Finite Group Databasebeta

← #24994552

Subgroups — #24994552

Order 11008.

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

normal subgroups only

This group has 682 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#19143636C211yesmaximal, characteristic
2#19134519C212yesmaximal, complemented
3#19132933C211yesmaximal, characteristic
4#19132501C212yesmaximal, complemented
5C43×#2409C211yesmaximal, characteristic
6#14582977—24no—
7#14582952—21no—
8C2×#13063244—24no—
9#14582863—21no—
10#14582863C2212yes—
11#14582863C2212yes—
12#14582625—21no—
13#14582623—21no—
14C23×Q8×C43C2211yescharacteristic
15C2×C22⋊Q8×C43—21no—
16C2×C4×D4×C43C2211yescharacteristic
17C43×#395—21no—
18C43×#393—21no—
19C43×#391—21no—
20C43×#386C412yes—
21C43×#385C412yes—
22C43×#380C2211yescharacteristic
23C2×#10621877—22no—
24C2×#10621877—22no—
25C2×#10621877D414yes—
26C4⋊(C43⋊C8)—22no—
27#13063244—48no—
28C22⋊(C43⋊C8)—22no—
29C22⋊(C43⋊C8)—22no—
30C22⋊(C43⋊C8)—22no—
31C22⋊(C43⋊C8)—22no—
32C22⋊(C43⋊C8)—22no—
33C4×C43⋊C8—22no—
34C22×Q8×C43—21no—
35C22×Q8×C43—22no—
36C22×Q8×C43—21no—
37C22×Q8×C43—22no—
38C22×Q8×C43—44no—
39C22×D4×C43C2×C411yescharacteristic
40C23×C4×C43C2×C412yes—
41C23×C4×C43D411yescharacteristic
42C23×C4×C43C2311yescharacteristic
43C22⋊Q8×C43—44no—
44C4×D4×C43—22no—
45C4×D4×C43—22no—
46C2×C4⋊C4×C43—22no—
47C2×C4⋊C4×C43—21no—
48C2×C4⋊C4×C43D411yescharacteristic
49C2×C22⋊C4×C43—22no—
50C2×C22⋊C4×C43C2×C411yescharacteristic
51C2×C22⋊C4×C43C2×C412yes—
52C2×C22⋊C4×C43D411yescharacteristic
53C2×C42×C43D411yescharacteristic
54#95×C43—22no—
55#95×C43—22no—
56#95×C43—21no—
57C24×C43C22⋊C411yescharacteristic
58C24×C43C22×C411yescharacteristic
59C2×Q8×C43—24no—
60C2×Q8×C43—28no—
61C2×Q8×C43—44no—
62C2×Q8×C43—42no—
63C2×Q8×C43—44no—
64C2×Q8×C43—48no—
65C2×Q8×C43—41no—
66C2×Q8×C43—42no—
67C2×Q8×C43—41no—
68C2×D4×C43—22no—
69C2×D4×C43—22no—
70C2×D4×C43—41no—
71C22×C4×C43—21no—
72C22×C4×C43—21no—
73C22×C4×C43—22no—
74C22×C4×C43—22no—
75C22×C4×C43—24no—
76C22×C4×C43—44no—
77C22×C4×C43—42no—
78C22×C4×C43—42no—
79C22×C4×C43C22⋊C411yescharacteristic
80C22×C4×C43C22⋊C411yescharacteristic
81C22×C4×C43C22⋊C411yescharacteristic
82C22×C4×C43C2×D411yescharacteristic
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43C2×D412yes—
85C22×C4×C43#4112yes—
86C4⋊C4×C43—22no—
87C4⋊C4×C43—42no—
88C4⋊C4×C43—44no—
89C22⋊C4×C43—22no—
90C22⋊C4×C43—22no—
91C22⋊C4×C43—42no—
92C22⋊C4×C43—44no—
93C42×C43—21no—
94C42×C43—21no—
95#10621877—44no—
96#10621877—44no—
97#10621877—82no—
98C2×C43⋊C8—24no—
99C2×C43⋊C8—42no—
100C23×C43—21no—
101C23×C43—21no—
102C23×C43—21no—
103C23×C43—21no—
104C23×C43—21no—
105C23×C43—41no—
106C23×C43—41no—
107C23×C43—41no—
108C23×C43—41no—
109C23×C43C2×C22⋊C411yescharacteristic
110C23×C43C4×D412yes—
111Q8×C43—48no—
112Q8×C43—44no—
113Q8×C43—44no—
114D4×C43—44no—
115C2×C4×C43—21no—
116C2×C4×C43—21no—
117C2×C4×C43—24no—
118C2×C4×C43—22no—
119C2×C4×C43—24no—
120C2×C4×C43—28no—
121C2×C4×C43—21no—
122C2×C4×C43—21no—
123C2×C4×C43—41no—
124C2×C4×C43—42no—
125C2×C4×C43—42no—
126C2×C4×C43—44no—
127C2×C4×C43—42no—
128C2×C4×C43—42no—
129C2×C4×C43—41no—
130C2×C4×C43—41no—
131C2×C4×C43—42no—
132C2×C4×C43—42no—
133C2×C4×C43—42no—
134C2×C4×C43—81no—
135C2×C4×C43#10112yescomplemented
136C2×C4×C43#12011yescharacteristic
137C2×C4×C43#12112yes—
138C2×C4×C43#12311yescharacteristic
139C43⋊C8—44no—
140C43⋊C8—82no—
141#9506—431nomaximal, Sylow
142C22×C43—21no—
143C22×C43—21no—
144C22×C43—22no—
145C22×C43—22no—
146C22×C43—22no—
147C22×C43—42no—
148C22×C43—42no—
149C22×C43—42no—
150C22×C43—41no—
151C22×C43—41no—
152C22×C43—41no—
153C22×C43—81no—
154C22×C43#38611yescharacteristic
155C22×C43C2×#10111yescharacteristic
156C22×C43#41311yescharacteristic
157C4×C43—22no—
158C4×C43—21no—
159C4×C43—22no—
160C4×C43—44no—
161C4×C43—41no—
162C4×C43—41no—
163C4×C43—82no—
164C4×C43B(4,𝔽2)12yes—
165#2975—431no—
166#2409D4311yescharacteristic
167#1784—432no—
168#1465—431no—
169#1305—432no—
170C2×C43—21no—
171C2×C43—21no—
172C2×C43—42no—
173C2×C43—81no—
174C2×C43#188911yescharacteristic
175C2×C43#190111yescharacteristic
176C2×C43#191511yescharacteristic
177C23×Q8C2×D4311yescharacteristic
178C2×C22⋊Q8—21no—
179C2×C4×D4C2×D4311yescharacteristic
180#436—864no—
181#431—861no—
182C2×#101—864no—
183#407—432no—
184#407—432no—
185#407—861no—
186#395—21no—
187#393—21no—
188#391—21no—
189#386Dic4312yes—
190#385Dic4312yes—
191#380C2×D4311yescharacteristic
192#324—861no—
193#323—861no—
194C43#950611yescharacteristic, complemented, Sylow
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—44no—
200C22×D4C2×Dic4311yescharacteristic
201C23×C4C2×Dic4312yes—
202C23×C4C22⋊D4311yescharacteristic
203C23×C4C22×D4311yescharacteristic
204C2×M16—434no—
205C2×M16—862no—
206C2×M16—862no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22⋊D4311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic4311yescharacteristic
215C2×C22⋊C4C2×Dic4312yes—
216C2×C22⋊C4C22⋊D4311yescharacteristic
217C2×C42C22⋊D4311yescharacteristic
218C4⋊C8—862no—
219#101—1728no—
220C22⋊C8—862no—
221C22⋊C8—862no—
222C22⋊C8—862no—
223C22⋊C8—862no—
224C22⋊C8—862no—
225C4×C8—862no—
226#95—22no—
227#95—22no—
228#95—21no—
229C24C22⋊Dic4311yescharacteristic
230C24C22×Dic4311yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4C22⋊Dic4311yescharacteristic
252C22×C4C22⋊Dic4311yescharacteristic
253C22×C4C22⋊Dic4311yescharacteristic
254C22×C4D4×D4312yes—
255C22×C4#1062190012yes—
256C22×C4C2×C22⋊D4311yescharacteristic
257C22×C4C2×C22⋊D4311yescharacteristic
258M16—3442no—
259M16—1724no—
260M16—1724no—
261C2×C8—864no—
262C2×C8—1722no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×C22⋊Dic4311yescharacteristic
282C23D4×Dic4312yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4#1306324412yescomplemented
308C2×C4#1306329311yescharacteristic
309C2×C4#1306329712yes—
310C2×C4#1306330611yescharacteristic
311C8—3442no—
312C8—1724no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1458277911yescharacteristic
326C22C2×#1306324411yescharacteristic
327C22#1458288311yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—44no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1458308512yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#1913500411yescharacteristic
341C2#1913506411yescharacteristic
342C2#1913514811yescharacteristic
subgroupG/H
centerC22#14582779
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#2409C2
socleC22×C43#386
prime radicalC22×D4×C43C2×C4
cosocleC23×C4×C43C23
hypercenter#2409D43
hypocenterC43#9506
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #19134519C2semidirect
4. #19132501C2semidirect
135. C2×C4×C43#101semidirect
194. C43#9506semidirect
307. C2×C4#13063244semidirect

Maximal subgroups

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

subgroupindex
#191436362
#191345192
#191329332
#191325012
C43×#24092
#950643