Finite Group Databasebeta

← #363127

Subgroups — #363127

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2962C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5C2×#1947C211yesmaximal, characteristic
6#11242C212yesmaximal
7#11239C212yesmaximal
8#10870C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10827C212yesmaximal
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13#2962—24no—
14#2962C418yescomplemented
15C2×#446—21no—
16C2×#446—22no—
17C2×C8×Q8—21no—
18C22×#108C2212yes—
19C22×#107C2212yes—
20C2×#424C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#405C2212yes—
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1992—22no—
29#1990—21no—
30#1990—21no—
31#1989—22no—
32#1988—21no—
33#1987—21no—
34#1950—24no—
35#1947—21no—
36#1947—22no—
37#1947—21no—
38C2×#343C2212yes—
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#343C2211yescharacteristic
44C2×#341C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49C2×#340C2212yes—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125D412yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#446—216no—
76#446—28no—
77C8×Q8—28no—
78C2×#108—24no—
79C2×#108—24no—
80C2×#108—42no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—24no—
84C2×#107—42no—
85#424—24no—
86#424C2×C418yes—
87#423—22no—
88#423—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C414yes—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D412yes—
97C2×C4⋊C8D414yes—
98#405—24no—
99#405C2×C418yes—
100C4×M16—22no—
101C4×M16—22no—
102C4×M16—22no—
103C4×M16C2×C414yes—
104C2×#97—22no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C2×#95—21no—
119#343—22no—
120#343—22no—
121#343—24no—
122#343—22no—
123#343—42no—
124#343Q814yescomplemented
125#341—24no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4C2411yescharacteristic
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16C22×C412yes—
169C2×M16C22×C412yes—
170C2×M16C2×Q812yes—
171C2×M16C2×Q812yes—
172C2×M16#4112yes—
173C2×M16#4112yes—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8C22×C414yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D414yes—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8C4⋊C414yes—
190C4⋊Q8C4⋊C414yes—
191#125—24no—
192#125C4⋊C418yes—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201#117—22no—
202#117—22no—
203#117C22×C414yes—
204#117C2×Q814yes—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4#4112yes—
235C2×C22⋊C4—22no—
236C2×C42—22no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—82no—
240C2×C42C22×C412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248#108—48no—
249#107—44no—
250#107—44no—
251C4⋊C8—28no—
252C4⋊C8—28no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8C22⋊C414yes—
256C4⋊C8C22⋊C414yes—
257C4⋊C8C22⋊C418yes—
258#97—28no—
259#97—24no—
260#97—42no—
261C4×C8—28no—
262C4×C8—24no—
263C4×C8—42no—
264#95—41no—
265#95—41no—
266C24C2×#4111yescharacteristic
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—81no—
272C2×Q8—81no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#11712yes—
295C22×C4#11712yes—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C42−1+411yescharacteristic
314C22×C42−1+411yescharacteristic
315C22×C42−1+411yescharacteristic
316M16—22no—
317M16—22no—
318M16—22no—
319M16—22no—
320M16—22no—
321M16—22no—
322M16—22no—
323M16—22no—
324M16—22no—
325M16—41no—
326M16—41no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—42no—
332M16—42no—
333M16—42no—
334M16—42no—
335M16C4×Q814yes—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8C2×C22⋊C414yes—
360C2×C8C2×C22⋊C414yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4C2×C4⋊C414yes—
384C4⋊C4C2×C4⋊C414yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C42—24no—
388C42—24no—
389C42—82no—
390C42—82no—
391C42C2×C22⋊C412yes—
392C42C2×C22⋊C412yes—
393C42C2×C4⋊C412yes—
394C42C2×C4⋊C412yes—
395C42#12112yes—
396C42#12112yes—
397C42C22⋊Q812yes—
398C42C22⋊Q812yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23C2×#11711yescharacteristic
404C23C2×#12111yescharacteristic
405C23C2×C22⋊Q811yescharacteristic
406C23#52911yescharacteristic
407C23#53311yescharacteristic
408C23#54111yescharacteristic
409C23#54211yescharacteristic
410Q8—48no—
411Q8—84no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—162no—
442C2×C4—162no—
443C2×C4#38012yes—
444C2×C4#38012yes—
445C2×C4#38012yes—
446C2×C4#38012yes—
447C2×C4#47212yes—
448C2×C4#47312yes—
449C2×C4C22×C22⋊C411yescharacteristic
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×C4×Q811yescharacteristic
452C2×C4C4×#4111yescharacteristic
453C2×C4#51911yescharacteristic
454C2×C4#52011yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#53311yescharacteristic
459C2×C4#54111yescharacteristic
460C2×C4#54211yescharacteristic
461C2×C4#54611yescharacteristic
462C2×C4#54611yescharacteristic
463C2×C4#54711yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#54811yescharacteristic
466C2×C4#54811yescharacteristic
467C2×C4#54811yescharacteristic
468C2×C4#54811yescharacteristic
469C2×C4D4×Q811yescharacteristic
470C2×C4D4×Q811yescharacteristic
471C2×C4D4×Q811yescharacteristic
472C2×C4#55011yescharacteristic
473C2×C4#55711yescharacteristic
474C2×C4#55711yescharacteristic
475C2×C4#55711yescharacteristic
476C2×C4Q8211yescharacteristic
477C8—24no—
478C8—24no—
479C8—24no—
480C8—42no—
481C8—42no—
482C8—42no—
483C8—82no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22C2×#38011yescharacteristic
497C22#229511yescharacteristic
498C22#232011yescharacteristic
499C22#242611yescharacteristic
500C22#242911yescharacteristic
501C22#245111yescharacteristic
502C22#245311yescharacteristic
503C22C2×#47211yescharacteristic
504C22C2×#47311yescharacteristic
505C22#306811yescharacteristic
506C22#306911yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—44no—
511C4—44no—
512C4—82no—
513C4—82no—
514C4—162no—
515C4#181614yes—
516C4#233911yescharacteristic
517C4Q8×C22⋊C411yescharacteristic
518C4Q8×C4⋊C411yescharacteristic
519C4#235111yescharacteristic
520C4#242611yescharacteristic
521C4#242911yescharacteristic
522C4#245111yescharacteristic
523C4#245311yescharacteristic
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2#1393511yescharacteristic
529C2C2×#181611yescharacteristic
530C2#2067911yescharacteristic
531C2#2531211yescharacteristic
532C2#2531411yescharacteristic
533C2#2548911yescharacteristic
534C2#2549111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2962C4semidirect
124. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#213222
#213122
C2×#19502
C2×#19472
#112422
#112392
#108702
#108572
#108272
C4×#3432