Finite Group Databasebeta

← #6494078

Subgroups — #6494078

Order 512.

1996 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic, complemented
2C2×C8×#41C211yesmaximal, characteristic
3#35297C212yesmaximal
4#33209C212yesmaximal, complemented
5#33172C212yesmaximal, complemented
6#29845C214yesmaximal, complemented
7#28268C214yesmaximal, complemented
8#28226C214yesmaximal, complemented
9#24779C218yesmaximal, complemented
10#21864C212yesmaximal, complemented
11#20797C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×C4×#41C2211yescharacteristic
14#3170—28no—
15C2×#500C2211yescharacteristic
16C2×#494C2211yescharacteristic
17C2×#487C2212yes—
18C2×#467C2214yes—
19C2×#467C2218yes—
20C8×#41—28no—
21C2×C8×Q8C2212yes—
22C2×C4×Q16C2214yes—
23C2×C8×D4C2212yes—
24C2×C8×D4C2214yes—
25C2×#431C2211yescharacteristic
26C2×#431C2212yes—
27C2×#427C2214yes—
28C22×C4×C8C2212yes—
29C22×C4×C8C2211yescharacteristic
30#2857C2214yes—
31#2857C2214yes—
32#2625—22no—
33#2616—22no—
34#2584—22no—
35#2581C2214yes—
36#2562—22no—
37#2536C2214yes—
38#2485—24no—
39#2415C2218yes—
40#2323—24no—
41#2285C2212yes—
42#2066C22116yes—
43#2035C22116yes—
44#1990C2214yes—
45#1969C2218yes—
46#1956C2214yes—
47#1942C2218yes—
48#1873C2218yes—
49#1846C2214yes—
50#1845C2212yes—
51#1844C2212yes—
52#1808C2218yes—
53#1805C2218yes—
54C22×#41D411yescharacteristic
55C22×Q16C2314yes—
56C23×C8C2314yes—
57C23×C8C2312yes—
58#533—28no—
59#533—28no—
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×#125—22no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2314yes—
66C2×C22⋊Q8C2318yes—
67C4×#41—24no—
68C4×#41—24no—
69C2×C4×Q8—22no—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8C2314yes—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D412yes—
75C2×C4×D4C2312yes—
76C2×C4×D4C2311yescharacteristic
77C2×#117—22no—
78C2×#117D412yes—
79C2×#117D411yescharacteristic
80C2×#117C2314yes—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#500—24no—
84#494—24no—
85#487—28no—
86#467—216no—
87#467—416no—
88C8×Q8—28no—
89C4×Q16—216no—
90C8×D4—216no—
91C8×D4—28no—
92#431—24no—
93#431—28no—
94#427—48no—
95C2×#107C2314yes—
96C2×#107C2318yes—
97C2×#106C2318yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2314yes—
100C2×#103C2318yes—
101C2×#103C23116yes—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2314yes—
104C2×C4×C8—28no—
105C2×C4×C8D418yes—
106C2×C4×C8C2312yes—
107C2×C4×C8C2314yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2312yes—
111#400—24no—
112#398—24no—
113#397—28no—
114#395—24no—
115#394—24no—
116#394—28no—
117#391—24no—
118#391C2318yes—
119#389—28no—
120#389—24no—
121#389—24no—
122#389—24no—
123#389C2318yes—
124#388—28no—
125#388—28no—
126#387—28no—
127#384C2314yes—
128#382C2314yes—
129#380—24no—
130#380—24no—
131#380—24no—
132#380—22no—
133#380—22no—
134#380C2318yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137#340C23116yes—
138C2×#41—24no—
139C2×#41—28no—
140C22×Q8—22no—
141C22×Q8C2×D411yescharacteristic
142C22×Q8C2414yes—
143C22×D4C2×D411yescharacteristic
144C22×D4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2412yes—
150C23×C4C2411yescharacteristic
151C2×Q16—28no—
152C2×Q16—28no—
153C2×Q16—28no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—28no—
157C22×C8—28no—
158C22×C8—28no—
159C22×C8C2×D418yes—
160C22×C8C2414yes—
161C22×C8C2412yes—
162C22×C8C2412yes—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167#125—48no—
168C22⋊Q8—28no—
169C22⋊Q8—28no—
170C22⋊Q8—48no—
171C22⋊Q8—416no—
172C22⋊Q8—416no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—48no—
179C4×D4—28no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185#117—24no—
186#117—28no—
187#117—48no—
188#117—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2418yes—
203C2×C4⋊C4C2414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4C2×D412yes—
209C2×C22⋊C4C2×D412yes—
210C2×C22⋊C4C2×D412yes—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D414yes—
220C2×C42#4114yes—
221C2×C42C2412yes—
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—28no—
225#107—216no—
226#106—48no—
227C4⋊C8—28no—
228C4⋊C8—24no—
229#103—216no—
230#103—416no—
231C22⋊C8—24no—
232C22⋊C8—28no—
233C4×C8—28no—
234C4×C8—24no—
235C4×C8C2×D414yes—
236C4×C8C2×D414yes—
237C4×C8C2×D418yes—
238#95—24no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244C24C22×D411yescharacteristic
245C24C22×D412yes—
246#41—232no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—28no—
252C2×Q8—28no—
253C2×Q8—44no—
254C2×Q8—48no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4#13518yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4C2511yescharacteristic
290Q16—416no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—216no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—416no—
298C2×C8C22×D414yes—
299C2×C8C22×D414yes—
300C2×C8H(5,𝔽2)14yes—
301C2×C82−1+414yes—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—216no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—416no—
316C4⋊C4—48no—
317C4⋊C4—88no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—48no—
322C22⋊C4—416no—
323C22⋊C4—84no—
324C22⋊C4—88no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—48no—
330C42—84no—
331C42#12114yes—
332C42#12114yes—
333C42C22×D412yes—
334C42C22×D412yes—
335C42C2×#4112yes—
336C42C2×#4112yes—
337C23—22no—
338C23—21no—
339C23—22no—
340C23—22no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23C2×#12112yes—
348C23C2×#13514yes—
349C23C23×D411yescharacteristic
350Q8—28no—
351Q8—48no—
352Q8—48no—
353Q8—88no—
354D4—28no—
355D4—216no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—216no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4C2×#12112yes—
385C2×C4C2×#12112yes—
386C2×C4C2×#12112yes—
387C2×C4C2×#12112yes—
388C2×C4C2×#12112yes—
389C2×C4C2×#13514yes—
390C2×C4C2×#13514yes—
391C2×C4#57814yes—
392C2×C4C23×D411yescharacteristic
393C2×C4C22×#4111yescharacteristic
394C2×C4C2×H(5,𝔽2)12yes—
395C2×C4C2×2−1+412yes—
396C8—24no—
397C8—44no—
398C8#53718yes—
399C22—24no—
400C22—22no—
401C22—24no—
402C22—22no—
403C22—22no—
404C22—24no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22C22×#12111yescharacteristic
412C22#349811yescharacteristic
413C22#349911yescharacteristic
414C22C22×#13512yes—
415C22C2×#57812yes—
416C4—22no—
417C4—24no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—84no—
423C4—84no—
424C4—88no—
425C4#305014yes—
426C4#305914yes—
427C4C2×#53712yes—
428C4#351112yes—
429C2—44no—
430C2—42no—
431C2#3527311yescharacteristic
432C2C2×#305011yescharacteristic
433C2C2×#305911yescharacteristic
434C2#6241812yes—
435C2#6247312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3170C2semidirect
4. #33209C2semidirect
5. #33172C2semidirect
6. #29845C2semidirect
7. #28268C2semidirect
8. #28226C2semidirect
9. #24779C2semidirect
10. #21864C2semidirect
11. #20797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×C8×#412
#352972
#332092
#331722
#298452
#282682
#282262
#247792
#218642
#207972