Finite Group Databasebeta

← #29575473

Subgroups — #29575473

Order 15616.

488 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839860C211yesmaximal, characteristic
2#21444214C211yesmaximal, characteristic, complemented
3#21444211C211yesmaximal, characteristic
4#21444139C211yesmaximal, characteristic
5#21443663C211yesmaximal, characteristic, complemented
6#21443649C211yesmaximal, characteristic, complemented
7C61×#2983C211yesmaximal, characteristic, complemented
8C22×#11758201C411yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16839860—22no—
11#16839860—22no—
12#16839815C2211yescharacteristic
13#16839815C2211yescharacteristic
14C2×#14519710C411yescharacteristic
15C2×#14519701C411yescharacteristic
16C2×#14519697C411yescharacteristic
17C2×#14519688C2211yescharacteristic
18Q8⋊#92418—22no—
19#16839491—22no—
20#16839485—22no—
21C2×C4×Q8×C61C2211yescharacteristic
22C61×#446—22no—
23C61×#446—22no—
24C61×#432—22no—
25C61×#431C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C4×D61—21no—
29C2×#11758201—22no—
30C2×#11758201—22no—
31C2×#11758201—22no—
32C22×#92418C2×C411yescharacteristic
33C22×#92418C2×C411yescharacteristic
34C4×D244—21no—
35C4×D244—21no—
36C4×D244—21no—
37C4×D244—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758198—21no—
40#14519710—22no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43#14519701—22no—
44#14519697—22no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49C4×#92418—21no—
50C4×#92418—21no—
51C22×Q8×C61C2×C411yescharacteristic
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×Q8×C61D412yes—
55C4×Q8×C61D412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C42×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C4⋊C8×C61—24no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64C22⋊C8×C61D412yes—
65#97×C61—24no—
66C4×C8×C61—21no—
67C4×C8×C61D412yes—
68C23×D61—21no—
69C2×Q8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61C22⋊C412yes—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×Dic61—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244—21no—
86C2×D244M1612yes—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—82no—
90C2×C8×C61—22no—
91C2×C8×C61—22no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61C22⋊C412yes—
96C42×C61—22no—
97C42×C61—22no—
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61#4111yescharacteristic
102#11758201—28no—
103#11758198—41no—
104#11758198—41no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C2×#92418—21no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418—21no—
111C2×#92418—21no—
112C2×#92418—41no—
113C2×#92418—41no—
114C2×#92418M1612yes—
115C23×C61C4×D411yescharacteristic
116C22×D61—22no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—81no—
120C22×D61—81no—
121Q8×C61—22no—
122Q8×C61—22no—
123Q8×C61—41no—
124Q8×C61—41no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—81no—
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12311yescharacteristic
138C2×C4×C61C2×M1611yescharacteristic
139C2×C4×C61C2×M1611yescharacteristic
140C2×C4×C61#13111yescharacteristic
141C2×C4×C61#13111yescharacteristic
142C2×Dic61—41no—
143C2×Dic61—81no—
144D244—22no—
145D244—22no—
146D244—41no—
147D244—41no—
148C4×D61—82no—
149C4×D61—82no—
150C8×C61—42no—
151C8×C61—42no—
152#92418—22no—
153#92418—41no—
154#92418—81no—
155#92418—81no—
156#9094—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61#38611yescharacteristic
160C22×C61#43511yescharacteristic
161C22×C61#43611yescharacteristic
162C2×D61—42no—
163C2×D61—82no—
164C2×D61—81no—
165C2×D61—81no—
166C2×D61—82no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—42no—
172C4×C61—42no—
173C4×C61#40711yescharacteristic
174C4×C61#40811yescharacteristic
175C4×C61#43511yescharacteristic
176C4×C61#43611yescharacteristic
177Dic61—81no—
178#2983D6111yescharacteristic, complemented
179C2×#440—611no—
180#1479—611no—
181#1478—611no—
182#1460—611no—
183#1281—611no—
184#1275—611no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#187011yescharacteristic
188C2×C61#190311yescharacteristic
189C2×C61#190511yescharacteristic
190D61—82no—
191C22×SD16—611no—
192C2×C4×Q8C2×D6111yescharacteristic
193C2×C4×D4—611no—
194#446—22no—
195#446—22no—
196#440—1222no—
197#440—1222no—
198#432—22no—
199#431—611no—
200#431—611no—
201#431C2×D6111yescharacteristic
202#431C2×D6111yescharacteristic
203C2×#107—611no—
204C2×#103—611no—
205C2×#102—611no—
206C2×#97—611no—
207Q8⋊C8—1222no—
208D4⋊C8—1222no—
209#323—1222no—
210C61#909411yescharacteristic, complemented, Sylow
211C22×Q8C4×D6111yescharacteristic
212C22×D4—611no—
213C23×C4—1221no—
214C2×SD16—1222no—
215C2×SD16—1222no—
216C2×SD16—1222no—
217C22×C8—611no—
218C22×C8—611no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8D24412yes—
222C4×Q8C22⋊D6112yes—
223C4×D4—1221no—
224C4×D4—1221no—
225C4×D4—1221no—
226C4×D4—1221no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4C4×D6111yescharacteristic
230C2×C22⋊C4—1221no—
231C2×C42—21no—
232C2×C42C22×D6111yescharacteristic
233#107—1222no—
234C4⋊C8—24no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—1221no—
238C4⋊C8—1221no—
239#103—1222no—
240#102—1222no—
241C22⋊C8—21no—
242C22⋊C8—1221no—
243C22⋊C8—1221no—
244C22⋊C8C22⋊D6112yes—
245#97—24no—
246#97—1221no—
247#97—1221no—
248C4×C8—21no—
249C4×C8—1221no—
250C4×C8—1221no—
251C4×C8C22⋊D6112yes—
252C24—1221no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8#1175819812yes—
259C2×D4—612no—
260C2×D4—1221no—
261C2×D4—1221no—
262C2×D4—1221no—
263C2×D4—1221no—
264C2×D4—1221no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—1221no—
268C22×C4—4882no—
269C22×C4—2441no—
270C22×C4—2441no—
271C22×C4C2×C4×D6111yescharacteristic
272C22×C4#1175822211yescharacteristic
273C22×C4C2×C22⋊D6111yescharacteristic
274SD16—1228no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—612no—
278C2×C8—1221no—
279C2×C8—1221no—
280C2×C8—1221no—
281C2×C8—1221no—
282C2×C8—1221no—
283C2×C8—2441no—
284C2×C8—2441no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4#1175819812yes—
291C22⋊C4—2441no—
292C22⋊C4—2441no—
293C42—22no—
294C42—22no—
295C42C2×D24411yescharacteristic
296C42#1175822211yescharacteristic
297C42C2×C22⋊D6111yescharacteristic
298C42C2×C22⋊D6111yescharacteristic
299C23—1222no—
300C23—4881no—
301C23—4881no—
302C23—2441no—
303C23—2441no—
304C23C4×C22⋊D6111yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—41no—
308Q8—41no—
309D4—1222no—
310D4—1222no—
311D4—2441no—
312D4—2441no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—4881no—
323C2×C4—2441no—
324C2×C4C2×#1175819811yescharacteristic
325C2×C4C4×C22⋊D6111yescharacteristic
326C2×C4#1451974611yescharacteristic
327C2×C4#1451975111yescharacteristic
328C2×C4#1451975111yescharacteristic
329C2×C4#1451976011yescharacteristic
330C2×C4M16×D6111yescharacteristic
331C2×C4M16×D6111yescharacteristic
332C2×C4#1451978511yescharacteristic
333C2×C4#1451978511yescharacteristic
334C8—42no—
335C8—42no—
336C8—1222no—
337C8—4881no—
338C8—4881no—
339C8—2441no—
340C22—21no—
341C22—21no—
342C22—4882no—
343C22—4882no—
344C22—4881no—
345C22—4881no—
346C22—2442no—
347C22#1683964111yescharacteristic
348C22#1683983111yescharacteristic
349C22#1683983811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—4881no—
357C4#1683972511yescharacteristic
358C4#1683973011yescharacteristic
359C4#1683983111yescharacteristic
360C4#1683983811yescharacteristic
361C2—21no—
362C2—21no—
363C2—4882no—
364C2#2144611611yescharacteristic
365C2#2144629911yescharacteristic
366C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2983C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#2983D61
hypocenterC61#9094
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #21444214C2semidirect
5. #21443663C2semidirect
6. #21443649C2semidirect
7. C61×#2983C2semidirect
178. #2983D61semidirect
210. C61#9094semidirect

Maximal subgroups

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

subgroupindex
C2×#168398602
#214442142
#214442112
#214441392
#214436632
#214436492
C61×#29832
#909461