Finite Group Databasebeta

← #367405

Subgroups — #367405

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1991C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic
4#14709C211yesmaximal, characteristic
5C4×#382C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11315C212yesmaximal
8#10842C211yesmaximal, characteristic
9#10807C212yesmaximal
10#10806C212yesmaximal
11C4×#336C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C43C2211yescharacteristic
21#1991—24no—
22#1991C418yescomplemented
23#1835—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—22no—
41C4×#95—22no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125C2×C414yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4×C8—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8C2×C414yes—
70#382—22no—
71#382—22no—
72#382—22no—
73#382—22no—
74#382C2×C414yes—
75#382D414yes—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C43—22no—
90C43—22no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340C2×C418yes—
98#340C2×C418yes—
99#336—24no—
100#336—24no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8C22×C414yes—
125C22×C8#4114yes—
126#125—28no—
127#125—28no—
128#125—44no—
129#125—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—44no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—44no—
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42#4114yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C4×C8—28no—
186C4×C8—44no—
187#95—28no—
188#95—28no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—42no—
194#95—42no—
195#95—44no—
196#95—44no—
197#95—44no—
198C24C22×D411yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C4×D414yes—
251C22×C4#12114yes—
252C22×C4#12314yes—
253C22×C4#12412yes—
254C22×C4#12412yes—
255C22×C4#13514yes—
256C22×C4#13514yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4C2×#4111yescharacteristic
265C22×C42−1+412yes—
266C22×C42−1+412yes—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—416no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C42—24no—
299C42—24no—
300C42—48no—
301C42—48no—
302C42—42no—
303C42—42no—
304C42—84no—
305C42—84no—
306C42#11714yes—
307C42C4×D414yes—
308C42#12414yes—
309C42#12614yes—
310C23C2×C4×D411yescharacteristic
311C23C2×#12111yescharacteristic
312C23C2×#12311yescharacteristic
313C23C2×#12411yescharacteristic
314C23#53211yescharacteristic
315C23#53611yescharacteristic
316C23#53611yescharacteristic
317C23C2×#13512yes—
318C23C2×#13512yes—
319C23#57512yes—
320C23#57512yes—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38818yes—
378C2×C4#48814yes—
379C2×C4C2×#11711yescharacteristic
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C4×#4112yes—
382C2×C4#51912yes—
383C2×C4C2×#12411yescharacteristic
384C2×C4C2×#12611yescharacteristic
385C2×C4#52912yes—
386C2×C4#54112yes—
387C2×C4#54712yes—
388C2×C4#54812yes—
389C2×C4#55212yes—
390C2×C4#55212yes—
391C2×C4#55212yes—
392C2×C4#55612yes—
393C2×C4#55612yes—
394C2×C4#55612yes—
395C8—44no—
396C8—44no—
397C22C2×#38811yescharacteristic
398C22C4×#12411yescharacteristic
399C22#232711yescharacteristic
400C22#249911yescharacteristic
401C22#250311yescharacteristic
402C22#254511yescharacteristic
403C22#254711yescharacteristic
404C22C4×#13512yes—
405C22#294312yes—
406C22#303512yes—
407C22#303512yes—
408C22#309112yes—
409C22#309112yes—
410C22C2×#48811yescharacteristic
411C22#313111yescharacteristic
412C22#314011yescharacteristic
413C22#314211yescharacteristic
414C22#316812yes—
415C22#318312yes—
416C22#323212yes—
417C22#323212yes—
418C22#323312yes—
419C22#323312yes—
420C4—24no—
421C4—28no—
422C4—42no—
423C4—44no—
424C4—82no—
425C4—82no—
426C4—84no—
427C4—82no—
428C4—82no—
429C4—162no—
430C4#235612yes—
431C4#236212yes—
432C4#248912yes—
433C4#251412yes—
434C2#1400311yescharacteristic
435C2#2105812yes—
436C2C4×#48811yescharacteristic
437C2#2164811yescharacteristic
438C2#2224912yes—
439C2#2787912yes—
440C2#2796311yescharacteristic
441C2#2796411yescharacteristic
442C2#2865712yes—
443C2#2872011yescharacteristic
444C2#2872111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1991C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19912
C2×#18352
#147092
C4×#3822
#113162
#113152
#108422
#108072
#108062
C4×#3362