Finite Group Databasebeta

← #25257049

Subgroups — #25257049

Order 11008.

2310 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19137604C218yesmaximal
2#19137552C214yesmaximal, complemented
3Q8×C4⋊Dic43C214yesmaximal, complemented
4#19137337C214yesmaximal, complemented
5#19137218C212yesmaximal
6C4⋊Q8×Dic43C214yesmaximal, complemented
7C2×#14582769C211yesmaximal, characteristic
8#19136951C212yesmaximal
9C4×#13063284C211yesmaximal, characteristic
10C2×C43×#533C211yesmaximal, characteristic
11C2×Q8×Dic43C2218yes—
12C2×#13063284C2211yescharacteristic
13C2×#13063284C2211yescharacteristic
14C22×C4⋊Dic43C2212yes—
15C22×C4⋊Dic43C2214yes—
16#14582806C2218yes—
17#14582798C22116yes—
18#14582798C2218yes—
19#14582798C2218yes—
20#14582785C22116yes—
21#14582774C2218yes—
22#14582769C2214yes—
23#14582769C2212yes—
24#14582769C2212yes—
25#14582769C2214yes—
26#14582765C2214yes—
27C4⋊C4×Dic43C2218yes—
28C4⋊C4×Dic43C2218yes—
29C4×C4⋊Dic43C2212yes—
30C4×C4⋊Dic43C2218yes—
31C4×C4⋊Dic43C2212yes—
32C22⋊C4×Dic43C2218yes—
33C4×C22⋊Dic43C2214yes—
34#14582745C2212yes—
35C42×Dic43C2212yes—
36C43×#533C4116yescomplemented
37C2×C4⋊Q8×C43C2212yes—
38C2×#125×C43C2212yes—
39C2×C22⋊Q8×C43C2214yes—
40C2×C4×Q8×C43C2214yes—
41C2×#117×C43C2212yes—
42C22×C42×C43C2211yescharacteristic
43C23×Dic43C2314yes—
44Q8×Dic43—416no—
45#13063284—24no—
46#13063284Q818yescomplemented
47C2×C4⋊Dic43—28no—
48C2×C4⋊Dic43—48no—
49C2×C4⋊Dic43C2314yes—
50C2×C4⋊Dic43C2318yes—
51C2×C4⋊Dic43C2314yes—
52C2×C4⋊Dic43C2311yescharacteristic
53C2×C4⋊Dic43C2311yescharacteristic
54C2×C4⋊Dic43C2318yes—
55C2×C4⋊Dic43C23116yes—
56C2×C4⋊Dic43C2312yes—
57C2×C22⋊Dic43C2312yes—
58C2×C22⋊Dic43C2312yes—
59C2×C4×Dic43C2312yes—
60C2×C4×Dic43C2314yes—
61C2×C4×Dic43C2314yes—
62C2×C4×Dic43C23116yes—
63C2×C4×Dic43C2312yes—
64C43⋊#95C2314yes—
65C43⋊#95C23116yes—
66C43⋊#95C23116yes—
67C43⋊#95C2314yes—
68C22×Q8×C43C2314yes—
69C23×C4×C43C2312yes—
70C23×C4×C43C2311yescharacteristic
71C4⋊Q8×C43C2×C4116yescomplemented
72#125×C43C2×C4116yescomplemented
73C22⋊Q8×C43C2×C4132yes—
74C4×Q8×C43C2×C4132yescomplemented
75#117×C43C2×C4116yes—
76C2×C4⋊C4×C43C2318yes—
77C2×C4⋊C4×C43C2314yes—
78C2×C4⋊C4×C43C2314yes—
79C2×C22⋊C4×C43C2314yes—
80C2×C42×C43C2×C418yes—
81C2×C42×C43C2312yes—
82C2×C42×C43C2311yescharacteristic
83C2×C42×C43C2311yescharacteristic
84C2×C42×C43C2314yes—
85C24×C43C2411yescharacteristic
86C2×Q8×C43—28no—
87C2×Q8×C43—28no—
88C2×Q8×C43C22×C4116yes—
89C22×C4×C43—24no—
90C22×C4×C43—24no—
91C22×C4×C43C22×C418yes—
92C22×C4×C43C22×C414yes—
93C22×C4×C43C2×Q814yes—
94C22×C4×C43#4114yes—
95C22×C4×C43C2412yes—
96C22×C4×C43C2418yes—
97C22×C4×C43C2411yescharacteristic
98C22×C4×C43C2411yescharacteristic
99C22×C4×C43C2412yes—
100C22×Dic43—28no—
101C22×Dic43—48no—
102C22×Dic43C2412yes—
103C22×Dic43C2418yes—
104C22×Dic43C2412yes—
105C22×Dic43C2414yes—
106C4⋊C4×C43C22×C4116yes—
107C4⋊C4×C43C22×C4132yes—
108C4⋊C4×C43C22×C4116yes—
109C22⋊C4×C43C22×C4116yes—
110C42×C43C22×C418yes—
111C42×C43C22×C414yes—
112C42×C43C22×C414yes—
113C42×C43C22×C4116yes—
114C22⋊Dic43—24no—
115C22⋊Dic43C2×Q818yes—
116C4⋊Dic43—28no—
117C4⋊Dic43—28no—
118C4⋊Dic43—22no—
119C4⋊Dic43—22no—
120C4⋊Dic43—216no—
121C4⋊Dic43—48no—
122C4⋊Dic43—416no—
123C4⋊Dic43C2×Q818yes—
124C4×Dic43—24no—
125C4×Dic43—28no—
126C4×Dic43—28no—
127C4×Dic43—416no—
128C4×Dic43C2×Q818yes—
129C23×C43—22no—
130C23×C43—22no—
131C23×C43C23×C412yes—
132C23×C43C22×Q812yes—
133C23×C43C2×#4112yes—
134C23×C43C2511yescharacteristic
135Q8×C43—232no—
136C2×C4×C43—28no—
137C2×C4×C43—22no—
138C2×C4×C43—22no—
139C2×C4×C43—28no—
140C2×C4×C43—216no—
141C2×C4×C43C4×Q8116yescomplemented
142C2×C4×C43C23×C414yes—
143C2×C4×C43C23×C4116yes—
144C2×C4×C43C23×C412yes—
145C2×C4×C43C23×C412yes—
146C2×C4×C43C23×C414yes—
147C2×C4×C43C22×Q812yes—
148C2×C4×C43C22×Q812yes—
149C2×C4×C43C2×#4112yes—
150C2×C4×C43C2×#4112yes—
151C2×C4×C43H(5,𝔽2)14yes—
152C2×C4×C432−1+414yes—
153C2×Dic43—24no—
154C2×Dic43—24no—
155C2×Dic43—22no—
156C2×Dic43—22no—
157C2×Dic43—28no—
158C2×Dic43—24no—
159C2×Dic43—216no—
160C2×Dic43—24no—
161C2×Dic43—88no—
162C2×Dic43C22×Q818yes—
163#34214—431nomaximal, Sylow
164C22×C43—24no—
165C22×C43—24no—
166C22×C43—22no—
167C22×C43—22no—
168C22×C43C2×C4×Q814yes—
169C22×C43C24×C411yescharacteristic
170C22×C43C23×Q811yescharacteristic
171C22×C43C22×#4111yescharacteristic
172C22×C43C2×H(5,𝔽2)12yes—
173C22×C43C2×2−1+412yes—
174C4×C43—24no—
175C4×C43—216no—
176C4×C43C2×C4×Q814yes—
177C4×C43C2×C4×Q814yes—
178C4×C43#52018yes—
179Dic43—28no—
180Dic43—42no—
181Dic43—48no—
182Dic43—42no—
183C2×#533D4311yescharacteristic
184#2360—438no—
185#2355—434no—
186Q8×C4⋊C4—434no—
187#2330—434no—
188#2317—432no—
189C4×C4⋊Q8—434no—
190C2×#384—431no—
191#2274—432no—
192C4×#117—431no—
193C2×C43—24no—
194C2×C43C22×C4×Q811yescharacteristic
195C2×C43C2×#52012yes—
196C2×C43#350312yes—
197C2×C43#350811yescharacteristic
198C2×C43#350911yescharacteristic
199#533Dic43116yescomplemented
200C2×C4⋊Q8C2×D4312yes—
201C2×#125C2×D4312yes—
202C2×C22⋊Q8C2×D4314yes—
203C2×C4×Q8—438no—
204C2×C4×Q8C2×D4314yes—
205C2×#117—431no—
206C2×#117—431no—
207C2×#117C2×D4312yes—
208C22×C4⋊C4—432no—
209C22×C4⋊C4—434no—
210C22×C42C2×D4311yescharacteristic
211#391—438no—
212#389—438no—
213#389—438no—
214#389—4316no—
215#387—4316no—
216#385—438no—
217#384—432no—
218#384—434no—
219#384—432no—
220#384—434no—
221#382—434no—
222C4×C4⋊C4—438no—
223C4×C4⋊C4—438no—
224C4×C4⋊C4—432no—
225C4×C4⋊C4—438no—
226C4×C4⋊C4—432no—
227C4×C22⋊C4—434no—
228C4×C22⋊C4—438no—
229#376—432no—
230C43—432no—
231C43#3421411yescharacteristic, complemented, Sylow
232C22×Q8C22×D4314yes—
233C23×C4—434no—
234C23×C4C22×D4312yes—
235C23×C4C22×D4311yescharacteristic
236C4⋊Q8C2×Dic43116yescomplemented
237#125C2×Dic43116yescomplemented
238C22⋊Q8C2×Dic43132yes—
239C4×Q8—17216no—
240C4×Q8C2×Dic43132yescomplemented
241#117—438no—
242#117—864no—
243#117C2×Dic43116yes—
244C2×C4⋊C4—432no—
245C2×C4⋊C4—438no—
246C2×C4⋊C4—434no—
247C2×C4⋊C4—434no—
248C2×C4⋊C4—431no—
249C2×C4⋊C4—431no—
250C2×C4⋊C4—438no—
251C2×C4⋊C4—4316no—
252C2×C4⋊C4—868no—
253C2×C4⋊C4—1728no—
254C2×C4⋊C4C22×D4314yes—
255C2×C4⋊C4C22×D4318yes—
256C2×C4⋊C4C22×D4314yes—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—432no—
259C2×C22⋊C4C22×D4314yes—
260C2×C42—432no—
261C2×C42—432no—
262C2×C42—434no—
263C2×C42—434no—
264C2×C42—4316no—
265C2×C42C2×Dic4318yes—
266C2×C42C22×D4314yes—
267C2×C42C22×D4312yes—
268C2×C42C22×D4311yescharacteristic
269C2×C42C22×D4311yescharacteristic
270#95—434no—
271#95—434no—
272#95—4316no—
273#95—4316no—
274C24C23×D4311yescharacteristic
275C2×Q8—28no—
276C2×Q8—28no—
277C2×Q8C22×Dic43116yes—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—434no—
281C22×C4—432no—
282C22×C4—438no—
283C22×C4—432no—
284C22×C4—868no—
285C22×C4—1728no—
286C22×C4Q8×D4314yes—
287C22×C4#1062190214yes—
288C22×C4C22×Dic4318yes—
289C22×C4C22×Dic4314yes—
290C22×C4C23×D4318yes—
291C22×C4C23×D4312yes—
292C22×C4C23×D4311yescharacteristic
293C22×C4C23×D4311yescharacteristic
294C22×C4C23×D4312yes—
295C4⋊C4—438no—
296C4⋊C4—868no—
297C4⋊C4—868no—
298C4⋊C4—862no—
299C4⋊C4—862no—
300C4⋊C4—8616no—
301C4⋊C4—1728no—
302C4⋊C4—17216no—
303C4⋊C4C22×Dic43116yes—
304C4⋊C4C22×Dic43132yes—
305C4⋊C4C22×Dic43116yes—
306C22⋊C4—438no—
307C22⋊C4—864no—
308C22⋊C4C22×Dic43116yes—
309C42—438no—
310C42—864no—
311C42—868no—
312C42—868no—
313C42—17216no—
314C42C22×Dic4318yes—
315C42C22×Dic4314yes—
316C42C22×Dic4314yes—
317C42C22×Dic43116yes—
318C23—22no—
319C23—22no—
320C23C23×Dic4312yes—
321C23C2×Q8×D4312yes—
322C23C2×#1062190212yes—
323C23C24×D4311yescharacteristic
324Q8—232no—
325C2×C4—28no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—216no—
330C2×C4—438no—
331C2×C4—864no—
332C2×C4—864no—
333C2×C4—862no—
334C2×C4—862no—
335C2×C4—868no—
336C2×C4—864no—
337C2×C4—8616no—
338C2×C4—864no—
339C2×C4—3448no—
340C2×C4Q8×Dic43116yescomplemented
341C2×C4C23×Dic4314yes—
342C2×C4C23×Dic43116yes—
343C2×C4C23×Dic4312yes—
344C2×C4C23×Dic4312yes—
345C2×C4C23×Dic4314yes—
346C2×C4C2×Q8×D4312yes—
347C2×C4C2×Q8×D4312yes—
348C2×C4C2×#1062190212yes—
349C2×C4C2×#1062190212yes—
350C2×C4#1306337114yes—
351C2×C4#1306337214yes—
352C22—24no—
353C22—24no—
354C22—22no—
355C22—22no—
356C22C2×Q8×Dic4314yes—
357C22C24×Dic4311yescharacteristic
358C22C22×Q8×D4311yescharacteristic
359C22C22×#1062190211yescharacteristic
360C22C2×#1306337112yes—
361C22C2×#1306337212yes—
362C4—24no—
363C4—216no—
364C4—868no—
365C4—1728no—
366C4—1722no—
367C4—1722no—
368C4C2×Q8×Dic4314yes—
369C4C2×Q8×Dic4314yes—
370C4#1458338218yes—
371C2—24no—
372C2C22×Q8×Dic4311yescharacteristic
373C2C2×#1458338212yes—
374C2#1914840712yes—
375C2#1914847211yescharacteristic
376C2#1914848211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D43
commutatorC22×C43C24×C4
FittingC2×C43×#533C2
socle, cosocleC23×C43C25
prime radicalC24×C43C24
hypercenterC2×#533D43
hypocenterC43#34214
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #19137552C2semidirect
3. Q8×C4⋊Dic43C2semidirect
4. #19137337C2semidirect
6. C4⋊Q8×Dic43C2semidirect
36. C43×#533C4semidirect
46. #13063284Q8semidirect
71. C4⋊Q8×C43C2×C4semidirect
72. #125×C43C2×C4semidirect
74. C4×Q8×C43C2×C4semidirect
141. C2×C4×C43C4×Q8semidirect
199. #533Dic43semidirect
231. C43#34214semidirect
236. C4⋊Q8C2×Dic43semidirect
237. #125C2×Dic43semidirect
240. C4×Q8C2×Dic43semidirect
340. C2×C4Q8×Dic43semidirect

Maximal subgroups

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

subgroupindex
#191376042
#191375522
Q8×C4⋊Dic432
#191373372
#191372182
C4⋊Q8×Dic432
C2×#145827692
#191369512
C4×#130632842
C2×C43×#5332
#3421443