Finite Group Databasebeta

← #29576709

Subgroups — #29576709

Order 15616.

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

normal subgroups only

This group has 594 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#21456757C211yesmaximal, characteristic
2#21448542C211yesmaximal, characteristic
3#21446801C211yesmaximal, characteristic, complemented
4#21444486C211yesmaximal, characteristic
5#21443677C211yesmaximal, characteristic, complemented
6#21443647C211yesmaximal, characteristic, complemented
7C61×#1518C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic
9C2×#14519739—21no—
10#16840136C412yescomplemented
11#16840110C412yescomplemented
12#16839962—22no—
13#16839912—22no—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839663—21no—
17#16839640—22no—
18#16839640—22no—
19#16839608C2211yescharacteristic
20#16839521—21no—
21#16839521—21no—
22#16839505—22no—
23C61×#533C2211yescharacteristic
24C61×#431C2211yescharacteristic
25C61×#382C2211yescharacteristic
26C61×#328—22no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244D412yes—
30C22×C4×D61—22no—
31C2×#11758201—24no—
32C2×#11758194—21no—
33#14519774—21no—
34#14519774—21no—
35#14519774C2×C412yes—
36#14519774D412yes—
37#14519739—42no—
38#14519739—42no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47#14519697—24no—
48C22⋊#92418—21no—
49C4×#92418C2×C412yescomplemented
50#14519684—21no—
51#14519684—21no—
52C4⋊Q8×C61C2×C411yescharacteristic, complemented
53C4⋊Q8×C61C2×C411yescharacteristic, complemented
54#125×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57C4×Q8×C61—21no—
58C4×Q8×C61—21no—
59#117×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C4⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64C4×C8×C61—21no—
65#95×C61—21no—
66#95×C61D412yes—
67C23×D61—22no—
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70C22×C4×C61—22no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61—21no—
76C2×D244—22no—
77C2×D244—22no—
78C2×D244—22no—
79C2×D244—22no—
80C2×D244—22no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—41no—
85C2×D244—82no—
86C2×D244—82no—
87C2×D244C22⋊C414yes—
88C2×C4×D61—44no—
89C2×C4×D61—82no—
90C2×C4×D61—82no—
91C2×C8×C61—41no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C42×C61—41no—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61#4111yescharacteristic
105C42×C61#4111yescharacteristic
106#11758201—84no—
107#11758198—44no—
108#11758198—82no—
109#11758198—82no—
110C4×Dic61—21no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C4×Dic61SD1612yescomplemented
114#11758194—41no—
115#11758194—41no—
116C2×#92418—22no—
117C2×#92418—41no—
118C23×C61#12011yescharacteristic
119C22×D61—24no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—42no—
126Q8×C61—42no—
127Q8×C61—81no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12311yescharacteristic
141C2×C4×C61C2×SD1611yescharacteristic
142C2×C4×C61#13511yescharacteristic
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C61#13711yescharacteristic
145C2×Dic61—22no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151D244—44no—
152D244—42no—
153D244—42no—
154D244—42no—
155D244—42no—
156D244—82no—
157D244—82no—
158D244—82no—
159C4×D61—88no—
160C8×C61—81no—
161#92418—42no—
162#92418—81no—
163#9295—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#38611yescharacteristic
167C22×C61#45011yescharacteristic
168C22×C61#45211yescharacteristic
169C2×D61—44no—
170C2×D61—44no—
171C2×D61—44no—
172C2×D61—82no—
173C2×D61—82no—
174C2×D61—82no—
175C2×D61—82no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—82no—
181C4×C61—81no—
182C4×C61—81no—
183C4×C61C4×SD1611yescharacteristic
184C4×C61#44011yescharacteristic
185C4×C61#46111yescharacteristic
186C4×C61#46311yescharacteristic
187Dic61—42no—
188Dic61—81no—
189#3208—611no—
190#2328—611no—
191#1995—611no—
192#1545—611no—
193#1518D6111yescharacteristic, complemented
194#1286—611no—
195#1274—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#187711yescharacteristic
199C2×C61#190411yescharacteristic
200C2×C61#191111yescharacteristic
201D61—84no—
202#533C2×D6111yescharacteristic
203C2×#127—611no—
204C2×C4×D4—1221no—
205#492—612no—
206#486—612no—
207#460—1222no—
208#450—1222no—
209#431—611no—
210#431C2×D6111yescharacteristic
211#423—611no—
212#390—1221no—
213#386—1222no—
214#386—1222no—
215#382C2×D6111yescharacteristic
216#376—611no—
217#339—1221no—
218#339—1221no—
219#331—1222no—
220#328—22no—
221C61#929511yescharacteristic, complemented, Sylow
222C22×D4—612no—
223C22×D4—1221no—
224C22×D4—1221no—
225C23×C4—1222no—
226C2×SD16—1224no—
227C2×M16—1221no—
228C4⋊Q8C4×D6111yescharacteristic, complemented
229C4⋊Q8C4×D6111yescharacteristic, complemented
230#127—612no—
231#127—612no—
232#127—1221no—
233#127—1221no—
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—2442no—
240C4×D4—2442no—
241#117—21no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—1221no—
251C4⋊C8—1221no—
252C4⋊C8—1221no—
253#102—1224no—
254C22⋊C8—21no—
255C22⋊C8—1221no—
256C4×C8—21no—
257C4×C8—612no—
258#95—21no—
259#95—1221no—
260#95—1221no—
261#95C22⋊D6112yes—
262C24—1222no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—614no—
266C2×D4—1222no—
267C2×D4—1222no—
268C2×D4—1222no—
269C2×D4—1222no—
270C2×D4—1222no—
271C2×D4—4882no—
272C2×D4—4882no—
273C2×D4—2441no—
274C2×D4—2441no—
275C2×D4—2441no—
276C2×D4—2441no—
277C22×C4—22no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—4882no—
281C22×C4—4882no—
282C22×C4—2444no—
283C22×C4D4×D6111yescharacteristic
284C22×C4D4×D6111yescharacteristic
285C22×C4C2×C22⋊D6111yescharacteristic
286SD16—4884no—
287M16—2441no—
288M16—2441no—
289C2×C8—41no—
290C2×C8—1222no—
291C2×C8—2441no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—4882no—
302C22⋊C4—4882no—
303C22⋊C4—2444no—
304C42—41no—
305C42—612no—
306C42—1221no—
307C42—1221no—
308C42—1221no—
309C42C2×C4×D6111yescharacteristic
310C42#1175822211yescharacteristic
311C42D4×D6111yescharacteristic
312C42#1175822411yescharacteristic
313C23—1224no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23—2442no—
318C23—2442no—
319C23—2442no—
320C23#1451974511yescharacteristic
321Q8—42no—
322Q8—81no—
323D4—4882no—
324D4—4882no—
325D4—4882no—
326D4—2444no—
327D4—2442no—
328D4—2442no—
329D4—2442no—
330D4—2442no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—1222no—
339C2×C4—4888no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4C22⋊C4×D6111yescharacteristic
346C2×C4#1451973711yescharacteristic
347C2×C4C4×C22⋊D6111yescharacteristic
348C2×C4#1451974811yescharacteristic
349C2×C4#1451975011yescharacteristic
350C2×C4#1451975911yescharacteristic
351C2×C4SD16×D6111yescharacteristic
352C2×C4#1451980011yescharacteristic
353C2×C4#1451980411yescharacteristic
354C2×C4#1451980911yescharacteristic
355C8—81no—
356C8—4881no—
357C8—2442no—
358C22—21no—
359C22—21no—
360C22—4882no—
361C22—4882no—
362C22—4882no—
363C22—4882no—
364C22—2444no—
365C22—2444no—
366C22—2444no—
367C22#1683963811yescharacteristic
368C22#1683991111yescharacteristic
369C22#1683992311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—4881no—
378C4—2442no—
379C4#1683984711yescharacteristic
380C4#1683986511yescharacteristic
381C4#1683996611yescharacteristic
382C4#1683998011yescharacteristic
383C2—21no—
384C2—21no—
385C2—4884no—
386C2#2144615011yescharacteristic
387C2#2144630911yescharacteristic
388C2#2144634811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1518C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1518D61
hypocenterC61#9295
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
3. #21446801C2semidirect
5. #21443677C2semidirect
6. #21443647C2semidirect
7. C61×#1518C2semidirect
10. #16840136C4semidirect
11. #16840110C4semidirect
49. C4×#92418C2×C4semidirect
52. C4⋊Q8×C61C2×C4semidirect
53. C4⋊Q8×C61C2×C4semidirect
113. C4×Dic61SD16semidirect
193. #1518D61semidirect
221. C61#9295semidirect
228. C4⋊Q8C4×D61semidirect
229. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214485422
#214468012
#214444862
#214436772
#214436472
C61×#15182
#929561