Finite Group Databasebeta

← #361078

Subgroups — #361078

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1918C212yesmaximal
2C2×C4×#107C211yesmaximal, characteristic
3#13918C211yesmaximal, characteristic
4#13821C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11219C212yesmaximal
7#10907C212yesmaximal
8#10862C211yesmaximal, characteristic
9#10806C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382—21no—
19C2×C4×C4⋊C4—21no—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1918—28no—
24C4×#107C418yescomplemented
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1442—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42—22no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—42no—
63C2×#107C2×C414yes—
64C2×#107C2×C414yes—
65C2×#107D414yes—
66C2×#107Q814yes—
67C2×C4×C8—24no—
68C2×C4×C8C2×C414yes—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—42no—
73#382—44no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—42no—
78C4×C4⋊C4—44no—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2×C414yes—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95D411yescharacteristic
93C2×#95D412yes—
94C2×#95D411yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340C2×C414yes—
101#340C2×C414yes—
102#336—24no—
103#336—24no—
104#336—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C22×C412yes—
129C22×C8C22×C412yes—
130C22×C8C2×D412yes—
131C22×C8C2×Q812yes—
132C22×C8#4114yes—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×Q812yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4#4114yes—
170C2×C4⋊C4#4114yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—82no—
188C2×C42C22×C412yes—
189C2×C42#4112yes—
190#107—28no—
191#107—48no—
192#107C4⋊C4116yescomplemented
193C4×C8—28no—
194#95—28no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—42no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—42no—
203#95—44no—
204#95—42no—
205#95—81no—
206#95—81no—
207#95D814yes—
208C24C22×D411yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4C2×D814yes—
269C22×C4C2×D812yes—
270C22×C4#13512yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q811yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C42−1+412yes—
280C22×C42−1+411yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8C2×C4⋊C414yes—
297C2×C8#11714yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—416no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—88no—
319C4⋊C4C2×C4⋊C414yes—
320C4⋊C4C2×C4⋊C414yes—
321C4⋊C4#11714yes—
322C42—22no—
323C42—22no—
324C42—24no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—48no—
330C42—48no—
331C42—88no—
332C23C2×C4×D411yescharacteristic
333C23#52512yes—
334C23D4211yescharacteristic
335C23#54712yes—
336C23D4×Q811yescharacteristic
337C23C22×D811yescharacteristic
338C23C2×#13511yescharacteristic
339C23#57512yes—
340C23#57611yescharacteristic
341C23#57812yes—
342C23#57811yescharacteristic
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4C4×D814yes—
405C2×C4C22×C4⋊C411yescharacteristic
406C2×C4C2×#11711yescharacteristic
407C2×C4C2×#11711yescharacteristic
408C2×C4C4×#4111yescharacteristic
409C2×C4#51911yescharacteristic
410C2×C4#52011yescharacteristic
411C2×C4#52912yes—
412C2×C4#53212yes—
413C2×C4#53312yes—
414C2×C4#54112yes—
415C2×C4#54212yes—
416C2×C4#54711yescharacteristic
417C2×C4#54812yes—
418C2×C4#55112yes—
419C2×C4#55212yes—
420C2×C4#55212yes—
421C2×C4#55411yescharacteristic
422C2×C4#55712yes—
423C8—24no—
424C8—24no—
425C8—82no—
426C22#229011yescharacteristic
427C22#234411yescharacteristic
428C22D4×C4⋊C411yescharacteristic
429C22#246112yes—
430C22#246212yes—
431C22C2×C4×D811yescharacteristic
432C22#294311yescharacteristic
433C22#295011yescharacteristic
434C22#295111yescharacteristic
435C22#310012yes—
436C22#310212yes—
437C22#312412yes—
438C22#312712yes—
439C22#327712yes—
440C22D4×D811yescharacteristic
441C22#328311yescharacteristic
442C22#333412yes—
443C22#333612yes—
444C22#334712yes—
445C22#334912yes—
446C22Q8×D811yescharacteristic
447C22#338011yescharacteristic
448C22#339012yes—
449C4—24no—
450C4—42no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4—82no—
456C4—84no—
457C4—82no—
458C4—162no—
459C4#228511yescharacteristic
460C4#232011yescharacteristic
461C4#234311yescharacteristic
462C4#235111yescharacteristic
463C4#251912yes—
464C4#252012yes—
465C2#1388911yescharacteristic
466C2#2111311yescharacteristic
467C2#2113511yescharacteristic
468C2#2261111yescharacteristic
469C2#2261611yescharacteristic
470C2#2267511yescharacteristic
471C2C4⋊C4×D811yescharacteristic
472C2#2678412yes—
473C2#2679212yes—
474C2#2685612yes—
475C2#2686412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. C4×#107C4semidirect
192. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19182
C2×C4×#1072
#139182
#138212
#113282
#112192
#109072
#108622
#108062
#107972