Finite Group Databasebeta

← #468339

Subgroups — #468339

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11325C212yesmaximal
6#11267C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#11208C211yesmaximal, characteristic
10#11208C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15#2647—24no—
16#2647—24no—
17C2×#400—21no—
18C2×#400—21no—
19C2×#398—21no—
20C2×#398—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2058—24no—
30#2057—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446—21no—
42#1446—21no—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
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⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—42no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—42no—
76#400—28no—
77#400—28no—
78#398—28no—
79#398—28no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389D414yes—
87#389D414yes—
88#384—22no—
89#384—22no—
90#382—22no—
91#382—22no—
92#382—42no—
93#376—22no—
94#376—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
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×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—22no—
119#336—42no—
120#336Q818yescomplemented
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×Q814yes—
152C22×C8C2×Q814yes—
153C4⋊Q8—48no—
154C4⋊Q8—48no—
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—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
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—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4#4114yes—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×Q814yes—
216#107—48no—
217#106—48no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—42no—
229#95—42no—
230#95—82no—
231#95—82no—
232#95—82no—
233#95—82no—
234C24C22×D411yescharacteristic
235C2×Q8—48no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—88no—
239C2×Q8—88no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
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×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#12012yes—
301C22×C4#12012yes—
302C22×C4C4⋊Q812yes—
303C22×C4C4⋊Q812yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q812yes—
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+411yescharacteristic
311C22×C42−1+411yescharacteristic
312C22×C42−1+411yescharacteristic
313C22×C42−1+412yes—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C23C2×#12011yescharacteristic
364C23#52511yescharacteristic
365C23C2×C4⋊Q811yescharacteristic
366C23#53511yescharacteristic
367C23#53611yescharacteristic
368C23#53611yescharacteristic
369C23#53711yescharacteristic
370C23#57612yes—
371C23#57712yes—
372C23#57712yes—
373C23#57812yes—
374Q8—816no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#53611yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53711yescharacteristic
442C2×C4#53711yescharacteristic
443C2×C4#54112yes—
444C2×C4#54112yes—
445C2×C4#54412yes—
446C2×C4#54412yes—
447C2×C4#54612yes—
448C2×C4#54612yes—
449C2×C4D4×Q812yes—
450C2×C4D4×Q812yes—
451C2×C4#55412yes—
452C2×C4#55712yes—
453C2×C4#56212yes—
454C2×C4#56312yes—
455C8—44no—
456C8—44no—
457C22#240411yescharacteristic
458C22#243411yescharacteristic
459C22#258411yescharacteristic
460C22#261611yescharacteristic
461C22#263411yescharacteristic
462C22#266011yescharacteristic
463C22#282011yescharacteristic
464C22#300811yescharacteristic
465C22#300911yescharacteristic
466C22#301011yescharacteristic
467C22#301111yescharacteristic
468C22#311812yes—
469C22#312112yes—
470C22#316312yes—
471C22#316412yes—
472C22#322412yes—
473C22#322512yes—
474C22#324112yes—
475C22#324212yes—
476C22#324312yes—
477C22#324412yes—
478C22#325712yes—
479C22#325812yes—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4—82no—
487C4—81no—
488C4—82no—
489C4—84no—
490C4—82no—
491C4—162no—
492C4#270412yes—
493C4#270412yes—
494C4#271312yes—
495C4#271312yes—
496C2#1660811yescharacteristic
497C2#2394511yescharacteristic
498C2#2394611yescharacteristic
499C2#2582011yescharacteristic
500C2#2582111yescharacteristic
501C2#2932512yes—
502C2#2969811yescharacteristic
503C2#2969911yescharacteristic
504C2#3010012yes—
505C2#3031512yes—
506C2#3283312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
120. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#20582
C2×#20572
#142822
#113252
#112672
#112202
#112192
#112082
#107962