Finite Group Databasebeta

← #411165

Subgroups — #411165

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1956C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#15180C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11077C212yesmaximal
8#11010C211yesmaximal, characteristic
9#10926C212yesmaximal
10#10926C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#400—21no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1956—24no—
27C8×C4⋊C4—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—21no—
35#1442—21no—
36#1442—21no—
37#1442—21no—
38#1441—21no—
39#1441—21no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—42no—
86C2×C4×C8—42no—
87C2×C4×C8D414yes—
88#400—44no—
89#391—24no—
90#391—24no—
91#391—24no—
92#391—42no—
93#384—22no—
94#384—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—24no—
112#340—42no—
113#340—42no—
114#340—42no—
115#336—22no—
116#336—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8#4114yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4114yes—
162C4⋊Q8—28no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4SD1614yes—
197C2×C4⋊C4Q1614yes—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C4⋊C8—28no—
213#103—48no—
214#103—48no—
215#103—48no—
216#103—48no—
217C4×C8—28no—
218C4×C8—44no—
219C4×C8—44no—
220#95—22no—
221#95—22no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—82no—
233#95—82no—
234C24C22×D411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
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—42no—
310C22×C4—42no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12412yes—
314C22×C4#12412yes—
315C22×C4C2×SD1612yes—
316C22×C4C2×SD1612yes—
317C22×C4C2×SD1612yes—
318C22×C4C2×Q1612yes—
319C22×C4C2×Q1612yes—
320C22×C4C2×Q1612yes—
321C22×C4#13512yes—
322C22×C4#13512yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8#12414yes—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—24no—
378C42—41no—
379C42—41no—
380C42—42no—
381C23C2×#12111yescharacteristic
382C23C2×#12411yescharacteristic
383C23#53211yescharacteristic
384C23#54612yes—
385C23#55012yes—
386C23C22×SD1611yescharacteristic
387C23C22×Q1611yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×#13511yescharacteristic
390C23#57712yes—
391C23#57812yes—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
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#46514yes—
458C2×C4#46714yes—
459C2×C4#48714yes—
460C2×C4#52511yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4C2×#12411yescharacteristic
464C2×C4#52911yescharacteristic
465C2×C4#52912yes—
466C2×C4#52912yes—
467C2×C4#52911yescharacteristic
468C2×C4#53411yescharacteristic
469C2×C4#53511yescharacteristic
470C2×C4#53812yes—
471C2×C4#54012yes—
472C2×C4#55112yes—
473C2×C4#55112yes—
474C2×C4#55312yes—
475C2×C4#55312yes—
476C2×C4#55312yes—
477C2×C4#55912yes—
478C8—24no—
479C8—42no—
480C8—42no—
481C8—42no—
482C22#249012yes—
483C22#253411yescharacteristic
484C22#254811yescharacteristic
485C22#255412yes—
486C22#285011yescharacteristic
487C22C2×#46511yescharacteristic
488C22C2×#46711yescharacteristic
489C22#305711yescharacteristic
490C22#305911yescharacteristic
491C22C2×#48711yescharacteristic
492C22#313111yescharacteristic
493C22#313712yes—
494C22#316711yescharacteristic
495C22#317011yescharacteristic
496C22#317812yes—
497C22#329812yes—
498C22#329912yes—
499C22#330012yes—
500C22#330112yes—
501C22#339212yes—
502C22#339312yes—
503C22#339412yes—
504C22#339512yes—
505C4—24no—
506C4—24no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—162no—
513C4—162no—
514C4—162no—
515C4#239811yescharacteristic
516C4#251612yes—
517C4#259111yescharacteristic
518C4#259512yes—
519C4#259811yescharacteristic
520C4#263511yescharacteristic
521C2#1510711yescharacteristic
522C2#2774912yes—
523C2#2775212yes—
524C2#2809511yescharacteristic
525C2#2809811yescharacteristic
526C2#2873711yescharacteristic
527C2#2873811yescharacteristic
528C2#2887512yes—
529C2#2887912yes—
530C2#3304311yescharacteristic
531C2#3304611yescharacteristic
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×#19562
C2×C8×C4⋊C42
#151802
#151522
#113572
#111902
#110772
#110102
#109262
#108202