Finite Group Databasebeta

← #365476

Subgroups — #365476

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1949C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11327C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2369—24no—
18#2369—24no—
19C2×#391—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1993—24no—
31#1988C418yescomplemented
32#1949—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45C4⋊#95—22no—
46C23×Q8—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—22no—
74C2×#107—42no—
75C2×#106—22no—
76C2×#106—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83#391—216no—
84#389—28no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#389—42no—
91#382—22no—
92#382—22no—
93C4×C4⋊C4—28no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4D414yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95D412yes—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340C2×C418yes—
117#340C2×C418yes—
118#340Q818yescomplemented
119#336—42no—
120#336—42no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—81no—
125C22×Q8—81no—
126C22×Q8—81no—
127C22×Q8—81no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C22×C412yes—
154C22×C8C22×C412yes—
155C22×C8C2×Q814yes—
156C4⋊Q8—28no—
157C4⋊Q8—28no—
158C4⋊Q8—28no—
159C4⋊Q8C4⋊C4116yescomplemented
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
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—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4C2×Q814yes—
203C2×C4⋊C4#4114yes—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—44no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42—82no—
217C2×C42—82no—
218C2×C42C22×C412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—44no—
225#106—44no—
226C4⋊C8—28no—
227#97—44no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237C24C22×D411yescharacteristic
238C2×Q8—216no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—84no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12012yes—
301C22×C4#12012yes—
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q812yes—
304C22×C4#13714yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q811yescharacteristic
309C22×C4C22×Q812yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C42−1+412yes—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—216no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C4⋊C4C2×C4⋊C418yes—
356C4⋊C4C4×Q818yes—
357C4⋊C4C4⋊Q814yes—
358C42—48no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—84no—
364C42—88no—
365C42C2×C4⋊C414yes—
366C42C4×D414yes—
367C42C22⋊Q814yes—
368C42#12314yes—
369C23C2×C4×D411yescharacteristic
370C23C2×#12011yescharacteristic
371C23C2×C22⋊Q811yescharacteristic
372C23#54612yes—
373C23D4×Q812yes—
374C23C2×#13712yes—
375C23#57512yes—
376C23#57611yescharacteristic
377C23#57611yescharacteristic
378C23#57711yescharacteristic
379C23#57711yescharacteristic
380Q8—816no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#38514yes—
440C2×C4#38514yes—
441C2×C4C22×C4⋊C411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×C4×Q812yes—
444C2×C4#52012yes—
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C2×C4C2×C4⋊Q811yescharacteristic
448C2×C4#53211yescharacteristic
449C2×C4#53312yes—
450C2×C4#53511yescharacteristic
451C2×C4#53611yescharacteristic
452C2×C4#54212yes—
453C2×C4D4×Q812yes—
454C2×C4#55012yes—
455C2×C4#55212yes—
456C2×C4#55412yes—
457C2×C4#55712yes—
458C2×C4Q8212yes—
459C8—42no—
460C8—42no—
461C8—44no—
462C22C2×#38511yescharacteristic
463C22#236012yes—
464C22#243111yescharacteristic
465C22#243411yescharacteristic
466C22#245112yes—
467C22C4×#13712yes—
468C22#294811yescharacteristic
469C22#294911yescharacteristic
470C22#300411yescharacteristic
471C22#300511yescharacteristic
472C22#301211yescharacteristic
473C22#301211yescharacteristic
474C22#307812yes—
475C22#308111yescharacteristic
476C22#308111yescharacteristic
477C22#331412yes—
478C22#331612yes—
479C22#331912yes—
480C22#332112yes—
481C22#338512yes—
482C22#338712yes—
483C22#338812yes—
484C22#338912yes—
485C4—24no—
486C4—24no—
487C4—48no—
488C4—48no—
489C4—44no—
490C4—44no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4C4×C4⋊Q811yescharacteristic
496C4#232611yescharacteristic
497C4Q8×C4⋊C412yes—
498C4#242312yes—
499C4#256511yescharacteristic
500C4#256711yescharacteristic
501C2#1397611yescharacteristic
502C2#2087411yescharacteristic
503C2#2087911yescharacteristic
504C2#2297912yes—
505C2#2298112yes—
506C2#2557811yescharacteristic
507C2#2557911yescharacteristic
508C2#2584511yescharacteristic
509C2#2584711yescharacteristic
510C2#2645312yes—
511C2#2645412yes—
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
31. #1988C4semidirect
118. #340Q8semidirect
159. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#19932
C2×#19882
C2×#19492
#139542
#113282
#113272
#108852
#108842
#108352
#107032