Finite Group Databasebeta

← #370263

Subgroups — #370263

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3C2×#1942C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic
5C2×C8×C4⋊C4C211yesmaximal, characteristic
6#14045C212yesmaximal
7#11293C212yesmaximal
8#11228C212yesmaximal
9#10885C212yesmaximal
10#10862C211yesmaximal, characteristic
11#10862C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1942—24no—
26#1942—24no—
27#1942C418yescomplemented
28C4×#107—24no—
29C8×C4⋊C4—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1441—22no—
40#1436—22no—
41C4⋊#95—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—22no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107C2×C414yes—
69C2×#107C2×C414yes—
70C2×#107D414yes—
71C2×#107Q814yes—
72C2×#107Q814yes—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8C2×C414yes—
84C2×C4×C8D414yes—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#389C2×C418yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—42no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340C2×C418yes—
111#336—22no—
112#336—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C22×C412yes—
143C22×C8C22×C412yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×Q812yes—
148C22×C8C2×Q812yes—
149C22×C8#4112yes—
150C22×C8#4112yes—
151C22×C8#4112yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4Q1614yes—
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×Q814yes—
190C2×C4⋊C4C2×Q814yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—82no—
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—28no—
213#107—28no—
214#107—44no—
215#107C4⋊C4116yescomplemented
216C4⋊C8—28no—
217C4×C8—28no—
218C4×C8—28no—
219C4×C8—44no—
220#95—28no—
221#95—22no—
222#95—22no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—82no—
228#95—82no—
229#95—82no—
230C24C22×D411yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4C2×Q1612yes—
296C22×C4C2×Q1612yes—
297C22×C4C2×Q1612yes—
298C22×C4#13512yes—
299C22×C4#13512yes—
300C22×C4#13512yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×Q811yescharacteristic
305C22×C4C22×Q811yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C42−1+411yescharacteristic
313C22×C42−1+411yescharacteristic
314C22×C42−1+411yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8C2×C4⋊C414yes—
337C2×C8C4×D414yes—
338C2×C8C4×Q814yes—
339C2×C8#12114yes—
340C2×C8C22⋊Q814yes—
341C2×C8#12514yes—
342C2×C8C4⋊Q814yes—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C4⋊C4C2×C4⋊C418yes—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—88no—
377C23C2×C4×D411yescharacteristic
378C23C2×#12111yescharacteristic
379C23#53211yescharacteristic
380C23D4211yescharacteristic
381C23#54711yescharacteristic
382C23D4×Q811yescharacteristic
383C23D4×Q811yescharacteristic
384C23C22×Q1611yescharacteristic
385C23C2×#13511yescharacteristic
386C23C2×#13511yescharacteristic
387C23C2×#13511yescharacteristic
388C23#57611yescharacteristic
389C23#57611yescharacteristic
390C23#57611yescharacteristic
391C23#57811yescharacteristic
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4C4×Q1614yes—
461C2×C4#46714yes—
462C2×C4C22×C4⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×Q811yescharacteristic
465C2×C4C4×#4111yescharacteristic
466C2×C4#51911yescharacteristic
467C2×C4#52011yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4#52511yescharacteristic
471C2×C4C2×#12511yescharacteristic
472C2×C4#52911yescharacteristic
473C2×C4C2×C4⋊Q811yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53711yescharacteristic
476C2×C4#54111yescharacteristic
477C2×C4#54211yescharacteristic
478C2×C4#54711yescharacteristic
479C2×C4#54811yescharacteristic
480C2×C4#54811yescharacteristic
481C2×C4#55211yescharacteristic
482C2×C4#55311yescharacteristic
483C2×C4#55411yescharacteristic
484C2×C4#55411yescharacteristic
485C2×C4#55611yescharacteristic
486C2×C4#55711yescharacteristic
487C2×C4#55711yescharacteristic
488C2×C4#55711yescharacteristic
489C2×C4#55711yescharacteristic
490C8—42no—
491C8—42no—
492C8—42no—
493C8#38918yes—
494C22C4×#12111yescharacteristic
495C22D4×C4⋊C411yescharacteristic
496C22#236511yescharacteristic
497C22#241611yescharacteristic
498C22#245211yescharacteristic
499C22#250611yescharacteristic
500C22#254611yescharacteristic
501C22C2×C4×Q1611yescharacteristic
502C22C4×#13511yescharacteristic
503C22#294811yescharacteristic
504C22#295111yescharacteristic
505C22C2×#46711yescharacteristic
506C22#305011yescharacteristic
507C22#305811yescharacteristic
508C22#306111yescharacteristic
509C22#316811yescharacteristic
510C22#317011yescharacteristic
511C22#317511yescharacteristic
512C22#317611yescharacteristic
513C22#328011yescharacteristic
514C22#328011yescharacteristic
515C22D4×Q1611yescharacteristic
516C22#328711yescharacteristic
517C22#333611yescharacteristic
518C22#333811yescharacteristic
519C22#333911yescharacteristic
520C22#334111yescharacteristic
521C22#338011yescharacteristic
522C22#338011yescharacteristic
523C22#338011yescharacteristic
524C22#338011yescharacteristic
525C22Q8×Q1611yescharacteristic
526C22Q8×Q1611yescharacteristic
527C22#338311yescharacteristic
528C22#338311yescharacteristic
529C4—24no—
530C4—24no—
531C4—42no—
532C4—48no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—84no—
537C4—84no—
538C4—162no—
539C4C2×#38911yescharacteristic
540C4#232311yescharacteristic
541C4#235111yescharacteristic
542C4#237011yescharacteristic
543C4#241511yescharacteristic
544C4#245311yescharacteristic
545C4#250911yescharacteristic
546C4#258411yescharacteristic
547C2#1404611yescharacteristic
548C2C4×#46711yescharacteristic
549C2#2152411yescharacteristic
550C2#2268211yescharacteristic
551C2C4⋊C4×Q1611yescharacteristic
552C2#2315811yescharacteristic
553C2#2316011yescharacteristic
554C2#2477811yescharacteristic
555C2#2478111yescharacteristic
556C2#2648211yescharacteristic
557C2#2648511yescharacteristic
558C2#2799211yescharacteristic
559C2#2799511yescharacteristic
560C2#2867611yescharacteristic
561C2#2867911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1942C4semidirect
215. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×C4×#1072
C2×C8×C4⋊C42
#140452
#112932
#112282
#108852
#108622