Finite Group Databasebeta

← #501675

Subgroups — #501675

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic, complemented
2C2×#2217C211yesmaximal, characteristic, complemented
3C2×#2168C211yesmaximal, characteristic, complemented
4C22×#366C211yesmaximal, characteristic, complemented
5#24781C211yesmaximal, characteristic, complemented
6#24780C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13383C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic, complemented
11#13238C211yesmaximal, characteristic, complemented
12#13168C211yesmaximal, characteristic, complemented
13#12993C211yesmaximal, characteristic, complemented
14#12991C211yesmaximal, characteristic, complemented
15#12836C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#466C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26#2417—21no—
27#2416—21no—
28#2220—24no—
29#2217—24no—
30#2168—24no—
31C2×#367C2211yescharacteristic
32C2×#366—24no—
33C2×#366C2211yescharacteristic
34C2×#366C2211yescharacteristic
35C2×#366C2211yescharacteristic
36C2×#366C2211yescharacteristic
37C2×C4⋊C16C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46C2×#346C2211yescharacteristic
47#2068—21no—
48#2067—21no—
49#2036—21no—
50#2033—21no—
51#1970C2211yescharacteristic
52#1969C2211yescharacteristic
53#1942C2211yescharacteristic
54#1942C2211yescharacteristic
55#1874—21no—
56#1872—21no—
57#1808C2211yescharacteristic
58#1808C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1380C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—41no—
72C2×C4×D4D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C2×SD32—22no—
76C2×SD32—22no—
77C2×SD32—22no—
78C2×M32—22no—
79C2×M32—22no—
80C2×M32—22no—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16D414yescomplemented
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#467—22no—
89#467—22no—
90#467—22no—
91#467—41no—
92#466—22no—
93#466—22no—
94#466—22no—
95#466—41no—
96C8×D4—22no—
97C8×D4—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—21no—
101C2×#107—21no—
102C2×#107—41no—
103C2×#107—41no—
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×C4⋊C8D411yescharacteristic
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#398—41no—
121#396—41no—
122#396—41no—
123#394—41no—
124#391—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#386—41no—
129#386—41no—
130#380—21no—
131#380—21no—
132#380—41no—
133#367—22no—
134#366—22no—
135#366—22no—
136#366—22no—
137#366D414yescomplemented
138C4⋊C16—22no—
139#358—22no—
140#358—22no—
141#358—41no—
142#357—22no—
143#357—22no—
144#357—41no—
145C22⋊C16—22no—
146C22⋊C16—22no—
147#346—22no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4D811yescharacteristic
157C23×C4—41no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D412yes—
187C22×C8#4112yes—
188C22×C8C2411yescharacteristic
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—81no—
193#121—42no—
194#121—42no—
195#121—42no—
196#121—81no—
197#121—84no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4D811yescharacteristic
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42C2×D411yescharacteristic
228SD32—44no—
229M32—44no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—28no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16C2×D412yes—
240C2×C16C2×D412yes—
241C2×C16C2×D412yes—
242C2×C16#4112yes—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—42no—
254#107—42no—
255#107C2×D412yes—
256#107C2×D412yes—
257C4⋊C8—22no—
258#103—42no—
259#103—42no—
260#103—81no—
261#102—42no—
262#102—42no—
263#102—81no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—41no—
275C24C2×D811yescharacteristic
276C24C2×D811yescharacteristic
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—161no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—161no—
320C22×C4—161no—
321C22×C4#12112yes—
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4#13512yes—
327C22×C4C22×D411yescharacteristic
328Q16—41no—
329Q16—41no—
330Q16—41no—
331Q16—41no—
332D8—41no—
333D8—41no—
334D8—41no—
335D8—41no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8#12112yes—
353C2×C8C22×D411yescharacteristic
354C2×C8C22×D411yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
358C2×C82−1+411yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—161no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—161no—
387C22⋊C4—161no—
388C42—21no—
389C42—21no—
390C16—22no—
391C16—41no—
392C16—41no—
393C16#12114yescomplemented
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23#46612yes—
413C23#50912yes—
414C23#51012yes—
415C23C22×D811yescharacteristic
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420D4—44no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—164no—
426D4—164no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—164no—
468C2×C4—164no—
469C2×C4#46612yes—
470C2×C4C2×#12111yescharacteristic
471C2×C4D4211yescharacteristic
472C2×C4#54711yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57811yescharacteristic
476C8—22no—
477C8—41no—
478C8—41no—
479C8C2×#12111yescharacteristic
480C8#53711yescharacteristic
481C8#54811yescharacteristic
482C8#55311yescharacteristic
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22#222014yescomplemented
504C22C2×#46611yescharacteristic
505C22D4×D811yescharacteristic
506C22#333411yescharacteristic
507C22C2×#50911yescharacteristic
508C22C2×#51011yescharacteristic
509C22#341511yescharacteristic
510C22#341711yescharacteristic
511C4—22no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4—161no—
521C4—161no—
522C4#241611yescharacteristic
523C4#306111yescharacteristic
524C4#328511yescharacteristic
525C4#333911yescharacteristic
526C2—24no—
527C2—81no—
528C2—321no—
529C2#2478011yescharacteristic
530C2C2×#222011yescharacteristic
531C2#3361011yescharacteristic
532C2#3383011yescharacteristic
533C2#3383611yescharacteristic
534C2#3391011yescharacteristic
535C2#3391311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#24780C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2220C2semidirect
2. C2×#2217C2semidirect
3. C2×#2168C2semidirect
4. C22×#366C2semidirect
5. #24781C2semidirect
7. #13475C2semidirect
8. #13474C2semidirect
9. #13383C2semidirect
10. #13380C2semidirect
11. #13238C2semidirect
12. #13168C2semidirect
13. #12993C2semidirect
14. #12991C2semidirect
15. #12836C2semidirect
83. C22×C16D4semidirect
137. #366D4semidirect
393. C16#121semidirect
503. C22#2220semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#22172
C2×#21682
C22×#3662
#247812
#247802
#134752
#134742
#133832
#133802
#132382
#131682
#129932
#129912
#128362