Finite Group Databasebeta

← #29577329

Subgroups — #29577329

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454843C211yesmaximal, characteristic
2#21445807C211yesmaximal, characteristic
3#21444391C212yesmaximal, complemented
4#21443698C212yesmaximal, complemented
5C61×#2564C211yesmaximal, characteristic
6D4×#92418—24no—
7#16839815—21no—
8C2×#14519692—21no—
9C2×C22⋊#92418—21no—
10C2×C22⋊#92418C2212yes—
11#16839622C2212yes—
12#16839529—21no—
13#16839499—24no—
14C2×C22⋊Q8×C61C2211yescharacteristic
15C2×C4×D4×C61C2211yescharacteristic
16C61×#400C411yescharacteristic, complemented
17C61×#399C411yescharacteristic, complemented
18C61×#395C411yescharacteristic, complemented
19C61×#393C411yescharacteristic, complemented
20C61×#387—21no—
21C61×#386—21no—
22C61×#385—21no—
23C61×#380—21no—
24C2×#95×C61C2211yescharacteristic
25C22×#92418—22no—
26C22×#92418—22no—
27C2×C4⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C4⋊#92418—22no—
30#14519692—42no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33C22⋊#92418D414yes—
34C4×#92418—22no—
35#14519684—22no—
36C22×Q8×C61C2×C411yescharacteristic
37C22×D4×C61D411yescharacteristic
38C23×C4×C61—21no—
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C4×D4×C61—22no—
44C4×D4×C61—22no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61C2×C411yescharacteristic
62#95×C61C2×C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C2×Q8×C61—22no—
66C2×Q8×C61—81no—
67C2×D4×C61—22no—
68C2×D4×C61—22no—
69C2×D4×C61SD1612yes—
70C2×D4×C61Q1612yes—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—81no—
80C22×C4×C61—81no—
81C22×C4×C61—81no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4112yes—
89C22×Dic61—42no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—42no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—81no—
96C22⋊C4×C61—22no—
97C22⋊C4×C61—22no—
98C22⋊C4×C61—42no—
99C22⋊C4×C61—42no—
100C22⋊C4×C61—81no—
101C42×C61—21no—
102C42×C61—21no—
103C22⋊Dic61—42no—
104C22⋊Dic61—42no—
105C4⋊Dic61—82no—
106C2×#92418—24no—
107C2×#92418—82no—
108C2×#92418—82no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61#10312yes—
116C23×C61C2×C22⋊C411yescharacteristic
117C23×C61#12112yes—
118C23×C61C2×SD1611yescharacteristic
119C23×C61C2×Q1611yescharacteristic
120C23×C61#13512yes—
121Q8×C61—82no—
122D4×C61—24no—
123D4×C61—24no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—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—41no—
147C2×C4×C61—81no—
148C2×C4×C61—84no—
149C2×C4×C61#10312yes—
150C2×C4×C61C4×D412yes—
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61#12311yescharacteristic
153C2×Dic61—82no—
154C2×Dic61—82no—
155C2×Dic61—82no—
156#92418—44no—
157#9383—611nomaximal, Sylow
158C22×C61—22no—
159C22×C61—21no—
160C22×C61—22no—
161C22×C61—21no—
162C22×C61—22no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—42no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—81no—
170C22×C61#38611yescharacteristic
171C22×C61C2×#10311yescharacteristic
172C22×C61#41611yescharacteristic
173C22×C61#46512yes—
174C22×C61#46712yes—
175C4×C61—22no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—22no—
179C4×C61—42no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61—81no—
185Dic61—162no—
186#2972—611no—
187#2564D6111yescharacteristic
188#1799—611no—
189#1520—612no—
190#1294—612no—
191C2×C61—22no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—22no—
195C2×C61—42no—
196C2×C61#188411yescharacteristic
197C2×C61#189211yescharacteristic
198C2×C61#191311yescharacteristic
199C2×C22⋊Q8C2×D6111yescharacteristic
200C2×C4×D4C2×D6111yescharacteristic
201C8×D4—1224no—
202#431—1221no—
203C2×#99—1221no—
204C2×C22⋊C8—612no—
205C2×C22⋊C8—1221no—
206#400Dic6111yescharacteristic, complemented
207#399Dic6111yescharacteristic, complemented
208#395Dic6111yescharacteristic, complemented
209#393Dic6111yescharacteristic, complemented
210#387—21no—
211#386—21no—
212#385—21no—
213#380—21no—
214#380—612no—
215C2×#95C2×D6111yescharacteristic
216#342—1221no—
217#328—1224no—
218C61#938311yescharacteristic, complemented, Sylow
219C22×Q8C2×Dic6111yescharacteristic
220C22×D4C22⋊D6111yescharacteristic
221C23×C4—21no—
222C23×C4C22⋊D6111yescharacteristic
223C23×C4C22×D6111yescharacteristic
224C22×C8—1222no—
225C22×C8—1222no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1222no—
234C2×C4⋊C4C2×Dic6111yescharacteristic
235C2×C4⋊C4C2×Dic6111yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4C2×Dic6111yescharacteristic
241C2×C22⋊C4C22⋊D6111yescharacteristic
242C2×C42C22⋊D6111yescharacteristic
243C4⋊C8—1222no—
244#99—2442no—
245C22⋊C8—614no—
246C22⋊C8—1222no—
247C22⋊C8—1222no—
248C4×C8—1222no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—1222no—
255#95C2×Dic6111yescharacteristic
256#95C2×Dic6111yescharacteristic
257C24C22⋊Dic6111yescharacteristic
258C24C2×C22⋊D6111yescharacteristic
259C2×Q8—22no—
260C2×Q8—81no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1175820112yes—
264C2×D4#1175820212yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—2442no—
277C22×C4C22⋊Dic6111yescharacteristic
278C22×C4C22⋊Dic6111yescharacteristic
279C22×C4C22⋊Dic6111yescharacteristic
280C22×C4D4×D6112yes—
281C22×C4#1175822412yes—
282C22×C4C22×Dic6111yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284C2×C8—1224no—
285C2×C8—4882no—
286C2×C8—4882no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—4882no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—81no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C42—21no—
302C42—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1451970112yes—
310C23C2×C22⋊Dic6111yescharacteristic
311C23#1451974912yes—
312C23C2×#1175820111yescharacteristic
313C23C2×#1175820211yescharacteristic
314C23#1451979712yes—
315Q8—82no—
316D4—24no—
317D4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—84no—
343C2×C4—4882no—
344C2×C4—4882no—
345C2×C4—4882no—
346C2×C4#1451970112yes—
347C2×C4D4×Dic6112yes—
348C2×C4#1451974511yescharacteristic
349C2×C4#1451975811yescharacteristic
350C8—2444no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#1683964211yescharacteristic
364C22C2×#1451970111yescharacteristic
365C22#1683975811yescharacteristic
366C22#1683999312yes—
367C22#1684000112yes—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—9762no—
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#2144620311yescharacteristic
385C2#2144624311yescharacteristic
386C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2564C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#2564D61
hypocenterC61#9383
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #21444391C2semidirect
4. #21443698C2semidirect
16. C61×#400C4semidirect
17. C61×#399C4semidirect
18. C61×#395C4semidirect
19. C61×#393C4semidirect
206. #400Dic61semidirect
207. #399Dic61semidirect
208. #395Dic61semidirect
209. #393Dic61semidirect
218. C61#9383semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214458072
#214443912
#214436982
C61×#25642
#938361