Finite Group Databasebeta

← #29576013

Subgroups — #29576013

Order 15616.

504 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454381C211yesmaximal, characteristic
2#21451641C211yesmaximal, characteristic
3#21444500C211yesmaximal, characteristic, complemented
4#21444131C211yesmaximal, characteristic
5#21443679C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1548C211yesmaximal, characteristic, complemented
8C2×M16×D61—21no—
9C2×#14519734C2211yescharacteristic
10#16839815C2211yescharacteristic
11#16839787—22no—
12#16839787—22no—
13#16839786C2211yescharacteristic
14#16839785—21no—
15#16839700—21no—
16#16839697—21no—
17#16839681C412yes—
18#16839673C412yes—
19#16839658—21no—
20#16839628C2211yescharacteristic
21#16839619—21no—
22#16839511—22no—
23#16839487—21no—
24#16839485—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C61×#333—22no—
29C22×C4×D61D411yescharacteristic
30C2×#11758194—21no—
31C2×#11758189—22no—
32M16×D61—42no—
33M16×D61—42no—
34C2×C8×D61—22no—
35#14519734—21no—
36#14519734—21no—
37#14519734—21no—
38#14519734—21no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—22no—
41C2×#11758196C2×C411yescharacteristic
42C2×#11758196D411yescharacteristic
43C2×#11758198—22no—
44C2×#11758198C2×C411yescharacteristic
45C2×#11758198D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418D412yes—
50#14519706—24no—
51C22⋊#92418—21no—
52C4×#92418—21no—
53#14519684—21no—
54#14519684—22no—
55#14519684—22no—
56#14519684C2×C412yes—
57C22×Q8×C61C2×C412yes—
58C2×M16×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61D412yes—
69C22⋊C8×C61—21no—
70C4×C8×C61—21no—
71C23×D61C22⋊C411yescharacteristic
72C2×Q8×C61—22no—
73C2×Q8×C61—22no—
74C2×Q8×C61—82no—
75C22×C4×C61—22no—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61SD1612yes—
88C2×C4×D61Q1612yes—
89M16×C61—41no—
90M16×C61—41no—
91C2×C8×C61—22no—
92C2×C8×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C42×C61C2×D411yescharacteristic
98C42×C61C2×D411yescharacteristic
99C42×C61#4111yescharacteristic
100C42×C61#4111yescharacteristic
101#11758198—21no—
102#11758198—21no—
103#11758198—21no—
104#11758198—21no—
105#11758198—82no—
106C4⋊Dic61—41no—
107C4⋊Dic61—41no—
108#11758196—21no—
109#11758196—21no—
110#11758196—21no—
111#11758196—21no—
112#11758196—82no—
113C4×Dic61—21no—
114C4×Dic61—21no—
115#11758194—41no—
116#11758194—41no—
117C2×#92418—22no—
118C2×#92418—41no—
119#11758189—44no—
120C8×D61—44no—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61#10312yes—
127Q8×C61—82no—
128Q8×C61—82no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×C4×C61C2×SD1611yescharacteristic
141C2×C4×C61C2×Q1611yescharacteristic
142C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61#10312yes—
154C4×D61—22no—
155C4×D61—22no—
156C4×D61—22no—
157C4×D61—22no—
158C8×C61—42no—
159C8×C61—81no—
160#92418—42no—
161#92418—81no—
162#9191—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61#38611yescharacteristic
166C22×C61C2×#10311yescharacteristic
167C22×C61#41711yescharacteristic
168C2×D61—22no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61—42no—
172C2×D61—81no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—82no—
178C4×C61#45011yescharacteristic
179C4×C61#45111yescharacteristic
180C4×C61#48311yescharacteristic
181C4×C61#48411yescharacteristic
182Dic61—22no—
183Dic61—41no—
184Dic61—41no—
185Dic61—82no—
186#2905—611no—
187#2628—611no—
188#1548—611no—
189#1548D6111yescharacteristic, complemented
190#1458—611no—
191#1286—611no—
192#1286—611no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61#187211yescharacteristic
196C2×C61#190911yescharacteristic
197C2×C61#191411yescharacteristic
198D61—42no—
199C22×M16—1221no—
200C2×C4⋊Q8C2×D6111yescharacteristic
201C2×#117—611no—
202#431—611no—
203#431C2×D6111yescharacteristic
204#424—1222no—
205#424—1222no—
206#423—611no—
207#423—1221no—
208#423C2×D6111yescharacteristic
209#400—1221no—
210#399—1221no—
211#395—612no—
212#393—612no—
213#389—1221no—
214#382—611no—
215#380—1221no—
216#333—22no—
217#333—1222no—
218#324—1221no—
219#323—1221no—
220C61#919111yescharacteristic, complemented, Sylow
221C22×Q8C4×D6112yes—
222C23×C4—611no—
223C2×M16—21no—
224C2×M16—1222no—
225C2×M16—1221no—
226C2×M16—2442no—
227C2×M16—2442no—
228C22×C8—1222no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233#117—1221no—
234#117—1221no—
235#117—1221no—
236#117—1221no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—611no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1221no—
243C2×C22⋊C4—611no—
244C2×C22⋊C4—611no—
245C2×C22⋊C4—1222no—
246C2×C42—611no—
247C2×C42C22×D6111yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—612no—
251C4⋊C8—1224no—
252C4⋊C8—1221no—
253C4⋊C8—1221no—
254C4⋊C8C22⋊D6112yes—
255C22⋊C8—21no—
256C22⋊C8—1221no—
257C4×C8—21no—
258C4×C8—1221no—
259#95—612no—
260#95—1222no—
261#95—1222no—
262#95—1221no—
263C24—611no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—82no—
267C22×C4—22no—
268C22×C4—611no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1222no—
278C22×C4C2×C4×D6111yescharacteristic
279C22×C4D4×D6111yescharacteristic
280C22×C4D4×D6111yescharacteristic
281M16—41no—
282M16—41no—
283M16—2444no—
284M16—2441no—
285M16—2441no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—1222no—
289C2×C8—2444no—
290C2×C8—2441no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—1221no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C4⋊C4—4882no—
300C4⋊C4—2441no—
301C4⋊C4—2441no—
302C22⋊C4—1221no—
303C22⋊C4—1221no—
304C22⋊C4—1221no—
305C22⋊C4—1221no—
306C22⋊C4—4882no—
307C42—1221no—
308C42—1221no—
309C42#1175822211yescharacteristic
310C42D4×D6111yescharacteristic
311C42#1175822411yescharacteristic
312C42C2×C22⋊D6111yescharacteristic
313C23—612no—
314C23—1221no—
315C23—1221no—
316C23—2441no—
317C23—2441no—
318C23C22⋊C4×D6111yescharacteristic
319Q8—82no—
320Q8—82no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—612no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—1222no—
330C2×C4—1222no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4#1451973711yescharacteristic
341C2×C4C4×C22⋊D6111yescharacteristic
342C2×C4#1451974511yescharacteristic
343C2×C4#1451974811yescharacteristic
344C2×C4#1451975011yescharacteristic
345C2×C4#1451975911yescharacteristic
346C2×C4SD16×D6111yescharacteristic
347C2×C4Q16×D6111yescharacteristic
348C2×C4#1451980411yescharacteristic
349C2×C4#1451980911yescharacteristic
350C8—42no—
351C8—81no—
352C8—4881no—
353C8—2442no—
354C22—21no—
355C22—21no—
356C22—1222no—
357C22—4881no—
358C22—2442no—
359C22—2441no—
360C22—2441no—
361C22#1683963811yescharacteristic
362C22#103×D6111yescharacteristic
363C22#1683976211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—1222no—
370C4—4882no—
371C4—2441no—
372C4—2441no—
373C4#1683991111yescharacteristic
374C4#1683991611yescharacteristic
375C4#1684009111yescharacteristic
376C4#1684009611yescharacteristic
377C2—21no—
378C2—21no—
379C2—2442no—
380C2#2144612311yescharacteristic
381C2#2144633811yescharacteristic
382C2#2144636311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#386
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1548D61
hypocenterC61#9191
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. #21444500C2semidirect
5. #21443679C2semidirect
6. #21443678C2semidirect
7. C61×#1548C2semidirect
189. #1548D61semidirect
220. C61#9191semidirect

Maximal subgroups

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

subgroupindex
#214543812
#214516412
#214445002
#214441312
#214436792
#214436782
C61×#15482
#919161