Finite Group Databasebeta

← #25148712

Subgroups — #25148712

Order 11008.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144388C211yesmaximal, characteristic
2C2×#14583176C211yesmaximal, characteristic
3#19143544C211yesmaximal, characteristic, complemented
4#19136103C212yesmaximal, complemented
5#19135952C212yesmaximal
6#19135585C211yesmaximal, characteristic, complemented
7#19135377C212yesmaximal, complemented
8#19135370C211yesmaximal, characteristic, complemented
9#19135256C212yesmaximal
10C4×#13063241C211yesmaximal, characteristic
11C43×#3165C211yesmaximal, characteristic, complemented
12C2×C4×D172C2211yescharacteristic
13#14583196—22no—
14#14583186—21no—
15#14583176—21no—
16#14583176—21no—
17#14583176—22no—
18#14583176—21no—
19#14583030—21no—
20#14583030—22no—
21Q8×C43⋊C8—21no—
22C2×#13063260C2212yes—
23#14582942C2212yes—
24C2×#13063258C2212yes—
25#14582926C2212yes—
26#14582923C2211yescharacteristic
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28#14582903C2212yes—
29C2×#13063245C2212yes—
30C2×#13063241C2211yescharacteristic
31C2×#13063241C2211yescharacteristic
32#14582855C2212yes—
33C4×#10621877C2211yescharacteristic
34#14582668C2212yes—
35#14582668C2212yes—
36#14582667C2211yescharacteristic
37#14582667C2211yescharacteristic
38#14582663C2212yes—
39C43×#533C2211yescharacteristic
40C2×C4⋊Q8×C43C2211yescharacteristic
41C43×#501—22no—
42C43×#500—21no—
43C43×#498—21no—
44C43×#428C2212yes—
45C43×#428C2212yes—
46C4×M16×C43C2211yescharacteristic
47C22×D172C2311yescharacteristic
48C22×C4×D43—21no—
49C2×#10621877C2311yescharacteristic
50C2×#10621877C2312yes—
51C2×#10621877C2311yescharacteristic
52C22×C43⋊C8C2312yes—
53C4×D172—21no—
54C4×D172—21no—
55C4×D172—21no—
56C4×D172—21no—
57C2×C4⋊Dic43C2311yescharacteristic
58C2×(C2×D43)⋊C4—21no—
59#13063260—24no—
60#13063258—22no—
61#13063258—22no—
62#13063258—22no—
63#13063256—22no—
64C4⋊(C43⋊C8)—21no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)—22no—
67C4⋊(C43⋊C8)—22no—
68C4⋊(C43⋊C8)D411yescharacteristic
69C4⋊(C43⋊C8)D412yes—
70C4⋊(C43⋊C8)D411yescharacteristic
71#13063245—22no—
72#13063245—22no—
73#13063245—22no—
74#13063245—22no—
75#13063241—22no—
76#13063241Q814yescomplemented
77#13063236—21no—
78#13063236—22no—
79C4×C43⋊C8—21no—
80C4×C43⋊C8—22no—
81C22×Q8×C43—21no—
82C2×M16×C43C2311yescharacteristic
83C2×M16×C43C2311yescharacteristic
84C4⋊Q8×C43—21no—
85C4⋊Q8×C43—21no—
86C4⋊Q8×C43—21no—
87C4⋊Q8×C43—21no—
88C4⋊Q8×C43D411yescharacteristic
89C4⋊Q8×C43D411yescharacteristic
90#125×C43D412yes—
91C22⋊Q8×C43—21no—
92C22⋊Q8×C43—21no—
93C4×Q8×C43—21no—
94C4×Q8×C43—21no—
95#117×C43C2312yes—
96C2×C4⋊C4×C43—21no—
97C2×C4⋊C4×C43C2312yes—
98C2×C42×C43C2311yescharacteristic
99#107×C43—22no—
100#107×C43—22no—
101#106×C43—22no—
102#106×C43—22no—
103#97×C43—21no—
104C4×C8×C43—21no—
105C23×D43—21no—
106C2×Q8×C43—21no—
107C2×Q8×C43—21no—
108C2×Q8×C43—41no—
109C2×Q8×C43—41no—
110C2×Q8×C43—82no—
111C22×C4×C43—22no—
112C22×C4×C43#4111yescharacteristic
113C22×C4×C43#4111yescharacteristic
114C22×C4×C43C2411yescharacteristic
115C22×Dic43—21no—
116C2×D172—21no—
117C2×D172—21no—
118C2×D172—21no—
119C2×D172—21no—
120C2×D172—21no—
121C2×D172C2×Q812yes—
122C2×C4×D43—41no—
123C2×C4×D43—41no—
124C2×C4×D43—82no—
125M16×C43—21no—
126M16×C43—21no—
127M16×C43C2×Q812yes—
128M16×C43#4112yes—
129C2×C8×C43—21no—
130C2×C8×C43—21no—
131C4⋊C4×C43—22no—
132C4⋊C4×C43—22no—
133C4⋊C4×C43—21no—
134C4⋊C4×C43—21no—
135C4⋊C4×C43—22no—
136C4⋊C4×C43—22no—
137C4⋊C4×C43—41no—
138C4⋊C4×C43—41no—
139C4⋊C4×C43C2×D412yes—
140C4⋊C4×C43C2×D412yes—
141C22⋊C4×C43—22no—
142C42×C43—22no—
143C42×C43C2×D411yescharacteristic
144C42×C43C2×D411yescharacteristic
145C42×C43C2×D411yescharacteristic
146C42×C43C2×D411yescharacteristic
147(C2×D43)⋊C4—41no—
148(C2×D43)⋊C4—41no—
149C4⋊Dic43—21no—
150C4⋊Dic43—21no—
151#10621877—21no—
152#10621877—21no—
153#10621877—22no—
154#10621877—21no—
155#10621877—22no—
156#10621877C2×Q812yes—
157C2×C43⋊C8—21no—
158C2×C43⋊C8—22no—
159C2×C43⋊C8—22no—
160C2×C43⋊C8—21no—
161C2×C43⋊C8—22no—
162C2×C43⋊C8C2×D412yes—
163C2×C43⋊C8C2×D412yes—
164C23×C43C2×#4111yescharacteristic
165C22×D43—22no—
166C22×D43—41no—
167C22×D43—41no—
168C22×D43—81no—
169C22×D43—81no—
170Q8×C43—42no—
171Q8×C43—81no—
172Q8×C43—82no—
173Q8×C43—82no—
174C2×C4×C43—22no—
175C2×C4×C43—22no—
176C2×C4×C43—22no—
177C2×C4×C43—41no—
178C2×C4×C43—42no—
179C2×C4×C43—82no—
180C2×C4×C43#12111yescharacteristic
181C2×C4×C43#12111yescharacteristic
182C2×C4×C43#12111yescharacteristic
183C2×C4×C43#12111yescharacteristic
184C2×C4×C43C22×D411yescharacteristic
185C2×C4×C43C22×D411yescharacteristic
186C2×C4×C43C22×Q811yescharacteristic
187C2×C4×C43C2×#4111yescharacteristic
188C2×C4×C432−1+411yescharacteristic
189C2×C4×C432−1+411yescharacteristic
190C2×Dic43—41no—
191C2×Dic43—81no—
192D172—22no—
193D172—22no—
194D172—41no—
195D172—41no—
196C4×D43—82no—
197C4×D43—82no—
198C8×C43—22no—
199C8×C43—22no—
200C43⋊C8—22no—
201C43⋊C8—22no—
202C43⋊C8—22no—
203C43⋊C8—41no—
204C43⋊C8—42no—
205#25489—431nomaximal, Sylow
206C22×C43—21no—
207C22×C43—21no—
208C22×C43C2×#12111yescharacteristic
209C22×C43#53311yescharacteristic
210C22×C43#54111yescharacteristic
211C2×D43—42no—
212C2×D43—82no—
213C2×D43—81no—
214C2×D43—81no—
215C2×D43—82no—
216C4×C43—21no—
217C4×C43—21no—
218C4×C43—21no—
219C4×C43—42no—
220C4×C43—42no—
221C4×C43—82no—
222C4×C43#47212yes—
223C4×C43#54711yescharacteristic
224C4×C43#54811yescharacteristic
225C4×C43D4×Q811yescharacteristic
226C4×C43D4×Q811yescharacteristic
227Dic43—81no—
228#3165D4311yescharacteristic, complemented
229#3077—431no—
230C2×#474—431no—
231#2962—431no—
232#2089—432no—
233#2063—432no—
234#1997—431no—
235#1952—432no—
236#1950—431no—
237#1931—432no—
238C4×#100—431no—
239C2×C43—21no—
240C2×C43—21no—
241C2×C43#242911yescharacteristic
242C2×C43C2×#47211yescharacteristic
243C2×C43#306911yescharacteristic
244D43—82no—
245#533C2×D4311yescharacteristic
246C2×C4⋊Q8C2×D4311yescharacteristic
247C2×C4×D4—431no—
248#501—22no—
249#500—21no—
250#498—21no—
251#478—862no—
252#476—861no—
253#474—861no—
254#474—862no—
255#474—861no—
256#474—861no—
257#446—862no—
258#446—861no—
259C8×Q8—861no—
260C2×#108—432no—
261#428—432no—
262#428C2×D4312yes—
263#428C2×D4312yes—
264C2×#107—432no—
265#424—432no—
266#423—431no—
267C2×C4⋊C8—431no—
268#418—432no—
269C2×#102—432no—
270C2×#100—431no—
271C2×#100—431no—
272#405—432no—
273C4×M16—431no—
274C4×M16C2×D4311yescharacteristic
275#343—431no—
276#343—431no—
277#343—432no—
278#343—432no—
279#341—432no—
280C43#2548911yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×D4—431no—
283C23×C4—861no—
284C2×M16—431no—
285C2×M16—432no—
286C2×M16—431no—
287C2×M16C22×D4311yescharacteristic
288C2×M16C22×D4311yescharacteristic
289C22×C8—432no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D4311yescharacteristic
295C4⋊Q8C22⋊D4311yescharacteristic
296#125C22⋊D4312yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—861no—
302C4×D4—861no—
303C4×D4—861no—
304C4×D4—861no—
305#117C22×D4312yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—431no—
308C2×C4⋊C4C22×D4312yes—
309C2×C22⋊C4—861no—
310C2×C42C22×D4311yescharacteristic
311#108—864no—
312#107—22no—
313#107—22no—
314#107—862no—
315#107—862no—
316#107—862no—
317#106—22no—
318#106—22no—
319#106—862no—
320C4⋊C8—431no—
321C4⋊C8—431no—
322C4⋊C8—432no—
323C4⋊C8—862no—
324C4⋊C8—862no—
325C4⋊C8—861no—
326C4⋊C8—861no—
327#102—862no—
328#102—862no—
329#102—862no—
330#102—862no—
331#100—434no—
332#100—862no—
333#97—21no—
334#97—862no—
335#97—861no—
336C4×C8—21no—
337C4×C8—862no—
338C4×C8—861no—
339C24—861no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×D4—432no—
346C2×D4—861no—
347C2×D4—861no—
348C2×D4—861no—
349C2×D4—861no—
350C2×D4—861no—
351C22×C4—22no—
352C22×C4—861no—
353C22×C4—3442no—
354C22×C4—1721no—
355C22×C4—1721no—
356C22×C4#1062190011yescharacteristic
357C22×C4#1062190011yescharacteristic
358C22×C4C23×D4311yescharacteristic
359M16—21no—
360M16—21no—
361M16—432no—
362M16—861no—
363M16—861no—
364M16—862no—
365M16—861no—
366M16—862no—
367M16Q8×D4312yes—
368M16#1062190212yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—432no—
372C2×C8—432no—
373C2×C8—861no—
374C2×C8—862no—
375C2×C8—862no—
376C2×C8—861no—
377C2×C8—862no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—861no—
387C4⋊C4—861no—
388C4⋊C4C2×C22⋊D4312yes—
389C4⋊C4C2×C22⋊D4312yes—
390C22⋊C4—22no—
391C22⋊C4—1721no—
392C22⋊C4—1721no—
393C42—22no—
394C42D4×D4311yescharacteristic
395C42D4×D4311yescharacteristic
396C42C2×C22⋊D4311yescharacteristic
397C42C2×C22⋊D4311yescharacteristic
398C23—862no—
399C23—3441no—
400C23—3441no—
401C23—1721no—
402C23—1721no—
403C23C2×#1062190011yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408D4—862no—
409D4—862no—
410D4—1721no—
411D4—1721no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—3442no—
419C2×C4—3442no—
420C2×C4—3441no—
421C2×C4—1721no—
422C2×C4#1306329711yescharacteristic
423C2×C4#1306329711yescharacteristic
424C2×C4#1306329711yescharacteristic
425C2×C4#1306329711yescharacteristic
426C2×C4C2×D4×D4311yescharacteristic
427C2×C4C22×C22⋊D4311yescharacteristic
428C2×C4C2×Q8×D4311yescharacteristic
429C2×C4C2×#1062190211yescharacteristic
430C2×C4#1306337311yescharacteristic
431C2×C4#1306337311yescharacteristic
432C8—22no—
433C8—22no—
434C8—862no—
435C8—862no—
436C8—862no—
437C8—1721no—
438C8—1722no—
439C22—21no—
440C22—21no—
441C22—3442no—
442C22—3442no—
443C22—3441no—
444C22—3441no—
445C22—1722no—
446C22C2×#1306329711yescharacteristic
447C22#1458345411yescharacteristic
448C22#1458351011yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—3441no—
456C4#1458316712yes—
457C4#1458355211yescharacteristic
458C4#1458356011yescharacteristic
459C4Q8×C22⋊D4311yescharacteristic
460C4#1458356411yescharacteristic
461C2—21no—
462C2—21no—
463C2—3442no—
464C2#1913819711yescharacteristic
465C2C2×#1458316711yescharacteristic
466C2#1914432911yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#3165C2
socleC22×C43C2×#121
prime radicalC22×D172C23
hypercenter#3165D43
hypocenterC43#25489
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #19143544C2semidirect
4. #19136103C2semidirect
6. #19135585C2semidirect
7. #19135377C2semidirect
8. #19135370C2semidirect
11. C43×#3165C2semidirect
76. #13063241Q8semidirect
228. #3165D43semidirect
280. C43#25489semidirect

Maximal subgroups

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

subgroupindex
#191443882
C2×#145831762
#191435442
#191361032
#191359522
#191355852
#191353772
#191353702
#191352562
C4×#130632412
C43×#31652
#2548943