Finite Group Databasebeta

← #29575935

Subgroups — #29575935

Order 15616.

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

normal subgroups only

This group has 494 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#21455618C211yesmaximal, characteristic
2#21448794C211yesmaximal, characteristic
3#21444382C211yesmaximal, characteristic, complemented
4#21444319C211yesmaximal, characteristic, complemented
5#21444209C211yesmaximal, characteristic
6#21443720C211yesmaximal, characteristic, complemented
7C61×#1286C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840063—22no—
11#16840053C412yescomplemented
12#16840043C412yescomplemented
13#16839815C2211yescharacteristic
14#16839804—21no—
15#16839768—21no—
16#16839663—21no—
17#16839654—22no—
18#16839624—21no—
19C4⋊C4×Dic61—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839609—21no—
22#16839500—21no—
23#16839500—21no—
24#16839498—22no—
25#16839489—22no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C61×#382C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31#14519777C2×C411yescharacteristic
32#14519777C2×C411yescharacteristic
33#14519769—21no—
34#14519756—21no—
35#14519756—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244C2×C412yes—
41C4×D244D412yes—
42#14519736—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758198—21no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—22no—
50#14519711—22no—
51#14519709—22no—
52C4⋊#92418—21no—
53#14519699—22no—
54#14519699—22no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57#14519684—22no—
58#14519684D411yescharacteristic
59#14519684D411yescharacteristic
60C2×M16×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61C2×C412yes—
66C22⋊C8×C61—21no—
67C4×C8×C61—21no—
68#95×C61—21no—
69#95×C61D412yes—
70C23×D61—21no—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244C22⋊C412yes—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—82no—
87C2×Dic122—21no—
88C2×Dic122—21no—
89M16×C61—41no—
90M16×C61—41no—
91C2×C8×C61—22no—
92C2×C8×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C42×C61C22×C411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61#4111yescharacteristic
98C42×C61#4111yescharacteristic
99C22⋊Dic61—41no—
100#11758198—22no—
101#11758198—22no—
102#11758198—44no—
103#11758198—82no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—42no—
110C4⋊Dic61C22⋊C412yes—
111#11758196—41no—
112#11758196—41no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116C4×Dic61—82no—
117C2×#92418—41no—
118C23×C61#12011yescharacteristic
119C22×D61—22no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—41no—
123C22×D61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12311yescharacteristic
135C2×C4×C61#13511yescharacteristic
136C2×C4×C61#13511yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C2×C4×C61#13711yescharacteristic
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—41no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149D244—22no—
150D244—22no—
151D244—41no—
152D244—41no—
153C4×D61—82no—
154C4×D61—82no—
155Dic122—81no—
156Dic122—81no—
157C8×C61—42no—
158C8×C61—81no—
159#92418—81no—
160#9177—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61#38611yescharacteristic
164C22×C61#45211yescharacteristic
165C22×C61#45211yescharacteristic
166C2×D61—42no—
167C2×D61—42no—
168C2×D61—42no—
169C2×D61—82no—
170C2×D61—82no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—82no—
176C4×C61#41611yescharacteristic
177C4×C61#41911yescharacteristic
178C4×C61#48511yescharacteristic
179C4×C61#48511yescharacteristic
180Dic61—42no—
181Dic61—42no—
182Dic61—81no—
183Dic61—81no—
184Dic61—81no—
185#3077—611no—
186#2357—611no—
187#1518—611no—
188#1503—611no—
189#1478—611no—
190#1303—611no—
191#1286D6111yescharacteristic, complemented
192C2×C61—21no—
193C2×C61—21no—
194C2×C61#187311yescharacteristic
195C2×C61#190411yescharacteristic
196C2×C61#190611yescharacteristic
197D61—82no—
198#533—611no—
199C2×C4×D4—611no—
200#478—1222no—
201#476—612no—
202#474—612no—
203#431—611no—
204#431C2×D6111yescharacteristic
205#428—1221no—
206#423C2×D6111yescharacteristic
207#418—1221no—
208#390—1221no—
209#388—1222no—
210#382C2×D6111yescharacteristic
211#381—1221no—
212C4×C4⋊C4—611no—
213C4×C4⋊C4—1221no—
214C4×C22⋊C4—1221no—
215#328—1221no—
216#328—1221no—
217#328—1222no—
218D4⋊C8—1222no—
219C61#917711yescharacteristic, complemented, Sylow
220C22×D4—611no—
221C23×C4—1221no—
222C2×M16—21no—
223C4⋊Q8—611no—
224C4⋊Q8—611no—
225#125—1221no—
226C22⋊Q8—1221no—
227C22⋊Q8—1221no—
228C4×Q8—1221no—
229C4×Q8—1221no—
230C4×D4—612no—
231C4×D4—612no—
232C4×D4—1221no—
233C4×D4—1221no—
234#117—1221no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—1221no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C42—1222no—
242C2×C42—1221no—
243C2×C42C22×D6111yescharacteristic
244#107—1222no—
245#106—1222no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—1221no—
249C4⋊C8C4×D6112yes—
250#102—1222no—
251#102—1222no—
252C22⋊C8—21no—
253C22⋊C8—1221no—
254C4×C8—21no—
255C4×C8—1221no—
256#95—21no—
257#95—611no—
258#95—611no—
259#95—1222no—
260#95D24412yes—
261C24—1221no—
262C2×Q8—1221no—
263C2×Q8—1221no—
264C2×D4—612no—
265C2×D4—1221no—
266C2×D4—1221no—
267C2×D4—1221no—
268C2×D4—1221no—
269C2×D4—1221no—
270C22×C4—22no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1221no—
276C22×C4—4882no—
277C22×C4C2×D24411yescharacteristic
278C22×C4D4×D6111yescharacteristic
279C22×C4D4×D6111yescharacteristic
280M16—41no—
281M16—41no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—2441no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—612no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4—1221no—
291C4⋊C4—1221no—
292C4⋊C4—1221no—
293C4⋊C4—2442no—
294C4⋊C4—2441no—
295C4⋊C4—2441no—
296C22⋊C4—1222no—
297C22⋊C4—1222no—
298C22⋊C4—4882no—
299C22⋊C4—2444no—
300C22⋊C4—2441no—
301C42—4882no—
302C42—2441no—
303C42—2441no—
304C42—2441no—
305C42C2×C4×D6111yescharacteristic
306C42#1175822211yescharacteristic
307C42D4×D6111yescharacteristic
308C42#1175822611yescharacteristic
309C23—1222no—
310C23—2442no—
311C23—2442no—
312C23—2441no—
313C23—2441no—
314C23#1451974711yescharacteristic
315Q8—4881no—
316Q8—4881no—
317D4—1222no—
318D4—1222no—
319D4—2441no—
320D4—2441no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—1222no—
327C2×C4—1222no—
328C2×C4—4882no—
329C2×C4—4882no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4C22⋊C4×D6111yescharacteristic
339C2×C4#1451973911yescharacteristic
340C2×C4C4×D24411yescharacteristic
341C2×C4#1451975011yescharacteristic
342C2×C4#1451975211yescharacteristic
343C2×C4#1451976211yescharacteristic
344C2×C4#1451979611yescharacteristic
345C2×C4#1451979811yescharacteristic
346C2×C4#1451980611yescharacteristic
347C2×C4#1451980911yescharacteristic
348C8—42no—
349C8—81no—
350C8—4881no—
351C22—21no—
352C22—21no—
353C22—4882no—
354C22—4882no—
355C22—2442no—
356C22—2442no—
357C22—2442no—
358C22#1683964011yescharacteristic
359C22#1683992511yescharacteristic
360C22#1683992711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—4881no—
367C4—4881no—
368C4—4881no—
369C4—2442no—
370C4—2442no—
371C4#1683975511yescharacteristic
372C4#1683976911yescharacteristic
373C4#1684010511yescharacteristic
374C4#1684010611yescharacteristic
375C2—21no—
376C2—21no—
377C2—4882no—
378C2#2144613011yescharacteristic
379C2#2144630711yescharacteristic
380C2#2144632111yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1286D61
hypocenterC61#9177
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #21444382C2semidirect
4. #21444319C2semidirect
6. #21443720C2semidirect
7. C61×#1286C2semidirect
11. #16840053C4semidirect
12. #16840043C4semidirect
191. #1286D61semidirect
219. C61#9177semidirect

Maximal subgroups

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

subgroupindex
#214556182
#214487942
#214443822
#214443192
#214442092
#214437202
C61×#12862
#917761