no—
223C2×C61—22no—
224C2×C61—41no—
225C2×C61—161no—
226C2×C61#188711yescharacteristic
227C2×C61#189711yescharacteristic
228C2×C61#191311yescharacteristic
229D61—82no—
230C2×#123—1221no—
231C2×#121C2×D6111yescharacteristic
232C2×#120—1221no—
233C22×C22⋊C4—611no—
234#407—611no—
235#407C2×D6111yescharacteristic
236#399—612no—
237#392—612no—
238#390—1222no—
239#388—1222no—
240#388—1222no—
241#380—1221no—
242#379—1221no—
243C4×C22⋊C4—611no—
244C4×C22⋊C4—611no—
245C4×C22⋊C4C2×D6111yescharacteristic
246#327—22no—
247#327—1222no—
248#323—1222no—
249C61#941311yescharacteristic, complemented, Sylow
250C25—611no—
251C22×D4—21no—
252C22×D4—1222no—
253C22×D4C4×D6111yescharacteristic
254C23×C4—611no—
255C23×C4C22×D6111yescharacteristic
256C2×M16—21no—
257C2×M16—1221no—
258#123—2444no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#120—2444no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4C4×D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—612no—
270C2×C22⋊C4—612no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1222no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1221no—
281C2×C42—21no—
282C2×C42—1221no—
283C2×C42—1221no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288#95—612no—
289#95—612no—
290#95—1222no—
291#95C22⋊D6112yescomplemented
292C24—21no—
293C24—612no—
294C24—1221no—
295C24—1221no—
296C24—4881no—
297C24—2442no—
298C24—2442no—
299C24D4×D6111yescharacteristic
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C2×D4—4882no—
308C2×D4—2444no—
309C2×D4—2444no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—81no—
313C22×C4—612no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1221no—
317C22×C4—1222no—
318C22×C4—2442no—
319C22×C4C2×C4×D6111yescharacteristic
320C22×C4#1175822211yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4#1175822411yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325M16—81no—
326M16—4881no—
327C2×C8—41no—
328C2×C8—2441no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—4882no—
332C4⋊C4—2444no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—1224no—
337C22⋊C4—1222no—
338C22⋊C4—1222no—
339C22⋊C4—1222no—
340C22⋊C4—1222no—
341C22⋊C4—4882no—
342C22⋊C4—4882no—
343C22⋊C4—2444no—
344C22⋊C4—2444no—
345C22⋊C4—2442no—
346C42—81no—
347C42—4881no—
348C42—4881no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—1222no—
358C23—1222no—
359C23—1222no—
360C23—4881no—
361C23—4881no—
362C23—4888no—
363C23—2442no—
364C23—2442no—
365C23—2442no—
366C23—2442no—
367C23—2442no—
368C23—2442no—
369C23C22⋊C4×D6111yescharacteristic
370C23#1451974511yescharacteristic
371C23#1451975911yescharacteristic
372D4—81no—
373D4—81no—
374D4—82no—
375D4—82no—
376D4—4888no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—1224no—
390C2×C4—1222no—
391C2×C4—4881no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4#1451973711yescharacteristic
400C2×C4C4×C22⋊D6111yescharacteristic
401C2×C4#1451974811yescharacteristic
402C2×C4#1451975011yescharacteristic
403C8—81no—
404C8—4881no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—82no—
414C22—82no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—4882no—
419C22—4884no—
420C22—4884no—
421C22—2444no—
422C22—2442no—
423C22—2442no—
424C22—2442no—
425C22#1683963811yescharacteristic
426C22#1683977712yes—
427C22#1683992812yes—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—82no—
433C4—81no—
434C4—4882no—
435C4—2444no—
436C4—2442no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—161no—
441C2—4882no—
442C2#2144621411yescharacteristic
443C2#2144626411yescharacteristic
444C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#386
prime radical#21450751C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#9413
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21448414C2semidirect
3. #21444335C2semidirect
4. #21444145C2semidirect
5. #21443705C2semidirect
6. #21443704C2semidirect
7. C61×#1507C2semidirect
12. #16839697C4semidirect
13. #16839670C4semidirect
53. #14519684D4semidirect
68. #95×C61D4semidirect
218. #1507D61semidirect
249. C61#9413semidirect
291. #95C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507512
#214484142
#214443352
#214441452
#214437052
#214437042
C61×#15072
#941361