Finite Group Databasebeta

← #25015255

Subgroups — #25015255

Order 11008.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19134587C211yesmaximal, characteristic
2C4×#13063240C211yesmaximal, characteristic
3#19132713C214yesmaximal
4C43×#2487C211yesmaximal, characteristic
5#14582923—21no—
6C2×#13063242C2212yes—
7C2×#13063240C2212yes—
8C4×C22⋊Dic43—21no—
9#14582668—21no—
10#14582666—21no—
11C2×C4×D4×C43C2211yescharacteristic
12C22×C4⋊C4×C43C411yescharacteristic, complemented
13C43×#400—21no—
14C43×#399—21no—
15C43×#389—21no—
16C43×#386C2212yes—
17C43×#385C412yescomplemented
18C43×#380—21no—
19C43×#380C411yescharacteristic, complemented
20C2×#10621877—22no—
21C2×C22⋊Dic43—22no—
22C2×C4×Dic43—41no—
23C4⋊(C43⋊C8)—41no—
24C4⋊(C43⋊C8)—41no—
25#13063242—24no—
26#13063240—22no—
27#13063240Q814yes—
28C43⋊#95—41no—
29C22×D4×C43C2311yescharacteristic
30C23×C4×C43C2×C411yescharacteristic
31C23×C4×C43C2×C412yes—
32C23×C4×C43D411yescharacteristic
33C4×D4×C43—41no—
34C4×D4×C43—41no—
35C2×C4⋊C4×C43—21no—
36C2×C4⋊C4×C43—21no—
37C2×C4⋊C4×C43—21no—
38C2×C4⋊C4×C43—22no—
39C2×C4⋊C4×C43—41no—
40C2×C4⋊C4×C43—41no—
41C2×C4⋊C4×C43D411yescharacteristic
42C2×C22⋊C4×C43—22no—
43C2×C22⋊C4×C43C2×C412yes—
44C2×C22⋊C4×C43C2×C411yescharacteristic
45C2×C22⋊C4×C43D411yescharacteristic
46C2×C42×C43D411yescharacteristic
47#95×C43—22no—
48#95×C43—21no—
49#95×C43—22no—
50C24×C43C22×C411yescharacteristic
51C24×C43C2×D411yescharacteristic
52C2×D4×C43—41no—
53C2×D4×C43—41no—
54C2×D4×C43C2×Q812yes—
55C2×D4×C43#4112yes—
56C22×C4×C43—21no—
57C22×C4×C43—21no—
58C22×C4×C43—21no—
59C22×C4×C43—21no—
60C22×C4×C43—22no—
61C22×C4×C43—22no—
62C22×C4×C43—42no—
63C22×C4×C43—42no—
64C22×C4×C43—42no—
65C22×C4×C43—82no—
66C22×C4×C43C22⋊C411yescharacteristic
67C22×C4×C43C22⋊C411yescharacteristic
68C22×C4×C43C22⋊C411yescharacteristic
69C22×C4×C43C22⋊C411yescharacteristic
70C22×C4×C43C2×D411yescharacteristic
71C22×Dic43—42no—
72C4⋊C4×C43—41no—
73C4⋊C4×C43—42no—
74C4⋊C4×C43—42no—
75C4⋊C4×C43—41no—
76C4⋊C4×C43—41no—
77C4⋊C4×C43—82no—
78C22⋊C4×C43—41no—
79C22⋊C4×C43—41no—
80C22⋊C4×C43—42no—
81C22⋊C4×C43—82no—
82C42×C43—21no—
83C42×C43—21no—
84C22⋊Dic43—24no—
85C22⋊Dic43—42no—
86C4×Dic43—82no—
87#10621877—42no—
88#10621877—42no—
89C2×C43⋊C8—42no—
90C23×C43—21no—
91C23×C43—21no—
92C23×C43—21no—
93C23×C43—21no—
94C23×C43—41no—
95C23×C43—41no—
96C23×C43—41no—
97C23×C43—41no—
98C23×C43C2×C22⋊C411yescharacteristic
99C23×C43C4×Q812yes—
100C23×C43C22⋊Q812yes—
101D4×C43—42no—
102D4×C43—42no—
103C2×C4×C43—21no—
104C2×C4×C43—21no—
105C2×C4×C43—22no—
106C2×C4×C43—21no—
107C2×C4×C43—21no—
108C2×C4×C43—21no—
109C2×C4×C43—42no—
110C2×C4×C43—42no—
111C2×C4×C43—42no—
112C2×C4×C43—41no—
113C2×C4×C43—41no—
114C2×C4×C43—41no—
115C2×C4×C43—42no—
116C2×C4×C43—42no—
117C2×C4×C43—42no—
118C2×C4×C43—42no—
119C2×C4×C43—42no—
120C2×C4×C43—42no—
121C2×C4×C43—81no—
122C2×C4×C43—84no—
123C2×C4×C43#9911yescharacteristic
124C2×C4×C43#9911yescharacteristic
125C2×C4×C43#12411yescharacteristic
126C2×C4×C43C4⋊Q811yescharacteristic
127C2×Dic43—44no—
128C2×Dic43—82no—
129C2×Dic43—82no—
130C43⋊C8—82no—
131#13295—431nomaximal, Sylow
132C22×C43—21no—
133C22×C43—21no—
134C22×C43—22no—
135C22×C43—22no—
136C22×C43—42no—
137C22×C43—42no—
138C22×C43—42no—
139C22×C43—41no—
140C22×C43—41no—
141C22×C43—41no—
142C22×C43—41no—
143C22×C43—81no—
144C22×C43#39111yescharacteristic
145C22×C43C2×#9911yescharacteristic
146C22×C43#41011yescharacteristic
147C4×C43—21no—
148C4×C43—21no—
149C4×C43—21no—
150C4×C43—41no—
151C4×C43—41no—
152C4×C43—41no—
153C4×C43—82no—
154C4×C43—82no—
155C4×C43#35212yes—
156Dic43—84no—
157#2487D4311yescharacteristic
158#1800—431no—
159C4×#99—431no—
160#1390—434no—
161C2×C43—21no—
162C2×C43—21no—
163C2×C43—42no—
164C2×C43—42no—
165C2×C43#199611yescharacteristic
166C2×C43C2×#35211yescharacteristic
167C2×C43#212211yescharacteristic
168C2×C4×D4C2×D4311yescharacteristic
169C22×C4⋊C4Dic4311yescharacteristic, complemented
170#423—861no—
171C2×#100—432no—
172C2×#99—432no—
173#400—21no—
174#399—21no—
175#389—21no—
176#386C2×D4312yes—
177#385Dic4312yescomplemented
178#380—21no—
179#380Dic4311yescharacteristic, complemented
180C4×C22⋊C4—861no—
181#343—861no—
182#342—861no—
183C43#1329511yescharacteristic, complemented, Sylow
184C22×D4C22×D4311yescharacteristic
185C23×C4C2×Dic4311yescharacteristic
186C23×C4C2×Dic4312yes—
187C23×C4C22⋊D4311yescharacteristic
188C2×M16—862no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C22⋊D4311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—862no—
200C2×C22⋊C4C2×Dic4312yes—
201C2×C22⋊C4C2×Dic4311yescharacteristic
202C2×C22⋊C4C22⋊D4311yescharacteristic
203C2×C42—1721no—
204C2×C42C22⋊D4311yescharacteristic
205C4⋊C8—1721no—
206C4⋊C8—1721no—
207#100—864no—
208#99—434no—
209#99—862no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—1721no—
214C24C22×Dic4311yescharacteristic
215C24C2×C22⋊D4311yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4Q8×D4312yes—
219C2×D4#1062190212yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—1722no—
231C22×C4C22⋊Dic4311yescharacteristic
232C22×C4C22⋊Dic4311yescharacteristic
233C22×C4C22⋊Dic4311yescharacteristic
234C22×C4C22⋊Dic4311yescharacteristic
235C22×C4C2×C22⋊D4311yescharacteristic
236M16—1722no—
237M16—1722no—
238C2×C8—1722no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—82no—
249C22⋊C4—864no—
250C22⋊C4—1722no—
251C42—21no—
252C42—21no—
253C42—3442no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic4311yescharacteristic
263C23Q8×Dic4312yes—
264C23#1306330112yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—84no—
287C2×C4—3442no—
288C2×C4—3442no—
289C2×C4—1724no—
290C2×C4#1306324011yescharacteristic
291C2×C4#1306324011yescharacteristic
292C2×C4#1306331411yescharacteristic
293C2×C4#1306332311yescharacteristic
294C8—3442no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1458280611yescharacteristic
308C22C2×#1306324011yescharacteristic
309C22#1458287311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4—3444no—
319C4#1458268912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#1913558111yescharacteristic
325C2C2×#1458268911yescharacteristic
326C2#1913624511yescharacteristic
subgroupG/H
centerC22C2×#13063240
commutatorC24×C43C22×C4
FrattiniC22×D4C22×D43
FittingC43×#2487C2
socleC22×C43C2×#99
prime radical, cosocleC22×D4×C43C23
hypercenter#2487D43
hypocenterC43#13295
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊C4×C43C4semidirect
17. C43×#385C4semidirect
19. C43×#380C4semidirect
169. C22×C4⋊C4Dic43semidirect
177. #385Dic43semidirect
179. #380Dic43semidirect
183. C43#13295semidirect

Maximal subgroups

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

subgroupindex
#191345872
C4×#130632402
#191327132
C43×#24872
#1329543