Finite Group Databasebeta

← #29576090

Subgroups — #29576090

Order 15616.

578 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 105 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#21451564C211yesmaximal, characteristic
2#21448421C211yesmaximal, characteristic, complemented
3#21444388C211yesmaximal, characteristic
4#21444386C211yesmaximal, characteristic
5C2×#16839487C211yesmaximal, characteristic, complemented
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1289C211yesmaximal, characteristic, complemented
8C2×#14519761—21no—
9C2×#14519751C2211yescharacteristic
10C22×#11758196C2211yescharacteristic
11C2×C22⋊#92418C2211yescharacteristic
12#16839694C412yescomplemented
13#16839684—22no—
14#16839680C412yescomplemented
15#16839649—22no—
16#16839643—22no—
17#16839639—21no—
18#16839623—21no—
19#16839622C2211yescharacteristic
20C22⋊C4×Dic61—21no—
21#16839497—22no—
22#16839495—22no—
23#16839487—22no—
24C2×C22⋊C8×C61C2211yescharacteristic
25C61×#380C2211yescharacteristic
26C61×#380C2211yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31C22×#92418—21no—
32#14519761—42no—
33#14519761—42no—
34#14519751—21no—
35#14519751—21no—
36#14519751—21no—
37#14519751—21no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196C2×C412yes—
46C2×#11758196D412yes—
47C2×C22⋊Dic61—21no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×C4×Dic61—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519684—22no—
55#14519684—24no—
56#14519684—22no—
57#14519684D412yes—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C22⋊C8×C61—21no—
65C22⋊C8×C61—21no—
66#95×C61C2×C412yes—
67#95×C61D412yes—
68C24×C61C2×D411yescharacteristic
69C23×D61—21no—
70C22×C4×C61—22no—
71C22×C4×C61—22no—
72C22×C4×C61—81no—
73C22×C4×C61C22×C411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61#4111yescharacteristic
78C22×C4×C61#4111yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—81no—
96C22×Dic61C22⋊C412yes—
97C22×Dic61C22⋊C412yes—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—81no—
101C2×C8×C61—41no—
102C2×C8×C61—81no—
103C4⋊C4×C61—81no—
104C4⋊C4×C61—81no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111#11758198—44no—
112#11758198—41no—
113#11758198—41no—
114#11758198—82no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—44no—
118C4⋊Dic61—81no—
119#11758196—22no—
120#11758196—22no—
121#11758196—22no—
122#11758196—22no—
123C4×Dic61—82no—
124C2×#92418—41no—
125C2×#92418—81no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61#12011yescharacteristic
130C23×C61#12311yescharacteristic
131C23×C61#13511yescharacteristic
132C23×C61#13511yescharacteristic
133C23×C61AGL(1,ℤ/8)11yescharacteristic
134C23×C61AGL(1,ℤ/8)11yescharacteristic
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C22⋊D61—81no—
157C22⋊D61—81no—
158C22⋊D61—81no—
159C22⋊D61—81no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—81no—
177D244—81no—
178D244—81no—
179C8×C61—81no—
180#92418—81no—
181#9206—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61#38611yescharacteristic
193C22×C61#41611yescharacteristic
194C22×C61#41811yescharacteristic
195C22×C61#44911yescharacteristic
196C22×C61#44911yescharacteristic
197C22×C61#48211yescharacteristic
198C22×C61#48211yescharacteristic
199C2×D61—82no—
200C2×D61—82no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—82no—
209C4×C61—82no—
210Dic61—42no—
211Dic61—42no—
212Dic61—82no—
213Dic61—81no—
214#2618—611no—
215#2313—611no—
216#1519—611no—
217#1519—611no—
218C2×#324—611no—
219#1289—611no—
220#1289D6111yescharacteristic, complemented
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61#187511yescharacteristic
228C2×C61#191211yescharacteristic
229C2×C61#191311yescharacteristic
230D61—161no—
231C2×#123—1221no—
232C2×#121—611no—
233C22×C4⋊C4—611no—
234C2×C22⋊C8—611no—
235C2×C22⋊C8C2×D6111yescharacteristic
236#398—612no—
237#396—1222no—
238#394—612no—
239#387—1222no—
240#387—1222no—
241#386—1221no—
242#381—1221no—
243#380—611no—
244#380C2×D6111yescharacteristic
245#380C2×D6111yescharacteristic
246C4×C22⋊C4—1221no—
247#327—1222no—
248#327—1222no—
249#324—1222no—
250C61#920611yescharacteristic, complemented, Sylow
251C22×D4—611no—
252C22×D4—1221no—
253C23×C4—611no—
254C23×C4—611no—
255C23×C4C22×D6111yescharacteristic
256C22×C8—21no—
257C22×C8—1221no—
258#123—2442no—
259#123—2442no—
260#121—1221no—
261#121—1221no—
262#121—1221no—
263#121—1221no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—612no—
267C2×C4⋊C4—612no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1222no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C42—1222no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—1221no—
285C22⋊C8—1221no—
286#95—612no—
287#95—1222no—
288#95—1222no—
289#95—1224no—
290#95C4×D6112yes—
291#95D24412yes—
292C24—1221no—
293C24D4×D6111yescharacteristic
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—4881no—
297C2×D4—2441no—
298C2×D4—2441no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2441no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—81no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1222no—
318C22×C4—4881no—
319C22×C4C2×C4×D6111yescharacteristic
320C22×C4C2×D24411yescharacteristic
321C22×C4#1175822211yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4D4×D6111yescharacteristic
324C22×C4#1175822611yescharacteristic
325C2×C8—41no—
326C2×C8—81no—
327C2×C8—4881no—
328C2×C8—2441no—
329C4⋊C4—81no—
330C4⋊C4—81no—
331C4⋊C4—1222no—
332C4⋊C4—1222no—
333C4⋊C4—1222no—
334C4⋊C4—1222no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—4881no—
338C4⋊C4—2444no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—4882no—
344C22⋊C4—2444no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—4882no—
350C23—41no—
351C23—41no—
352C23—2442no—
353C23—2442no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23C22⋊C4×D6111yescharacteristic
358C23#1451974711yescharacteristic
359C23#1451976211yescharacteristic
360C23#1451979811yescharacteristic
361C23#1451980011yescharacteristic
362C23#1451980111yescharacteristic
363C23#1451980411yescharacteristic
364D4—4881no—
365D4—4881no—
366D4—4881no—
367D4—4881no—
368D4—4881no—
369D4—4881no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—4881no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2442no—
399C2×C4#1451973911yescharacteristic
400C2×C4C4×D24411yescharacteristic
401C2×C4#1451975011yescharacteristic
402C2×C4#1451975211yescharacteristic
403C8—81no—
404C8—4881no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—4882no—
416C22—4882no—
417C22—4881no—
418C22—4881no—
419C22—4881no—
420C22#1683964011yescharacteristic
421C22#1683975711yescharacteristic
422C22#1683976511yescharacteristic
423C22#1683990811yescharacteristic
424C22#1683991011yescharacteristic
425C22#1684008811yescharacteristic
426C22#1684008911yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—82no—
433C4—4882no—
434C4—4881no—
435C4—2442no—
436C4—2442no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—9761no—
444C2#2144614011yescharacteristic
445C2#2144635511yescharacteristic
446C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1289D61
hypocenterC61#9206
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21448421C2semidirect
5. C2×#16839487C2semidirect
6. #21443686C2semidirect
7. C61×#1289C2semidirect
12. #16839694C4semidirect
14. #16839680C4semidirect
220. #1289D61semidirect
250. C61#9206semidirect

Maximal subgroups

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

subgroupindex
#214515642
#214484212
#214443882
#214443862
C2×#168394872
#214436862
C61×#12892
#920661