Finite Group Databasebeta

← #355758

Subgroups — #355758

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2C2×#1918C212yesmaximal
3C4×#389C212yesmaximal
4#10907C212yesmaximal
5#10884C212yesmaximal
6#10880C212yesmaximal
7#10820C211yesmaximal, characteristic
8#10806C212yesmaximal
9C4×#336C211yesmaximal, characteristic
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×C4×C4⋊C4—22no—
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C43C2211yescharacteristic
20#1942C418yescomplemented
21#1918C4116yescomplemented
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1436—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4×#95—22no—
37C23×C8C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2311yescharacteristic
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42—22no—
47C22×C42—22no—
48C22×C42D412yes—
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2312yes—
53C2×#107—41no—
54C2×#107—41no—
55C2×#107C2×C414yes—
56C2×#107C2×C418yes—
57C2×#107C2×C414yes—
58C2×C4×C8—22no—
59C2×C4×C8—41no—
60C2×C4×C8—41no—
61C2×C4×C8—41no—
62C2×C4×C8—41no—
63C2×C4×C8C2×C414yes—
64#389—24no—
65#389—24no—
66#389C2×C418yes—
67#389C2×C418yes—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—42no—
70C4×C4⋊C4—42no—
71C4×C4⋊C4—48no—
72C4×C4⋊C4C2×C418yes—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C43—41no—
82C43—41no—
83#340—41no—
84#340—42no—
85#340—41no—
86#340—42no—
87#340C2×C418yes—
88#340C2×C418yes—
89#340C2×C418yes—
90#340C2×C418yes—
91#336—24no—
92#336C2×C418yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2411yescharacteristic
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C22×C414yes—
116C22×C8C22×C412yes—
117C22×C8C22×C414yes—
118C22×C8C22×C412yes—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—28no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—41no—
162C2×C42—42no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42—44no—
171C2×C42D814yes—
172C2×C42C22×C414yes—
173C2×C42C22×C412yes—
174C2×C42#4114yes—
175C2×C42#4112yes—
176#107—44no—
177#107C42116yescomplemented
178C4×C8—44no—
179C4×C8—44no—
180#95—28no—
181#95—28no—
182#95—28no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188C24C22×D411yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C4×D414yes—
235C22×C4C4×D414yes—
236C22×C4C4×D412yes—
237C22×C4C4×D412yes—
238C22×C4C2×D812yes—
239C22×C4C2×D814yes—
240C22×C4#13514yes—
241C22×C4#13512yes—
242C22×C4#13714yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C23×C412yes—
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4112yes—
247C22×C4H(5,𝔽2)11yescharacteristic
248C22×C4H(5,𝔽2)11yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C42−1+411yescharacteristic
251C22×C42−1+412yes—
252C22×C42−1+412yes—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8C2×C4214yes—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—416no—
286C4⋊C4—416no—
287C4⋊C4C2×C4218yes—
288C42—22no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—24no—
295C42—22no—
296C42—44no—
297C42—48no—
298C42—44no—
299C42—416no—
300C42—88no—
301C23C2×C4×D412yes—
302C23C2×C4×D411yescharacteristic
303C23#53411yescharacteristic
304C23#53612yes—
305C23#53711yescharacteristic
306C23C22×D811yescharacteristic
307C23C2×#13512yes—
308C23C2×#13511yescharacteristic
309C23C2×#13712yes—
310C23#57511yescharacteristic
311C23#57511yescharacteristic
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C4×D818yes—
364C2×C4C4×D814yes—
365C2×C4#44114yes—
366C2×C4C22×C4211yescharacteristic
367C2×C4C4×#4111yescharacteristic
368C2×C4C4×#4112yes—
369C2×C4#51811yescharacteristic
370C2×C4#51911yescharacteristic
371C2×C4#51911yescharacteristic
372C2×C4#51912yes—
373C2×C4#52012yes—
374C2×C4#52012yes—
375C2×C4#52012yes—
376C2×C4#52011yescharacteristic
377C2×C4#53811yescharacteristic
378C2×C4#53912yes—
379C2×C4#53912yes—
380C2×C4#53911yescharacteristic
381C2×C4#54112yes—
382C2×C4#54112yes—
383C2×C4#54212yes—
384C8—24no—
385C8—42no—
386C8—44no—
387C22C42×D411yescharacteristic
388C22#232212yes—
389C22#232312yes—
390C22#232511yescharacteristic
391C22#232711yescharacteristic
392C22C2×C4×D812yes—
393C22C2×C4×D811yescharacteristic
394C22C4×#13512yes—
395C22C4×#13511yescharacteristic
396C22C2×#44111yescharacteristic
397C22#294311yescharacteristic
398C22C4×#13712yes—
399C22C4×#13712yes—
400C22#318611yescharacteristic
401C22#319012yes—
402C22#319111yescharacteristic
403C22#322612yes—
404C22#323112yes—
405C22#323212yes—
406C22#323312yes—
407C22#324511yescharacteristic
408C22#324812yes—
409C22#325011yescharacteristic
410C4—28no—
411C4—24no—
412C4—24no—
413C4—48no—
414C4—44no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4#227411yescharacteristic
419C4#227611yescharacteristic
420C4#233211yescharacteristic
421C4#233412yes—
422C4#233611yescharacteristic
423C4#233612yes—
424C2#1379011yescharacteristic
425C2C42×D811yescharacteristic
426C2C4×#44111yescharacteristic
427C2#2207512yes—
428C2#2207712yes—
429C2#2211012yes—
430C2#2211512yes—
431C2#2218111yescharacteristic
432C2#2218311yescharacteristic
433C2#2224711yescharacteristic
434C2#2224911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×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
20. #1942C4semidirect
21. #1918C4semidirect
177. #107C42semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#19182
C4×#3892
#109072
#108842
#108802
#108202
#108062
C4×#3362