Finite Group Databasebeta

← #399605

Subgroups — #399605

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1863C212yesmaximal
4C2×#1863C211yesmaximal, characteristic
5#15194C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11328C212yesmaximal
9#11219C212yesmaximal
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2083—24no—
30#1863—28no—
31#1863—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—22no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103D414yes—
90C2×#103Q818yes—
91#389—216no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—42no—
98#384—28no—
99#384—22no—
100#384D414yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—24no—
116#340—42no—
117#336—24no—
118#336—22no—
119#336—22no—
120#336—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8C2×D412yes—
127C22×Q8C2×D412yes—
128C22×Q8C2×Q814yes—
129C22×Q8#4112yes—
130C22×Q8#4114yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8C2×D412yes—
158C22×C8C2×Q814yes—
159C4×Q8—216no—
160C4×Q8—28no—
161C4×Q8—28no—
162C4×Q8—48no—
163C4×Q8—48no—
164C4×Q8—48no—
165C4×Q8—48no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4Q1614yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C4⋊C4#4114yes—
214C2×C42—28no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229#107—48no—
230#107—44no—
231#103—28no—
232#103—216no—
233#103—44no—
234#95—28no—
235#95—24no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—81no—
240#95—82no—
241#95—82no—
242C24C22×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8#12114yes—
257C2×Q8C22⋊Q818yes—
258C2×Q8#12314yes—
259C2×Q8#12518yes—
260C2×Q8C4⋊Q814yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4#12012yes—
315C22×C4#12012yes—
316C22×C4C2×Q1614yes—
317C22×C4C2×Q1612yes—
318C22×C4#13712yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×Q812yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4112yes—
325C22×C4C2×#4112yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C42−1+412yes—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—216no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4#12314yes—
373C42—216no—
374C42—22no—
375C42—22no—
376C42—24no—
377C42—416no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C23C2×#12011yescharacteristic
383C23#52512yes—
384C23D4211yescharacteristic
385C23#54611yescharacteristic
386C23D4×Q812yes—
387C23C22×Q1611yescharacteristic
388C23C2×#13711yescharacteristic
389C23#57512yes—
390C23#57612yes—
391C23#57711yescharacteristic
392C23#57811yescharacteristic
393Q8—24no—
394Q8—28no—
395Q8—28no—
396Q8—24no—
397Q8—24no—
398Q8#39818yes—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4#45114yes—
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×C22⋊Q812yes—
459C2×C4C2×#12311yescharacteristic
460C2×C4C2×#12311yescharacteristic
461C2×C4#52512yes—
462C2×C4C2×#12512yes—
463C2×C4C2×C4⋊Q811yescharacteristic
464C2×C4#53511yescharacteristic
465C2×C4#53611yescharacteristic
466C2×C4#53912yes—
467C2×C4#53912yes—
468C2×C4D4211yescharacteristic
469C2×C4#54611yescharacteristic
470C2×C4#54712yes—
471C2×C4D4×Q812yes—
472C2×C4#55212yes—
473C2×C4#55412yes—
474C2×C4#55412yes—
475C8—44no—
476C8—42no—
477C8—81no—
478C22#240811yescharacteristic
479C22#243411yescharacteristic
480C22#244912yes—
481C22#247312yes—
482C22#284911yescharacteristic
483C22C2×#45111yescharacteristic
484C22#300511yescharacteristic
485C22#300811yescharacteristic
486C22#301311yescharacteristic
487C22#310412yes—
488C22#311712yes—
489C22#311912yes—
490C22#312412yes—
491C22#326812yes—
492C22#327811yescharacteristic
493C22D4×Q1611yescharacteristic
494C22#329911yescharacteristic
495C22#330912yes—
496C22#332811yescharacteristic
497C22Q8×Q1612yes—
498C22#338512yes—
499C22#338812yes—
500C22#339012yes—
501C4—28no—
502C4—24no—
503C4—44no—
504C4—42no—
505C4—44no—
506C4—84no—
507C4—82no—
508C4—84no—
509C4—82no—
510C4—161no—
511C4C2×#39811yescharacteristic
512C4#244912yes—
513C4#247312yes—
514C4#248611yescharacteristic
515C4#250211yescharacteristic
516C4#262311yescharacteristic
517C2#1478611yescharacteristic
518C2#2424311yescharacteristic
519C2#2425011yescharacteristic
520C2#2584111yescharacteristic
521C2#2584611yescharacteristic
522C2#2638112yes—
523C2#2638712yes—
524C2#2730012yes—
525C2#2731612yes—
526C2#3301011yescharacteristic
527C2#3301211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#20832
C2×#18632
#151942
#113712
#113462
#113282
#112192
#107892