Finite Group Databasebeta

← #24988371

Subgroups — #24988371

Order 11008.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#14583614C211yesmaximal, characteristic
2#19133281C212yesmaximal, complemented
3#19133019C212yesmaximal, complemented
4#19132916C211yesmaximal, characteristic, complemented
5C43×#1800C211yesmaximal, characteristic, complemented
6#14583614—24no—
7#14583614—24no—
8C2×#13063325C2211yescharacteristic
9C2×#13063311—22no—
10C2×#13063299C412yes—
11C2×#13063296—21no—
12C2×C4×D172C2211yescharacteristic
13C2×#13063284C412yes—
14C2×#13063283—21no—
15#14582952C2212yes—
16#14582644—24no—
17#14582626—24no—
18#14582623—22no—
19C2×C4×D4×C43C2211yescharacteristic
20C43×#423C2212yes—
21C2×#100×C43—21no—
22C43×#343—21no—
23C22×Dic86—21no—
24C22×D172C2×C411yescharacteristic
25C22×C22⋊D43—22no—
26C22×C4×D43—21no—
27#13063325—22no—
28#13063325D414yes—
29#13063311—216no—
30#13063299—24no—
31#13063299—24no—
32#13063296—28no—
33C4×D172—22no—
34C4×D172D414yes—
35#13063284—24no—
36#13063284—24no—
37#13063283—28no—
38C2×C4⋊Dic43—21no—
39C2×C4⋊Dic43C2×C411yescharacteristic
40C2×C4⋊Dic43C2×C412yes—
41C2×C22⋊Dic43—22no—
42C2×(C2×D43)⋊C4—22no—
43C2×(C2×D43)⋊C4—21no—
44C2×C4×Dic43—22no—
45C4⋊(C43⋊C8)—22no—
46C22⋊(C43⋊C8)—22no—
47C4×C43⋊C8—22no—
48C22×D4×C43D411yescharacteristic
49C23×C4×C43C2×C412yes—
50C2×M16×C43—22no—
51C4×D4×C43—22no—
52C4×D4×C43—22no—
53C2×C4⋊C4×C43D411yescharacteristic
54C2×C22⋊C4×C43D412yes—
55C2×C42×C43C2311yescharacteristic
56C4⋊C8×C43—22no—
57C4⋊C8×C43—22no—
58#100×C43—42no—
59C24×C43C22⋊C412yes—
60C23×D43—21no—
61C2×D4×C43—22no—
62C2×D4×C43—22no—
63C2×D4×C43—21no—
64C2×D4×C43—21no—
65C22×C4×C43—22no—
66C22×C4×C43—22no—
67C22×C4×C43—44no—
68C22×C4×C43C22⋊C412yes—
69C22×C4×C43C22×C411yescharacteristic
70C22×C4×C43C2×D411yescharacteristic
71C22×C4×C43C2×D411yescharacteristic
72C2×C22⋊D43—28no—
73C2×C22⋊D43—84no—
74C22×Dic43—21no—
75C22×Dic43—22no—
76C2×D172—21no—
77C2×D172—42no—
78C2×D172C22⋊C412yes—
79C2×C4×D43—24no—
80C2×C4×D43—42no—
81C2×C4×D43—41no—
82C2×C4×D43—41no—
83C2×Dic86—24no—
84C2×Dic86—82no—
85M16×C43—42no—
86M16×C43—42no—
87C2×C8×C43—42no—
88C4⋊C4×C43—22no—
89C22⋊C4×C43—24no—
90C42×C43C2×D412yes—
91C42×C43C2×D412yes—
92C22⋊Dic43—28no—
93(C2×D43)⋊C4—28no—
94(C2×D43)⋊C4—24no—
95C4⋊Dic43—21no—
96C4⋊Dic43—22no—
97C4⋊Dic43—24no—
98C4⋊Dic43C22⋊C412yes—
99C4⋊Dic43C22⋊C414yes—
100C4×Dic43—28no—
101C2×C43⋊C8—42no—
102C23×C43—22no—
103C23×C43—22no—
104C23×C43—22no—
105C23×C43—42no—
106C23×C43—42no—
107C23×C43C2×C22⋊C411yescharacteristic
108C22×D43—22no—
109C22×D43—42no—
110C22×D43—42no—
111C22×D43—41no—
112C22×D43—41no—
113D4×C43—42no—
114D4×C43—42no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—41no—
118C2×C4×C43—44no—
119C2×C4×C43—44no—
120C2×C4×C43—42no—
121C2×C4×C43C2×C22⋊C411yescharacteristic
122C2×C4×C43C2×C22⋊C411yescharacteristic
123C2×C4×C43#12012yes—
124C2×C4×C43#12012yes—
125C2×C4×C43#13512yes—
126C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C43#13711yescharacteristic
128C22⋊D43—88no—
129C2×Dic43—22no—
130C2×Dic43—24no—
131C2×Dic43—44no—
132C2×Dic43—41no—
133C2×Dic43—41no—
134D172—24no—
135D172—42no—
136C4×D43—44no—
137C4×D43—44no—
138Dic86—84no—
139C8×C43—82no—
140C43⋊C8—82no—
141#8397—431nomaximal, Sylow
142C22×C43—21no—
143C22×C43—21no—
144C22×C43—42no—
145C22×C43—42no—
146C22×C43—42no—
147C22×C43—82no—
148C22×C43—82no—
149C22×C43#37911yescharacteristic
150C22×C43#41611yescharacteristic
151C22×C43#41711yescharacteristic
152C2×D43—42no—
153C2×D43—42no—
154C2×D43—42no—
155C2×D43—82no—
156C2×D43—82no—
157C4×C43—22no—
158C4×C43—21no—
159C4×C43—42no—
160C4×C43#41312yes—
161C4×C43#44912yes—
162C4×C43#45212yes—
163Dic43—44no—
164Dic43—42no—
165C2×#555—431no—
166#1800D4311yescharacteristic, complemented
167#1549—432no—
168#1486—432no—
169#1460—431no—
170C2×C43—21no—
171C2×C43—21no—
172C2×C43—82no—
173C2×C43#178511yescharacteristic
174C2×C43#178811yescharacteristic
175C2×C43#179511yescharacteristic
176D43—82no—
177#555—864no—
178#555—864no—
179C2×C4⋊Q8—431no—
180C2×#124—862no—
181C2×#121—432no—
182C2×#121—861no—
183C2×C4×D4—431no—
184C2×C4×D4C2×D4311yescharacteristic
185C2×#117—432no—
186C2×#117—861no—
187#431—432no—
188#423C2×D4312yes—
189C2×#100—21no—
190#343—21no—
191#332—864no—
192D4⋊C8—864no—
193#323—862no—
194C43#839711yescharacteristic, complemented, Sylow
195C22×Q8—861no—
196C22×D4—431no—
197C22×D4—862no—
198C22×D4C22⋊D4311yescharacteristic
199C23×C4—861no—
200C23×C4C4×D4312yes—
201C2×M16—22no—
202C4⋊Q8—434no—
203C4⋊Q8—862no—
204#124—8616no—
205#121—864no—
206#121—864no—
207#121—868no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—434no—
211C4×D4—862no—
212#117—868no—
213#117—864no—
214#117—864no—
215C2×C4⋊C4—431no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—861no—
218C2×C4⋊C4C22⋊D4311yescharacteristic
219C2×C22⋊C4—862no—
220C2×C22⋊C4—862no—
221C2×C22⋊C4—861no—
222C2×C22⋊C4D17212yes—
223C2×C42—862no—
224C2×C42C22×D4311yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—862no—
228#100—42no—
229C22⋊C8—862no—
230C4×C8—862no—
231C24—861no—
232C24(C2×D43)⋊C412yes—
233C2×Q8—864no—
234C2×Q8—3442no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—432no—
240C2×D4—868no—
241C2×D4—861no—
242C2×D4—3444no—
243C2×D4—1722no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—44no—
247C22×C4—861no—
248C22×C4—862no—
249C22×C4—864no—
250C22×C4—1722no—
251C22×C4—1721no—
252C22×C4—1721no—
253C22×C4(C2×D43)⋊C412yes—
254C22×C4C2×C4×D4311yescharacteristic
255C22×C4C2×D17211yescharacteristic
256C22×C4C2×C22⋊D4311yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—1722no—
261C4⋊C4—22no—
262C4⋊C4—432no—
263C4⋊C4—434no—
264C4⋊C4—861no—
265C4⋊C4—862no—
266C4⋊C4—864no—
267C22⋊C4—24no—
268C22⋊C4—868no—
269C22⋊C4—868no—
270C22⋊C4—864no—
271C42—868no—
272C42D4×D4312yes—
273C42D4×D4312yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—862no—
280C23—1722no—
281C23—1722no—
282C23—1721no—
283C23—1721no—
284C23C2×(C2×D43)⋊C411yescharacteristic
285Q8—3444no—
286D4—42no—
287D4—42no—
288D4—864no—
289D4—3448no—
290D4—1722no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—862no—
298C2×C4—864no—
299C2×C4—1724no—
300C2×C4—1721no—
301C2×C4—1721no—
302C2×C4—1724no—
303C2×C4—1724no—
304C2×C4C22⋊C4×D4311yescharacteristic
305C2×C4C22⋊C4×D4311yescharacteristic
306C2×C4#1306329312yes—
307C2×C4#1306329512yes—
308C2×C4#1306334512yes—
309C2×C4#1306335111yescharacteristic
310C2×C4#1306335611yescharacteristic
311C8—82no—
312C8—3442no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—3442no—
321C22—3442no—
322C22—1722no—
323C22—1722no—
324C22—1722no—
325C22#1458275411yescharacteristic
326C22#1458289311yescharacteristic
327C22#1458289811yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—1722no—
332C4—1724no—
333C4#1458288012yes—
334C4#1458304212yes—
335C4#1458305912yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—3442no—
340C2#1913452011yescharacteristic
341C2#1913453311yescharacteristic
342C2#1913456611yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1800C2
socleC22×C43#379
prime radicalC2×#14583614C2
cosocleC2×C42×C43C23
hypercenter#1800D43
hypocenterC43#8397
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #19133281C2semidirect
3. #19133019C2semidirect
4. #19132916C2semidirect
5. C43×#1800C2semidirect
166. #1800D43semidirect
194. C43#8397semidirect

Maximal subgroups

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

subgroupindex
C2×#145836142
#191332812
#191330192
#191329162
C43×#18002
#839743