Finite Group Databasebeta

← #25098436

Subgroups — #25098436

Order 11008.

1866 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583666C211yesmaximal, characteristic, complemented
2C22×#13063242C212yesmaximal, complemented
3#19143040C211yesmaximal, characteristic, complemented
4#19135059C218yesmaximal, complemented
5#19134334C212yesmaximal, complemented
6C2×C43×#535C211yesmaximal, characteristic
7#14583666—28no—
8C2×#13063332—24no—
9C22×#10621877C2212yes—
10C22×#10621877C2212yes—
11C22×#10621877C2214yescomplemented
12C2×#13063242—28no—
13C2×#13063242C2218yes—
14#14582866—24no—
15#14582863C2214yescomplemented
16#14582668C2218yescomplemented
17C22×#41×C43C2211yescharacteristic
18C23×D4×C43C2211yescharacteristic
19C43×#535—24no—
20C43×#535—24no—
21C2×#127×C43C412yes—
22C2×#124×C43C412yes—
23C2×#121×C43C2214yes—
24C2×#120×C43C414yes—
25C2×#117×C43C2211yescharacteristic
26#13063332—232no—
27C2×#10621877—22no—
28C2×#10621877—24no—
29C2×#10621877—24no—
30C2×#10621877—216no—
31C2×#10621877—22no—
32C2×#10621877—24no—
33C2×#10621877—24no—
34C2×#10621877D418yescomplemented
35C2×#10621877C2314yes—
36C2×#10621877C2314yes—
37C22×C43⋊C8—22no—
38C22×C43⋊C8—24no—
39C22×C43⋊C8—22no—
40#13063242—216no—
41C22⋊(C43⋊C8)—28no—
42C25×C43C2×C412yes—
43C2×#41×C43—24no—
44C2×#41×C43—24no—
45C2×#41×C43D414yes—
46C2×#41×C43D414yes—
47C22×Q8×C43C2×C411yescharacteristic
48C22×D4×C43—22no—
49C22×D4×C43—22no—
50C22×D4×C43—24no—
51C22×D4×C43—24no—
52C22×D4×C43C2×C412yes—
53C22×D4×C43C2×C412yes—
54C22×D4×C43C2×C411yescharacteristic
55C22×D4×C43C2×C418yes—
56C22×D4×C43C2312yes—
57C23×C4×C43C2312yes—
58C23×C4×C43C2311yescharacteristic
59#127×C43—28no—
60#124×C43—28no—
61#121×C43—28no—
62#121×C43—28no—
63#120×C43—216no—
64#117×C43—22no—
65#117×C43—22no—
66C2×C4⋊C4×C43C2312yes—
67C2×C22⋊C4×C43C2×C412yes—
68C2×C22⋊C4×C43C2×C418yes—
69C2×C42×C43C2×C412yes—
70C24×C43—24no—
71C24×C43—28no—
72C24×C43—84no—
73C24×C43C22×C414yes—
74C24×C43C22×C412yes—
75C24×C43C22×C411yescharacteristic
76#41×C43—28no—
77#41×C43—28no—
78#41×C43—28no—
79#41×C43—28no—
80C2×Q8×C43—24no—
81C2×Q8×C43C22⋊C414yes—
82C2×D4×C43—24no—
83C2×D4×C43—24no—
84C2×D4×C43—28no—
85C2×D4×C43—22no—
86C2×D4×C43—22no—
87C2×D4×C43—24no—
88C2×D4×C43—24no—
89C2×D4×C43—216no—
90C2×D4×C43—28no—
91C2×D4×C43—44no—
92C2×D4×C43—44no—
93C2×D4×C43—44no—
94C2×D4×C43—44no—
95C2×D4×C43—88no—
96C2×D4×C43C22⋊C418yes—
97C2×D4×C43C22⋊C414yes—
98C22×C4×C43—24no—
99C22×C4×C43—24no—
100C22×C4×C43C22×C412yes—
101C22×C4×C43C22×C414yes—
102C22×C4×C43C22×C411yescharacteristic
103C22×C4×C43C2×D414yes—
104C22×C4×C43C2×D414yes—
105C22×C4×C43C2×D412yes—
106C22×C4×C43C2×D412yes—
107C22×C4×C43C2×D414yes—
108C22×C4×C43#4114yes—
109C22×C4×C43C2411yescharacteristic
110C4⋊C4×C43—22no—
111C4⋊C4×C43—22no—
112C22⋊C4×C43—216no—
113C22⋊C4×C43—24no—
114C42×C43—24no—
115#10621877—216no—
116#10621877—28no—
117#10621877—24no—
118#10621877—24no—
119#10621877—24no—
120#10621877—44no—
121#10621877C2×D418yes—
122C2×C43⋊C8—216no—
123C2×C43⋊C8—28no—
124C2×C43⋊C8—24no—
125C2×C43⋊C8—24no—
126C2×C43⋊C8—44no—
127C23×C43—24no—
128C23×C43—24no—
129C23×C43—28no—
130C23×C43—42no—
131C23×C43—44no—
132C23×C43—48no—
133C23×C43—48no—
134C23×C43—88no—
135C23×C43—816no—
136C23×C43C2×C22⋊C414yes—
137C23×C43C2×C22⋊C412yes—
138C23×C43C4×D414yes—
139C23×C43C23×C411yescharacteristic
140Q8×C43—28no—
141D4×C43—216no—
142D4×C43—28no—
143D4×C43—42no—
144D4×C43—44no—
145D4×C43—42no—
146D4×C43—48no—
147D4×C43—816no—
148C2×C4×C43—24no—
149C2×C4×C43—24no—
150C2×C4×C43—28no—
151C2×C4×C43—24no—
152C2×C4×C43—24no—
153C2×C4×C43—44no—
154C2×C4×C43C2×C22⋊C414yes—
155C2×C4×C43C2×C22⋊C412yes—
156C2×C4×C43C4×D414yes—
157C2×C4×C43#12014yes—
158C2×C4×C43#12114yes—
159C2×C4×C43#12114yes—
160C2×C4×C43#12314yes—
161C2×C4×C43C22×D412yes—
162C2×C4×C43C22×D411yescharacteristic
163C2×C4×C43C22×D411yescharacteristic
164C2×C4×C43C2×#4112yes—
165C43⋊C8—28no—
166C43⋊C8—44no—
167#20990—431nomaximal, Sylow
168C22×C43—22no—
169C22×C43—24no—
170C22×C43—28no—
171C22×C43—24no—
172C22×C43—44no—
173C22×C43—48no—
174C22×C43—44no—
175C22×C43—84no—
176C22×C43—88no—
177C22×C43—816no—
178C22×C43#38614yes—
179C22×C43C22×C22⋊C411yescharacteristic
180C22×C43C2×C4×D412yes—
181C4×C43—24no—
182C4×C43—44no—
183C4×C43#38614yes—
184C4×C43C2×#12011yescharacteristic
185C4×C43C2×#12112yes—
186C4×C43C2×#12311yescharacteristic
187C2×#568—431no—
188C2×#535D4311yescharacteristic
189C22×#100—432no—
190#2879—431no—
191#1900—438no—
192#1742—432no—
193C2×C43—22no—
194C2×C43—24no—
195C2×C43—44no—
196C2×C43—84no—
197C2×C43C2×#38611yescharacteristic
198C2×C43#288812yes—
199C22×#41C2×D4311yescharacteristic
200C23×D4C2×D4311yescharacteristic
201#568—868no—
202C2×#131—864no—
203C22×M16—432no—
204C22×M16—432no—
205C22×M16—434no—
206#535—24no—
207#535—24no—
208C2×#127Dic4312yes—
209C2×#124Dic4312yes—
210C2×#121C2×D4314yes—
211C2×#120Dic4314yes—
212C2×#117C2×D4311yescharacteristic
213C2×#100—438no—
214C2×#100—868no—
215#408—864no—
216#407—434no—
217#343—438no—
218C43#2099011yescharacteristic, complemented, Sylow
219C25C2×Dic4312yes—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41C22⋊D4314yes—
223C2×#41C22⋊D4314yes—
224C22×Q8C2×Dic4311yescharacteristic
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C2×Dic4312yes—
230C22×D4C2×Dic4312yes—
231C22×D4C2×Dic4311yescharacteristic
232C22×D4C2×Dic4318yes—
233C22×D4C22×D4312yes—
234C23×C4C22×D4312yes—
235C23×C4C22×D4311yescharacteristic
236#131—8632no—
237C2×M16—438no—
238C2×M16—434no—
239C2×M16—434no—
240C2×M16—8616no—
241C2×M16—864no—
242C2×M16—864no—
243C2×M16—862no—
244C2×M16—862no—
245C2×M16—864no—
246C2×M16—864no—
247C22×C8—864no—
248C22×C8—862no—
249C22×C8—862no—
250#127—28no—
251#124—28no—
252#121—28no—
253#121—28no—
254#120—216no—
255#117—22no—
256#117—22no—
257C2×C4⋊C4C22×D4312yes—
258C2×C22⋊C4C2×Dic4312yes—
259C2×C22⋊C4C2×Dic4318yes—
260C2×C42C2×Dic4312yes—
261#100—8616no—
262C22⋊C8—868no—
263C24—24no—
264C24—28no—
265C24—84no—
266C24C22×Dic4314yes—
267C24C22×Dic4312yes—
268C24C22×Dic4311yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8C22⋊Dic4314yes—
275C2×D4—28no—
276C2×D4—216no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—88no—
289C2×D4C22⋊Dic4318yes—
290C2×D4C22⋊Dic4314yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4D4×D4314yes—
294C22×C4#1062190014yes—
295C22×C4C22×Dic4312yes—
296C22×C4C22×Dic4314yes—
297C22×C4C22×Dic4311yescharacteristic
298C22×C4C2×C22⋊D4314yes—
299C22×C4C2×C22⋊D4314yes—
300C22×C4C2×C22⋊D4312yes—
301C22×C4C2×C22⋊D4312yes—
302C22×C4C23×D4311yescharacteristic
303M16—438no—
304M16—8616no—
305M16—868no—
306M16—864no—
307M16—864no—
308M16—864no—
309M16—1724no—
310C2×C8—8616no—
311C2×C8—868no—
312C2×C8—864no—
313C2×C8—864no—
314C2×C8—1724no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C22⋊C4—24no—
318C22⋊C4—216no—
319C42—24no—
320C23—24no—
321C23—24no—
322C23—28no—
323C23—42no—
324C23—44no—
325C23—48no—
326C23—48no—
327C23—88no—
328C23—816no—
329C23C2×C22⋊Dic4314yes—
330C23C2×C22⋊Dic4312yes—
331C23D4×Dic4314yes—
332C23C23×Dic4311yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336D4—42no—
337D4—44no—
338D4—42no—
339D4—48no—
340D4—816no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4C2×C22⋊Dic4314yes—
348C2×C4C2×C22⋊Dic4312yes—
349C2×C4D4×Dic4314yes—
350C2×C4#1306329314yes—
351C2×C4#1306329714yes—
352C2×C4#1306329714yes—
353C2×C4#1306330614yes—
354C2×C4C2×D4×D4312yes—
355C2×C4C22×C22⋊D4311yescharacteristic
356C2×C4C22×C22⋊D4311yescharacteristic
357C2×C4C2×#1062190012yes—
358C8—868no—
359C8—1724no—
360C22—22no—
361C22—24no—
362C22—28no—
363C22—24no—
364C22—44no—
365C22—48no—
366C22—44no—
367C22—84no—
368C22—88no—
369C22—816no—
370C22#1458277914yes—
371C22C22×C22⋊Dic4311yescharacteristic
372C22C2×D4×Dic4312yes—
373C4—24no—
374C4—44no—
375C4#1458277914yes—
376C4C2×#1306329311yescharacteristic
377C4C2×#1306329712yes—
378C4C2×#1306330611yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—84no—
383C2C2×#1458277911yescharacteristic
384C2#1914307412yes—
subgroupG/H
centerC22C22×C22⋊Dic43
commutatorC23×C43C23×C4
FrattiniC22×C4C23×D43
Fitting, prime radicalC2×C43×#535C2
socleC22×C43C22×C22⋊C4
cosocleC22×C4×C43C24
hypercenterC2×#535D43
hypocenterC43#20990
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#14583666C2semidirect
2. C22×#13063242C2semidirect
3. #19143040C2semidirect
4. #19135059C2semidirect
5. #19134334C2semidirect
11. C22×#10621877C22semidirect
15. #14582863C22semidirect
16. #14582668C22semidirect
34. C2×#10621877D4semidirect
218. C43#20990semidirect

Maximal subgroups

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

subgroupindex
C2×#145836662
C22×#130632422
#191430402
#191350592
#191343342
C2×C43×#5352
#2099043