Finite Group Databasebeta

← #25012839

Subgroups — #25012839

Order 11008.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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#19134523C211yesmaximal, characteristic
2#19134513C211yesmaximal, characteristic
3#19132713C214yesmaximal, complemented
4C43×#2297C211yesmaximal, characteristic
5C2×#13063244—21no—
6C2×#13063242C2212yes—
7C2×#13063240—21no—
8C2×#13063240C2212yes—
9#14582863—21no—
10#14582762—21no—
11C22×#41×C43C2211yescharacteristic
12C2×#117×C43C411yescharacteristic, complemented
13C22×C42×C43C411yescharacteristic, complemented
14C43×#391—21no—
15C43×#390—21no—
16C43×#386C412yescomplemented
17C43×#386C2212yes—
18C4×C22⋊C4×C43—21no—
19C4×C22⋊C4×C43—21no—
20C2×#10621877—22no—
21C2×C22⋊Dic43—22no—
22#13063244—42no—
23#13063242—24no—
24#13063240—24no—
25#13063240—42no—
26C22⋊(C43⋊C8)—42no—
27C43⋊#95—42no—
28C2×#41×C43—24no—
29C2×#41×C43—42no—
30C2×#41×C43—42no—
31C22×Q8×C43D411yescharacteristic
32C22×D4×C43C2×C411yescharacteristic
33C22×D4×C43D411yescharacteristic
34C22×D4×C43C2311yescharacteristic
35C23×C4×C43C2×C411yescharacteristic
36C23×C4×C43C2×C412yes—
37C23×C4×C43D411yescharacteristic
38C23×C4×C43D411yescharacteristic
39#117×C43—42no—
40C2×C4⋊C4×C43—21no—
41C2×C22⋊C4×C43—22no—
42C2×C22⋊C4×C43—22no—
43C2×C22⋊C4×C43C2×C412yes—
44C2×C42×C43—21no—
45C2×C42×C43—21no—
46C2×C42×C43—21no—
47C2×C42×C43—42no—
48#95×C43—22no—
49#95×C43—22no—
50C24×C43C22⋊C411yescharacteristic
51C24×C43C22×C411yescharacteristic
52C24×C43C2×D411yescharacteristic
53#41×C43—416no—
54C2×Q8×C43—22no—
55C2×Q8×C43—22no—
56C2×Q8×C43—41no—
57C2×D4×C43—22no—
58C2×D4×C43—22no—
59C2×D4×C43—22no—
60C2×D4×C43—41no—
61C2×D4×C43—41no—
62C2×D4×C43—41no—
63C2×D4×C43—41no—
64C2×D4×C43—41no—
65C2×D4×C43#4114yes—
66C22×C4×C43—22no—
67C22×C4×C43—22no—
68C22×C4×C43—22no—
69C22×C4×C43—22no—
70C22×C4×C43—22no—
71C22×C4×C43—22no—
72C22×C4×C43—22no—
73C22×C4×C43—22no—
74C22×C4×C43—22no—
75C22×C4×C43—41no—
76C22×C4×C43—41no—
77C22×C4×C43—42no—
78C22×C4×C43—44no—
79C22×C4×C43C22⋊C411yescharacteristic
80C22×C4×C43C22⋊C411yescharacteristic
81C22×C4×C43C22⋊C411yescharacteristic
82C22×C4×C43C2×D411yescharacteristic
83C22×Dic43—42no—
84C4⋊C4×C43—42no—
85C22⋊C4×C43—42no—
86C22⋊C4×C43—44no—
87C22⋊C4×C43—44no—
88C42×C43—42no—
89C42×C43—42no—
90C42×C43—42no—
91C22⋊Dic43—42no—
92C22⋊Dic43—42no—
93#10621877—42no—
94#10621877—42no—
95C2×C43⋊C8—42no—
96C23×C43—21no—
97C23×C43—21no—
98C23×C43—21no—
99C23×C43—21no—
100C23×C43—21no—
101C23×C43—21no—
102C23×C43—21no—
103C23×C43—42no—
104C23×C43—42no—
105C23×C43—42no—
106C23×C43C2×C22⋊C411yescharacteristic
107C23×C43#11712yes—
108C23×C43#12312yes—
109Q8×C43—44no—
110D4×C43—44no—
111D4×C43—44no—
112D4×C43—44no—
113C2×C4×C43—28no—
114C2×C4×C43—21no—
115C2×C4×C43—21no—
116C2×C4×C43—21no—
117C2×C4×C43—21no—
118C2×C4×C43—21no—
119C2×C4×C43—21no—
120C2×C4×C43—24no—
121C2×C4×C43—21no—
122C2×C4×C43—21no—
123C2×C4×C43—21no—
124C2×C4×C43—44no—
125C2×C4×C43—44no—
126C2×C4×C43—42no—
127C2×C4×C43—44no—
128C2×C4×C43—42no—
129C2×C4×C43—44no—
130C2×C4×C43—42no—
131C2×C4×C43—42no—
132C2×C4×C43—42no—
133C2×C4×C43#9912yes—
134C2×C4×C43#9912yes—
135C2×C4×C43#12312yes—
136C2×Dic43—82no—
137C2×Dic43—84no—
138C43⋊C8—82no—
139#12853—431nomaximal, Sylow
140C22×C43—22no—
141C22×C43—22no—
142C22×C43—22no—
143C22×C43—42no—
144C22×C43—42no—
145C22×C43—42no—
146C22×C43—42no—
147C22×C43—42no—
148C22×C43—42no—
149C22×C43—41no—
150C22×C43—42no—
151C22×C43—81no—
152C22×C43—81no—
153C22×C43#38111yescharacteristic
154C22×C43C2×#9911yescharacteristic
155C22×C43C2×#9911yescharacteristic
156C4×C43—22no—
157C4×C43—22no—
158C4×C43—22no—
159C4×C43—41no—
160C4×C43—44no—
161C4×C43—82no—
162Dic43—162no—
163#2297D4311yescharacteristic
164#1785—431no—
165#1782—431no—
166#1390—434no—
167C2×C43—21no—
168C2×C43—21no—
169C2×C43—42no—
170C2×C43—42no—
171C2×C43—81no—
172C2×C43#181911yescharacteristic
173C2×C43#212012yes—
174C22×#41C2×D4311yescharacteristic
175C2×#117Dic4311yescharacteristic, complemented
176C22×C42Dic4311yescharacteristic, complemented
177C2×#101—861no—
178C2×#100—432no—
179C2×#99—432no—
180C2×#99—861no—
181#407—861no—
182#391—21no—
183#390—21no—
184#386Dic4312yescomplemented
185#386C2×D4312yes—
186#381—861no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C43#1285311yescharacteristic, complemented, Sylow
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8C22⋊D4311yescharacteristic
194C22×D4C2×Dic4311yescharacteristic
195C22×D4C22⋊D4311yescharacteristic
196C22×D4C22×D4311yescharacteristic
197C23×C4C2×Dic4311yescharacteristic
198C23×C4C2×Dic4312yes—
199C23×C4C22⋊D4311yescharacteristic
200C23×C4C22⋊D4311yescharacteristic
201C2×M16—862no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—862no—
207C2×C22⋊C4C2×Dic4312yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—42no—
212#101—1722no—
213#100—864no—
214#99—864no—
215#99—1722no—
216C22⋊C8—1722no—
217#95—22no—
218#95—22no—
219#95—1722no—
220C24C22⋊Dic4311yescharacteristic
221C24C22×Dic4311yescharacteristic
222C24C2×C22⋊D4311yescharacteristic
223#41—416no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4#1062189814yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—1722no—
250C22×C4C22⋊Dic4311yescharacteristic
251C22×C4C22⋊Dic4311yescharacteristic
252C22×C4C22⋊Dic4311yescharacteristic
253C22×C4C2×C22⋊D4311yescharacteristic
254M16—1722no—
255M16—1722no—
256C2×C8—1722no—
257C4⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—1722no—
262C22⋊C4—1722no—
263C42—42no—
264C42—42no—
265C42—42no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C2×C22⋊Dic4311yescharacteristic
277C23#1306328412yes—
278C23#1306330812yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—3442no—
304C2×C4—3444no—
305C2×C4#1306324012yes—
306C2×C4#1306324012yes—
307C2×C4#1306330812yes—
308C8—3442no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22#1458276211yescharacteristic
323C22C2×#1306324011yescharacteristic
324C22C2×#1306324011yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—44no—
330C4—82no—
331C4—6882no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2#1913466911yescharacteristic
338C2#1913623612yes—
subgroupG/H
centerC22C2×#13063240
commutatorC24×C43C22×C4
FrattiniC22×D4C22×D43
FittingC43×#2297C2
socleC22×C43C2×#99
prime radicalC22×#41×C43C22
cosocleC22×D4×C43C23
hypercenter#2297D43
hypocenterC43#12853
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #19132713C2semidirect
12. C2×#117×C43C4semidirect
13. C22×C42×C43C4semidirect
16. C43×#386C4semidirect
175. C2×#117Dic43semidirect
176. C22×C42Dic43semidirect
184. #386Dic43semidirect
189. C43#12853semidirect

Maximal subgroups

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

subgroupindex
#191345232
#191345132
#191327132
C43×#22972
#1285343