Finite Group Databasebeta

← #25080574

Subgroups — #25080574

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144404C211yesmaximal, characteristic
2#19144404C211yesmaximal, characteristic
3C8×#10621898C211yesmaximal, characteristic
4#19134050C214yesmaximal, complemented
5#19134043C214yesmaximal, complemented
6#19133354C211yesmaximal, characteristic, complemented
7#19133280C211yesmaximal, characteristic, complemented
8#19133276C211yesmaximal, characteristic, complemented
9C2×C43×#487C211yesmaximal, characteristic, complemented
10C4×#10621898C2211yescharacteristic
11#14583191—21no—
12#14583191C2212yes—
13#14583186—21no—
14#14583186C2212yes—
15#14583181—21no—
16#14583181C2212yes—
17#14583176—21no—
18#14583176C2212yes—
19C8×Dic86C2212yes—
20C8×D172C2212yes—
21C8×C22⋊D43—22no—
22#14582952C2211yescharacteristic
23#14582950—21no—
24#14582939—21no—
25#14582939—21no—
26#14582923C2211yescharacteristic
27#14582923C2211yescharacteristic
28#14582898—21no—
29#14582898—21no—
30C4×C8×D43—21no—
31#14582652C2214yes—
32#14582648C2214yes—
33#14582645C2214yes—
34C2×C4⋊Q8×C43C2211yescharacteristic
35C2×C4⋊Q8×C43C2211yescharacteristic
36C43×#487—21no—
37C43×#487—21no—
38C43×#487C2214yes—
39C2×#103×C43—22no—
40C2×C4×C8×C43C2211yescharacteristic
41C2×#10621898D411yescharacteristic
42C2×#10621877—21no—
43C2×#10621877—21no—
44C2×C8×D43—22no—
45C4×Dic86D411yescharacteristic
46C4×Dic86C2311yescharacteristic
47C4×D172D411yescharacteristic
48C4×D172C2311yescharacteristic
49C4×C22⋊D43—21no—
50C4×C22⋊D43—21no—
51#13063284D411yescharacteristic
52#13063282—21no—
53C42×D43—21no—
54#13063258—22no—
55#13063258—22no—
56#13063256—22no—
57#13063256—22no—
58C4⋊(C43⋊C8)—21no—
59C4⋊(C43⋊C8)—21no—
60C4⋊(C43⋊C8)C2312yes—
61C4⋊(C43⋊C8)C2312yes—
62C4⋊(C43⋊C8)C2312yes—
63Dic43⋊C8—22no—
64#13063248—22no—
65#13063248—22no—
66#13063245—22no—
67#13063245—22no—
68C22⋊(C43⋊C8)—21no—
69(C2×D43)⋊C8—22no—
70C4×C43⋊C8—21no—
71C8×Dic43—22no—
72C22×Q8×C43—21no—
73C22×Q8×C43—21no—
74C22×C8×C43D412yes—
75C4⋊Q8×C43—21no—
76C4⋊Q8×C43—21no—
77C4⋊Q8×C43—21no—
78C4⋊Q8×C43—21no—
79C4⋊Q8×C43—21no—
80C4⋊Q8×C43—21no—
81C4⋊Q8×C43C2312yes—
82C4⋊Q8×C43C2312yes—
83C2×C4⋊C4×C43—21no—
84C2×C4⋊C4×C43—21no—
85C2×C4⋊C4×C43—21no—
86C2×C4⋊C4×C43—21no—
87C2×C42×C43C2311yescharacteristic
88#103×C43—24no—
89#103×C43—24no—
90C4×C8×C43D411yescharacteristic
91C4×C8×C43D411yescharacteristic
92C4×C8×C43C2312yes—
93C2×Q8×C43—22no—
94C2×Q8×C43—22no—
95C2×Q8×C43—44no—
96C2×Q8×C43—41no—
97C2×Q8×C43—41no—
98C2×Q8×C43—82no—
99C22×C4×C43—41no—
100C22×C4×C43—41no—
101C22×C4×C43C2×D411yescharacteristic
102C22×C4×C43C2×D411yescharacteristic
103C22×C4×C43#4111yescharacteristic
104C2×C22⋊D43—21no—
105#10621898—22no—
106#10621898Q1614yes—
107C2×D172C2×D411yescharacteristic
108C2×C4×D43—21no—
109C2×C4×D43—21no—
110C2×C4×D43—21no—
111C2×Dic86C2×D411yescharacteristic
112C2×C8×C43—42no—
113C2×C8×C43C2×D412yes—
114C2×C8×C43C2×D412yes—
115C4⋊C4×C43—22no—
116C4⋊C4×C43—22no—
117C4⋊C4×C43—22no—
118C4⋊C4×C43—22no—
119C4⋊C4×C43—41no—
120C4⋊C4×C43—41no—
121C4⋊C4×C43—41no—
122C4⋊C4×C43—41no—
123C42×C43C2×D411yescharacteristic
124C42×C43C2×D411yescharacteristic
125C42×C43#4111yescharacteristic
126C42×C43C2411yescharacteristic
127C22⋊Dic43—21no—
128(C2×D43)⋊C4—21no—
129(C2×D43)⋊C4—21no—
130C4⋊Dic43C2×D411yescharacteristic
131C4⋊Dic43C2×D411yescharacteristic
132Dic43⋊C4—21no—
133Dic43⋊C4—21no—
134C4×Dic43—21no—
135C4×Dic43—21no—
136C4×Dic43—21no—
137#10621877—42no—
138#10621877—42no—
139C2×C43⋊C8—22no—
140C2×C43⋊C8—22no—
141C2×C43⋊C8—22no—
142C8×D43—44no—
143C23×C43#12111yescharacteristic
144C22×D43—21no—
145Q8×C43—44no—
146Q8×C43—82no—
147Q8×C43—82no—
148Q8×C43—82no—
149C2×C4×C43—22no—
150C2×C4×C43—22no—
151C2×C4×C43—42no—
152C2×C4×C43—42no—
153C2×C4×C43—82no—
154C2×C4×C43—82no—
155C2×C4×C43#12111yescharacteristic
156C2×C4×C43#12111yescharacteristic
157C2×C4×C43#12111yescharacteristic
158C2×C4×C43C2×Q1612yes—
159C2×C4×C43#13512yes—
160C2×C4×C43C22×D411yescharacteristic
161C2×C4×C43C22×D411yescharacteristic
162C2×C4×C43C2×#4111yescharacteristic
163C22⋊D43—24no—
164C2×Dic43—21no—
165C2×Dic43—21no—
166C2×Dic43—21no—
167D172—21no—
168D172C2×Q1612yes—
169C4×D43—24no—
170C4×D43—42no—
171C4×D43—42no—
172Dic86—21no—
173Dic86C2×Q1612yes—
174C8×C43—22no—
175C8×C43—22no—
176C43⋊C8—42no—
177C43⋊C8—42no—
178C43⋊C8—42no—
179#19277—431nomaximal, Sylow
180C22×C43—21no—
181C22×C43—21no—
182C22×C43#46712yes—
183C22×C43C2×#12111yescharacteristic
184C2×D43—22no—
185C2×D43—41no—
186C2×D43—41no—
187C4×C43—21no—
188C4×C43—21no—
189C4×C43—82no—
190C4×C43—82no—
191C4×C43#46712yes—
192C4×C43#47112yes—
193C4×C43C22×Q1611yescharacteristic
194C4×C43C2×#13511yescharacteristic
195C4×C43#57512yes—
196Dic43—22no—
197Dic43—41no—
198Dic43—41no—
199C2×#487D4311yescharacteristic, complemented
200#3079—431no—
201#3079—431no—
202C8×#41—431no—
203#1684—434no—
204#1683—434no—
205#1567—431no—
206#1549—431no—
207#1548—431no—
208C2×C43—22no—
209C2×C43#304011yescharacteristic
210C2×C43C2×#46711yescharacteristic
211C2×C43C2×#47111yescharacteristic
212D43—42no—
213C2×C4⋊Q8C2×D4311yescharacteristic
214C2×C4⋊Q8C2×D4311yescharacteristic
215C4×#41—431no—
216#487—21no—
217#487—21no—
218#487C2×D4314yes—
219#477—432no—
220#477—861no—
221#476—432no—
222#476—861no—
223#475—432no—
224#475—861no—
225#474—432no—
226#474—861no—
227C8×Q8—432no—
228C8×D4—432no—
229C8×D4—862no—
230#431—431no—
231#431—861no—
232#428—861no—
233#428—861no—
234#423—431no—
235#423—431no—
236#417—861no—
237#417—861no—
238C2×#103—22no—
239C2×C4×C8—861no—
240C2×C4×C8C2×D4311yescharacteristic
241#335—434no—
242#333—434no—
243#332—434no—
244C43#1927711yescharacteristic, complemented, Sylow
245C2×#41—431no—
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—861no—
249C2×M16—861no—
250C22×C8—862no—
251C22×C8C22⋊D4312yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22×D4312yes—
259C4⋊Q8C22×D4312yes—
260C4×Q8—431no—
261C4×Q8—431no—
262C4×D4—431no—
263C4×D4—431no—
264C4×D4—861no—
265C4×D4—861no—
266#117—431no—
267#117—861no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42—861no—
273C2×C42C22×D4311yescharacteristic
274#107—862no—
275#107—862no—
276#106—862no—
277#106—862no—
278C4⋊C8—432no—
279C4⋊C8—432no—
280C4⋊C8—432no—
281C4⋊C8—862no—
282C4⋊C8—861no—
283C4⋊C8—861no—
284#103—24no—
285#103—24no—
286#103—862no—
287#103—862no—
288#102—862no—
289#102—862no—
290C22⋊C8—861no—
291C22⋊C8—862no—
292C4×C8—862no—
293C4×C8—861no—
294C4×C8C22⋊D4311yescharacteristic
295C4×C8C22⋊D4311yescharacteristic
296C4×C8C22×D4312yes—
297#41—434no—
298#41—862no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—431no—
306C2×D4—431no—
307C2×D4—861no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—861no—
311C22×C4—861no—
312C22×C4—861no—
313C22×C4D4×D4311yescharacteristic
314C22×C4#1062190011yescharacteristic
315C22×C4C2×C22⋊D4311yescharacteristic
316M16—1722no—
317M16—1722no—
318C2×C8—42no—
319C2×C8—862no—
320C2×C8—862no—
321C2×C8—862no—
322C2×C8—1724no—
323C2×C8C2×C22⋊D4312yes—
324C2×C8C2×C22⋊D4312yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—431no—
334C4⋊C4—431no—
335C4⋊C4—861no—
336C4⋊C4—861no—
337C22⋊C4—861no—
338C22⋊C4—861no—
339C22⋊C4—861no—
340C42—861no—
341C42—861no—
342C42—861no—
343C42D4×D4311yescharacteristic
344C42#1062190011yescharacteristic
345C42C2×C22⋊D4311yescharacteristic
346C42C23×D4311yescharacteristic
347C23—861no—
348C23#1306329611yescharacteristic
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—432no—
354Q8—861no—
355D4—432no—
356D4—864no—
357D4—861no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—861no—
365C2×C4—864no—
366C2×C4—861no—
367C2×C4—861no—
368C2×C4—1722no—
369C2×C4—1722no—
370C2×C4#1306329611yescharacteristic
371C2×C4#1306329611yescharacteristic
372C2×C4#1306329611yescharacteristic
373C2×C4Q16×D4312yes—
374C2×C4#1306334812yes—
375C2×C4C2×D4×D4311yescharacteristic
376C2×C4C22×C22⋊D4311yescharacteristic
377C2×C4C2×#1062190011yescharacteristic
378C8—22no—
379C8—22no—
380C8—1722no—
381C8—1722no—
382C8—1722no—
383C22—21no—
384C22—21no—
385C22—862no—
386C22—1721no—
387C22—1721no—
388C22#1458313712yes—
389C22C2×#1306329611yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—862no—
395C4—1721no—
396C4—1721no—
397C4#1458313712yes—
398C4#1458315912yes—
399C4C2×Q16×D4311yescharacteristic
400C4C2×#1306334811yescharacteristic
401C4#1458369712yes—
402C2—22no—
403C2—1722no—
404C2#1914408511yescharacteristic
405C2C2×#1458313711yescharacteristic
406C2C2×#1458315911yescharacteristic
subgroupG/H
centerC22C2×#13063296
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×C43×#487C2
socleC22×C43C2×#121
prime radicalC2×#10621898D4
hypercenterC2×#487D43
hypocenterC43#19277
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #19134050C2semidirect
5. #19134043C2semidirect
6. #19133354C2semidirect
7. #19133280C2semidirect
8. #19133276C2semidirect
9. C2×C43×#487C2semidirect
199. C2×#487D43semidirect
244. C43#19277semidirect

Maximal subgroups

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

subgroupindex
#191444042
C8×#106218982
#191340502
#191340432
#191333542
#191332802
#191332762
C2×C43×#4872
#1927743