Finite Group Databasebeta

← #25262917

Subgroups — #25262917

Order 11008.

1962 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19137902C216yesmaximal, complemented
2#19137889C213yesmaximal, complemented
3#19137886C213yesmaximal, complemented
4#19137871C216yesmaximal
5#19137867C213yesmaximal
6#19137865C213yesmaximal
7C42×Dic86C213yesmaximal, complemented
8C4×#13063282C213yesmaximal
9C42×Q8×C43C211yesmaximal, characteristic, complemented
10C2×C4×Dic86C2213yes—
11C2×C4×Dic86C2216yes—
12C2×#13063282C2213yes—
13C2×#13063282C2213yes—
14C2×C42×D43C2211yescharacteristic
15#14582840C2216yes—
16#14582839C2213yes—
17#14582837C2213yes—
18#14582836C2211yescharacteristic
19#14582834C2213yes—
20#14582833C2216yes—
21#14582832C2216yes—
22#14582830C2216yes—
23#14582829C2216yes—
24#14582823C2212yes—
25#14582821C2216yes—
26#14582820C2213yes—
27#14582819C2216yes—
28#14582818C2213yes—
29#14582812C2213yes—
30#14582810C2211yescharacteristic
31C4×Dic43⋊C4C2216yes—
32C4×Dic43⋊C4C2216yes—
33C4×Dic43⋊C4C2216yes—
34C4×Dic43⋊C4C2216yes—
35C4×Dic43⋊C4C2216yes—
36C4×C4⋊Dic43C2213yes—
37C4×C4⋊Dic43C2216yes—
38C4×(C2×D43)⋊C4C2216yes—
39C4×(C2×D43)⋊C4C2216yes—
40#14582745C2213yes—
41#14582745C2213yes—
42C42×Dic43C2213yes—
43C2×C4×Q8×C43C2213yes—
44C4×C4⋊C4×C43C2213yes—
45C4×C4⋊C4×C43C2216yes—
46C43×C43C2213yes—
47C22×Dic86C2313yes—
48C22×C4×D43C2313yes—
49C4×Dic86—212no—
50C4×Dic86—224no—
51#13063282—212no—
52#13063282—212no—
53C2×C4⋊Dic43C2316yes—
54C2×C4⋊Dic43C2313yes—
55C2×Dic43⋊C4C2316yes—
56C2×Dic43⋊C4C2313yes—
57C2×Dic43⋊C4C2316yes—
58C2×Dic43⋊C4C2316yes—
59C2×Dic43⋊C4C2313yes—
60C2×Dic43⋊C4C2316yes—
61C2×(C2×D43)⋊C4C2313yes—
62C2×(C2×D43)⋊C4C2313yes—
63C2×(C2×D43)⋊C4C2316yes—
64C2×C4×Dic43C2313yes—
65C2×C4×Dic43C2316yes—
66C2×C4×Dic43C2316yes—
67C2×C4×Dic43C2313yes—
68C2×C4×Dic43C2313yes—
69C42×D43Q818yescomplemented
70C43⋊#95C2313yes—
71C43⋊#95C2316yes—
72C43⋊#95C2313yes—
73C43⋊#95C2316yes—
74C43⋊#95C2316yes—
75C43⋊#95C2316yes—
76C43⋊#95C2316yes—
77C43⋊#95C2316yes—
78C22×Q8×C43C2311yescharacteristic
79C4×Q8×C43—212no—
80C2×C4⋊C4×C43C2316yes—
81C2×C4⋊C4×C43C2313yes—
82C2×C42×C43C2316yes—
83C2×C42×C43C2313yes—
84C2×C42×C43C2316yes—
85C2×C42×C43C2313yes—
86C2×C42×C43C2311yescharacteristic
87#95×C43C2313yes—
88#95×C43C2311yescharacteristic
89#95×C43C2312yes—
90C23×D43C2411yescharacteristic
91C2×Q8×C43—26no—
92C22×C4×C43C2416yes—
93C22×C4×C43C2413yes—
94C22×C4×C43C2413yes—
95C22×C4×C43C2413yes—
96C22×Dic43C2413yes—
97C22×Dic43C2413yes—
98C22×Dic43C2413yes—
99C22×Dic43C2416yes—
100C2×C4×D43—26no—
101C2×C4×D43—26no—
102C2×C4×D43C2×Q8112yes—
103C2×Dic86—26no—
104C2×Dic86—212no—
105C4⋊C4×C43—212no—
106C4⋊C4×C43—26no—
107C42×C43—212no—
108C42×C43—26no—
109C42×C43—212no—
110C42×C43C2×Q814yes—
111C42×C43#41112yes—
112(C2×D43)⋊C4—26no—
113(C2×D43)⋊C4—26no—
114(C2×D43)⋊C4—212no—
115C4⋊Dic43—26no—
116C4⋊Dic43—212no—
117Dic43⋊C4—26no—
118Dic43⋊C4—26no—
119Dic43⋊C4—212no—
120Dic43⋊C4—212no—
121Dic43⋊C4—212no—
122Dic43⋊C4—212no—
123C4×Dic43—26no—
124C4×Dic43—212no—
125C4×Dic43—212no—
126C4×Dic43C2×Q8112yes—
127C4×Dic43#41112yes—
128C23×C43C2511yescharacteristic
129C22×D43—23no—
130C22×D43—23no—
131C22×D43C22×Q812yes—
132Q8×C43—28no—
133C2×C4×C43—26no—
134C2×C4×C43—26no—
135C2×C4×C43—26no—
136C2×C4×C43—26no—
137C2×C4×C43—26no—
138C2×C4×C43C22×Q816yes—
139C2×C4×C43C2×#4116yes—
140C2×C4×C43C2×#4116yes—
141C2×C4×C43C2×#4116yes—
142C2×C4×C43C2×#4116yes—
143C2×Dic43—26no—
144C2×Dic43—26no—
145C2×Dic43—23no—
146C2×Dic43—23no—
147C2×Dic43—26no—
148C2×Dic43—26no—
149C2×Dic43—23no—
150C2×Dic43—23no—
151C2×Dic43C22×Q816yes—
152C2×Dic43C2×#41112yes—
153C4×D43—224no—
154Dic86—224no—
155#34552—431nomaximal, Sylow
156C22×C43C23×Q811yescharacteristic
157C22×C43C22×#4113yes—
158C22×C43C22×#4113yes—
159C2×D43—26no—
160C2×D43—43no—
161C2×D43—41no—
162C4×C43—26no—
163C4×C43—26no—
164C4×C43—46no—
165C4×C43#533112yes—
166Dic43—26no—
167Dic43—212no—
168Dic43—43no—
169Dic43—43no—
170#2402—436no—
171#2399—433no—
172#2399—433no—
173#2396—433no—
174#2396—436no—
175#2396—433no—
176C42×Q8—433no—
177C42×Q8D4311yescharacteristic, complemented
178C4×#117—433no—
179C2×C43C2×#53313yes—
180C2×C43#344413yes—
181C2×C43#344411yescharacteristic
182D43—42no—
183C2×C4×Q8—436no—
184C2×C4×Q8—433no—
185C2×C4×Q8C2×D4313yes—
186C2×#117—433no—
187C2×#117—433no—
188C22×C42—431no—
189#400—433no—
190#400—433no—
191#400—436no—
192#399—433no—
193#399—431no—
194#398—436no—
195#398—436no—
196#398—436no—
197#398—436no—
198#396—436no—
199#396—432no—
200#395—433no—
201#395—436no—
202#395—433no—
203#393—431no—
204#393—433no—
205C4×C4⋊C4—433no—
206C4×C4⋊C4—436no—
207C4×C4⋊C4—436no—
208C4×C4⋊C4—436no—
209C4×C4⋊C4—436no—
210C4×C4⋊C4—436no—
211C4×C4⋊C4—436no—
212C4×C4⋊C4C2×D4313yes—
213C4×C4⋊C4C2×D4316yes—
214C4×C22⋊C4—436no—
215C4×C22⋊C4—436no—
216#376—433no—
217#376—433no—
218C43—433no—
219C43C2×D4313yes—
220C43#3455211yescharacteristic, complemented, Sylow
221C22×Q8—433no—
222C22×Q8C22×D4311yescharacteristic
223C23×C4—433no—
224C4×Q8—212no—
225C4×Q8—8624no—
226C4×Q8—8612no—
227#117—8612no—
228#117—8612no—
229C2×C4⋊C4—436no—
230C2×C4⋊C4—433no—
231C2×C4⋊C4—433no—
232C2×C4⋊C4—436no—
233C2×C4⋊C4—436no—
234C2×C4⋊C4—436no—
235C2×C4⋊C4—436no—
236C2×C4⋊C4—433no—
237C2×C4⋊C4C22×D4316yes—
238C2×C4⋊C4C22×D4313yes—
239C2×C22⋊C4—436no—
240C2×C22⋊C4—433no—
241C2×C22⋊C4—433no—
242C2×C42—433no—
243C2×C42—433no—
244C2×C42—438no—
245C2×C42—433no—
246C2×C42—436no—
247C2×C42—436no—
248C2×C42C22×D4313yes—
249C2×C42C22×D4316yes—
250C2×C42C22×D4316yes—
251C2×C42C22×D4313yes—
252C2×C42C22×D4311yescharacteristic
253#95—436no—
254#95—436no—
255#95—436no—
256#95—436no—
257#95—436no—
258#95—433no—
259#95—433no—
260#95—436no—
261#95C22×D4313yes—
262#95C22×D4311yescharacteristic
263#95C22×D4312yes—
264C24—431no—
265C2×Q8—26no—
266C2×Q8—8612no—
267C2×Q8—866no—
268C22×C4—433no—
269C22×C4—436no—
270C22×C4—4312no—
271C22×C4—433no—
272C22×C4—433no—
273C22×C4—866no—
274C22×C4—866no—
275C22×C4C23×D4316yes—
276C22×C4C23×D4313yes—
277C22×C4C23×D4313yes—
278C22×C4C23×D4313yes—
279C4⋊C4—212no—
280C4⋊C4—26no—
281C4⋊C4—8612no—
282C4⋊C4—8612no—
283C4⋊C4—8612no—
284C4⋊C4—8612no—
285C4⋊C4—866no—
286C4⋊C4—866no—
287C4⋊C4—8612no—
288C4⋊C4—866no—
289C22⋊C4—8612no—
290C22⋊C4—866no—
291C22⋊C4—866no—
292C42—26no—
293C42—212no—
294C42—212no—
295C42—4312no—
296C42—4312no—
297C42—8612no—
298C42—866no—
299C42—8612no—
300C42#10621898112yes—
301C42Q8×D4314yes—
302C23—432no—
303C23—863no—
304C23—863no—
305C23C24×D4311yescharacteristic
306Q8—28no—
307Q8—8624no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—436no—
314C2×C4—4312no—
315C2×C4—866no—
316C2×C4—863no—
317C2×C4—866no—
318C2×C4—863no—
319C2×C4—8624no—
320C2×C4—866no—
321C2×C4—863no—
322C2×C4—863no—
323C2×C4—866no—
324C2×C4C2×Q8×D4316yes—
325C2×C4#41×D4316yes—
326C2×C4C2×#1062189816yes—
327C2×C4C2×#1062189816yes—
328C2×C4C2×#1062189816yes—
329C22—866no—
330C22—1723no—
331C22—1721no—
332C22C22×Q8×D4311yescharacteristic
333C22C2×#41×D4313yes—
334C22C22×#1062189813yes—
335C4—26no—
336C4—26no—
337C4—46no—
338C4—866no—
339C4—8612no—
340C4—1723no—
341C4—1723no—
342C4#14583453112yes—
343C2—1722no—
344C2C2×#1458345313yes—
345C2#1914788813yes—
346C2#1914789111yescharacteristic
subgroupG/H
centerC2×C42C22×D43
commutator, socle, cosocleC23×C43C25
FrattiniC23C24×D43
FittingC42×Q8×C43C2
prime radicalC23×D43C24
hypercenterC42×Q8D43
hypocenterC43#34552
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
1. #19137902C2semidirect
2. #19137889C2semidirect
3. #19137886C2semidirect
7. C42×Dic86C2semidirect
9. C42×Q8×C43C2semidirect
69. C42×D43Q8semidirect
177. C42×Q8D43semidirect
220. C43#34552semidirect

Maximal subgroups

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

subgroupindex
#191379022
#191378892
#191378862
#191378712
#191378672
#191378652
C42×Dic862
C4×#130632822
C42×Q8×C432
#3455243