Finite Group Databasebeta

← #4466164

Subgroups — #4466164

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic
3#17097C214yesmaximal
4#16891C212yesmaximal, complemented
5#16377C214yesmaximal, complemented
6#16279C214yesmaximal, complemented
7#16093C212yesmaximal, complemented
8#15944C212yesmaximal, complemented
9#15671C212yesmaximal
10#15637C214yesmaximal
11#15465C212yesmaximal
12#14887C212yesmaximal
13#14489C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2811—28no—
19#2621—28no—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2212yes—
27C2×#393C2212yes—
28C2×#393C2212yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2211yescharacteristic
35C2×#387C2214yes—
36C2×#382C2211yescharacteristic
37#1446C2214yes—
38#1446C2214yes—
39#1446C2214yes—
40#1446C2212yes—
41#1445C2212yes—
42#1445C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1437C2212yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×Q8C2312yes—
73C24×C4C2311yescharacteristic
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—44no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—42no—
79C2×C22⋊Q8—42no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×#117—24no—
83C2×#117—44no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#400—44no—
104#398—216no—
105#397—216no—
106#397—44no—
107#395—28no—
108#395—216no—
109#395—44no—
110#393—44no—
111#393—44no—
112#391—28no—
113#391—24no—
114#391—28no—
115#391—216no—
116#391—44no—
117#391D418yes—
118#387—48no—
119#382—24no—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—28no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8C2×D414yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C4⋊Q8—416no—
188C22⋊Q8—416no—
189C22⋊Q8—416no—
190#117—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42C2×D414yes—
243#95—28no—
244#95—24no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
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—42no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—42no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95C2×D4116yes—
272C24—21no—
273C24—22no—
274C24—21no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24H(5,𝔽2)12yes—
279C24C2511yescharacteristic
280C2×Q8—216no—
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—88no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×Q8—84no—
299C2×Q8—88no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)14yes—
371C22×C42−1+418yes—
372C22×C42−1+414yes—
373C4⋊C4—44no—
374C4⋊C4—416no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—416no—
379C4⋊C4—44no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4—84no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C42—28no—
389C42—44no—
390C42—84no—
391C42—84no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23C23×D411yescharacteristic
408C23C2×H(5,𝔽2)12yes—
409C23C2×H(5,𝔽2)11yescharacteristic
410C23C2×H(5,𝔽2)11yescharacteristic
411C23C2×2−1+412yes—
412C23C2×2−1+414yes—
413C23#58514yes—
414Q8—88no—
415Q8—88no—
416Q8—816no—
417Q8—816no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—416no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4#53614yes—
487C22—48no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—84no—
494C22C2×#53611yescharacteristic
495C22#348411yescharacteristic
496C22#348411yescharacteristic
497C22#348612yes—
498C22#350012yes—
499C22#353514yes—
500C22#353612yes—
501C22#353612yes—
502C22#354012yes—
503C22#354412yes—
504C22#354512yes—
505C22#355012yes—
506C22#355514yes—
507C22#355514yes—
508C22#356114yes—
509C4—24no—
510C4—44no—
511C4—44no—
512C4—48no—
513C4—44no—
514C4—42no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4—84no—
521C2—82no—
522C2#3671811yescharacteristic
523C2#4279312yes—
524C2#4635712yes—
525C2#5174911yescharacteristic
526C2#5207211yescharacteristic
527C2#5479914yes—
528C2#5803814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2811C2semidirect
4. #16891C2semidirect
5. #16377C2semidirect
6. #16279C2semidirect
7. #16093C2semidirect
8. #15944C2semidirect
13. #14489C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#26212
#170972
#168912
#163772
#162792
#160932
#159442
#156712
#156372
#154652
#148872
#144892