Finite Group Databasebeta

← #4453092

Subgroups — #4453092

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#2399C211yesmaximal, characteristic
3#16828C214yesmaximal
4#16827C214yesmaximal
5#16525C214yesmaximal
6#16524C214yesmaximal
7#15202C212yesmaximal
8#15194C212yesmaximal
9#15158C212yesmaximal
10#15149C212yesmaximal
11#13954C212yesmaximal
12#13911C212yesmaximal
13#13797C211yesmaximal, characteristic
14C22×C4×Q8C2212yes—
15#2647—28no—
16#2399—28no—
17C2×#400C2214yes—
18C2×#398C2214yes—
19C2×#398C2212yes—
20C2×#398C2212yes—
21C2×#398C2214yes—
22C2×#395C2212yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1446C2214yes—
40#1445C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69C23×Q8C2312yes—
70C2×C4×Q8—28no—
71C2×C4×Q8—44no—
72C2×C4×Q8—44no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2314yes—
96C22×C42C2311yescharacteristic
97#400—216no—
98#398—216no—
99#398—28no—
100#398—28no—
101#398—48no—
102#395—28no—
103#395—28no—
104#395—48no—
105#389—28no—
106#389—28no—
107#389—44no—
108#389—44no—
109#389—44no—
110#389—44no—
111#384—24no—
112#382—24no—
113#382—48no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—28no—
117C4×C4⋊C4Q8116yescomplemented
118#376D418yes—
119C2×#95C2312yes—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2311yescharacteristic
125C2×#95C2311yescharacteristic
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2414yes—
158C23×C4C2411yescharacteristic
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C4×Q8—432no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
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—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×Q818yes—
221C2×C4⋊C4C2×Q818yes—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42C2×D414yes—
234C2×C42C2×D414yes—
235C2×C42C2×D414yes—
236C2×C42C2×Q8116yes—
237C2×C42C2×Q818yes—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95C2×D418yes—
264#95C2×D418yes—
265#95C2×Q8116yes—
266C24C2511yescharacteristic
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D414yes—
347C22×C4C22×D414yes—
348C22×C4C22×D412yes—
349C22×C4C22×Q818yes—
350C22×C4C22×Q818yes—
351C22×C4C22×Q814yes—
352C22×C4C22×Q814yes—
353C22×C4C22×Q814yes—
354C22×C4H(5,𝔽2)14yes—
355C22×C4H(5,𝔽2)14yes—
356C22×C42−1+414yes—
357C22×C42−1+412yes—
358C22×C42−1+412yes—
359C4⋊C4—216no—
360C4⋊C4—216no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C42—28no—
378C42—28no—
379C42—416no—
380C42—48no—
381C42—48no—
382C42C4⋊Q8116yes—
383C23C23×D411yescharacteristic
384C23C23×Q812yes—
385C23C2×H(5,𝔽2)12yes—
386C23C2×H(5,𝔽2)12yes—
387C23C2×2−1+411yescharacteristic
388C23C2×2−1+411yescharacteristic
389C23C2×2−1+412yes—
390C23#58512yes—
391C23#58512yes—
392Q8—432no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4C2×C4⋊Q818yes—
473C2×C4C2×C4⋊Q814yes—
474C2×C4#53514yes—
475C2×C4#53614yes—
476C2×C4#54418yes—
477C2×C4#54418yes—
478C22C22×C4⋊Q811yescharacteristic
479C22C2×#53511yescharacteristic
480C22C2×#53611yescharacteristic
481C22C2×#54412yes—
482C22C2×#54412yes—
483C22#348812yes—
484C22#348912yes—
485C22#350112yes—
486C22#350212yes—
487C22#350312yes—
488C22#350412yes—
489C22#353612yes—
490C22#354012yes—
491C22#354112yes—
492C22#354212yes—
493C22#354811yescharacteristic
494C22#355412yes—
495C22#355512yes—
496C22#355611yescharacteristic
497C22#355711yescharacteristic
498C22#355711yescharacteristic
499C4—28no—
500C4—42no—
501C4—44no—
502C4—44no—
503C4—44no—
504C4—42no—
505C4—48no—
506C4—48no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4#266018yes—
512C2C2×#266011yescharacteristic
513C2#3556612yes—
514C2#3727112yes—
515C2#3727411yescharacteristic
516C2#3737711yescharacteristic
517C2#4249312yes—
518C2#4249412yes—
519C2#4282112yes—
520C2#4282212yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
117. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#23992
#168282
#168272
#165252
#165242
#152022
#151942
#151582
#151492
#139542
#139112
#137972