Finite Group Databasebeta

← #360465

Subgroups — #360465

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1930C212yesmaximal
2#20801C211yesmaximal, characteristic
3#20647C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5#20441C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11145C212yesmaximal
9#10896C212yesmaximal
10#10858C211yesmaximal, characteristic
11#10853C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#419C2212yes—
18C22×#103C2212yes—
19C22×#101C2211yescharacteristic
20C22×#101C2211yescharacteristic
21C2×#405C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26#1930—28no—
27#1847—22no—
28#1846—21no—
29#1846—21no—
30#1805—21no—
31#1805—21no—
32#1805—22no—
33#1805—21no—
34#1805—21no—
35#1803—22no—
36#1771—22no—
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×#101C418yescomplemented
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47C2×#340C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C414yes—
64C2×#117C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#428—24no—
71#428—24no—
72#428—42no—
73#428—42no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#106—22no—
77C2×#106—22no—
78#423—22no—
79#423—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8D412yes—
87#419—24no—
88#419—42no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—42no—
95C2×#101—22no—
96C2×#101—22no—
97C2×#101C2×C414yes—
98C2×#101C2×C414yes—
99C2×#101D414yes—
100C2×#101Q814yes—
101#405—24no—
102#405—42no—
103C4×M16—22no—
104C4×M16—22no—
105C4×M16C2×C414yes—
106C4×M16C2×C414yes—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—22no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—22no—
113#384D412yes—
114#382—21no—
115#382—21no—
116#382—21no—
117#382—21no—
118#382D411yescharacteristic
119#382D411yescharacteristic
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C2×#95—21no—
125C2×#95—21no—
126#343—24no—
127#343—42no—
128#343C2×C414yes—
129#343C2×C414yes—
130#341—24no—
131#341—24no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336—22no—
137#336—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16C22×C412yes—
179C2×M16C22×C412yes—
180C2×M16C22×C412yes—
181C2×M16C22×C412yes—
182C2×M16C2×D412yes—
183C2×M16C2×D412yes—
184C2×M16C2×Q812yes—
185C2×M16C2×Q812yes—
186C2×M16#4112yes—
187C2×M16#4112yes—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C22⋊C4—22no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—42no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—44no—
248#107—44no—
249#106—44no—
250#106—44no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—24no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256#103—44no—
257#103—44no—
258#103—44no—
259#103—44no—
260#101—28no—
261#101C4⋊C4116yescomplemented
262#97—24no—
263#97—24no—
264#97—44no—
265C4×C8—24no—
266C4×C8—24no—
267C4×C8—44no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276C24C2×#4111yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8C2×C4⋊C414yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4#11712yes—
326C22×C4#11712yes—
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12612yes—
332C22×C4#12612yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—41no—
359M16—41no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16C2×C4⋊C414yes—
365M16C2×C4⋊C414yes—
366M16#11714yes—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—44no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—82no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—42no—
411C42—42no—
412C42—44no—
413C42—44no—
414C42C4×D412yes—
415C42C4×D412yes—
416C42C4×D412yes—
417C42C4×D412yes—
418C42#12312yes—
419C42#12312yes—
420C42#12412yes—
421C42#12412yes—
422C23—22no—
423C23—22no—
424C23C2×#11711yescharacteristic
425C23C2×#12111yescharacteristic
426C23C2×#12611yescharacteristic
427C23#53211yescharacteristic
428C23#53311yescharacteristic
429C23#54011yescharacteristic
430C23#54111yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—162no—
468C2×C4—162no—
469C2×C4#38812yes—
470C2×C4#38812yes—
471C2×C4#38812yes—
472C2×C4#38812yes—
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#52011yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×#12411yescharacteristic
481C2×C4#53211yescharacteristic
482C2×C4#53311yescharacteristic
483C2×C4#54211yescharacteristic
484C2×C4#54211yescharacteristic
485C2×C4D4211yescharacteristic
486C2×C4D4211yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#54711yescharacteristic
489C2×C4#54711yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4D4×Q811yescharacteristic
493C2×C4D4×Q811yescharacteristic
494C2×C4#55011yescharacteristic
495C2×C4#55211yescharacteristic
496C2×C4#55311yescharacteristic
497C2×C4#55411yescharacteristic
498C2×C4#55411yescharacteristic
499C2×C4#55511yescharacteristic
500C2×C4#55611yescharacteristic
501C8—24no—
502C8—24no—
503C8—24no—
504C8—42no—
505C8—42no—
506C8—42no—
507C8—82no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22#228511yescharacteristic
517C22C2×#38811yescharacteristic
518C22#231411yescharacteristic
519C22#242811yescharacteristic
520C22#242911yescharacteristic
521C22#249611yescharacteristic
522C22#249711yescharacteristic
523C22#306712yes—
524C22#306912yes—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—162no—
534C4D4×C4⋊C411yescharacteristic
535C4D4×C4⋊C411yescharacteristic
536C4#235211yescharacteristic
537C4#235411yescharacteristic
538C4#242211yescharacteristic
539C4#242311yescharacteristic
540C4#249111yescharacteristic
541C4#249211yescharacteristic
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2#1387611yescharacteristic
547C2#2107511yescharacteristic
548C2#2107711yescharacteristic
549C2#2540912yes—
550C2#2549412yes—
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
39. C4×#101C4semidirect
261. #101C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19302
#208012
#206472
C2×#18052
#204412
C2×C4×#1012
#112902
#111452
#108962
#108582
#108532