Finite Group Databasebeta

← #6465484

Subgroups — #6465484

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C212yesmaximal, complemented
2C2×#3037C212yesmaximal
3#34070C211yesmaximal, characteristic
4C2×#2327C211yesmaximal, characteristic, complemented
5#28261C214yesmaximal, complemented
6#28260C214yesmaximal, complemented
7#28258C218yesmaximal, complemented
8#23547C214yesmaximal, complemented
9#20792C211yesmaximal, characteristic, complemented
10#20649C212yesmaximal, complemented
11#20647C212yesmaximal, complemented
12C22×#137C2214yes—
13C2×#532C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×C4×#41C2212yes—
16C22×C4×Q8C2212yes—
17#3039—216no—
18#3037—216no—
19C2×#464C2218yes—
20C2×#462C2218yes—
21C2×#462C2214yes—
22C2×#460C2214yes—
23C2×#461C2218yes—
24C2×#423C2212yes—
25C2×#423C2212yes—
26C2×#423C2212yes—
27C2×#417C2214yes—
28C2×#419C2214yes—
29#2328C2212yes—
30#2327—24no—
31#2327C2214yes—
32#2327C2214yes—
33#2327C2212yes—
34#2326C2214yes—
35#2325—24no—
36#2297C2212yes—
37C2×#391C2212yes—
38#2290—24no—
39C2×#387—24no—
40#2285C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#376C2211yescharacteristic
43#2056C22116yes—
44#2055C22116yes—
45#1843C2214yes—
46#1843C2214yes—
47#1805C2218yes—
48#1805C2218yes—
49#1804C2218yes—
50#1803C2218yes—
51#1736C2214yes—
52C22×#41C2312yes—
53C23×Q8C2312yes—
54C2×#137—216no—
55C2×#137—216no—
56C2×#137—48no—
57C22×Q16C2318yes—
58C22×SD16C2318yes—
59C22×M16C2314yes—
60#532—24no—
61#532D418yes—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8D418yescomplemented
65C2×C4⋊Q8C2314yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#124C2312yes—
69C2×#121—22no—
70C4×#41—28no—
71C4×#41—28no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—28no—
75C2×C4×Q8—28no—
76C2×C4×Q8C2314yes—
77C2×C4×Q8C2314yes—
78C2×C4×Q8C2314yes—
79C2×C4×D4—24no—
80C2×C4×D4—22no—
81C2×C4×D4C2314yes—
82C2×#117—22no—
83C2×#117C2312yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2312yes—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#464—232no—
93#462—216no—
94#462—232no—
95#461—232no—
96#460—216no—
97#423—28no—
98#423—28no—
99#423—44no—
100C2×C4⋊C8C2314yes—
101C2×C4⋊C8C2318yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2318yes—
104#419—216no—
105#417—216no—
106C2×#103C23116yes—
107C2×#103C2318yes—
108C2×#102C2318yes—
109#391—24no—
110#391C2318yes—
111#391C2314yes—
112#390C2314yes—
113#388—28no—
114#387—28no—
115#387—216no—
116#387—216no—
117#386—24no—
118#386—28no—
119#385—28no—
120#384C2312yes—
121#382—22no—
122#382C2314yes—
123#382C2312yes—
124#380—22no—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—24no—
127#376D414yes—
128#376C2312yes—
129#376C2312yes—
130C2×#95—22no—
131C2×#95—24no—
132C2×#95D414yes—
133#340C23116yes—
134C2×#41—24no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41C2×D418yes—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8C2×D418yes—
147C22×Q8C2412yes—
148C22×Q8C2414yes—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C2412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160#137—464no—
161C2×Q16—216no—
162C2×Q16—216no—
163C2×Q16—48no—
164C2×SD16—216no—
165C2×SD16—216no—
166C2×SD16—48no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C22×C8C2418yes—
171C4⋊Q8—216no—
172C4⋊Q8—28no—
173C4⋊Q8—22no—
174C4⋊Q8C2×D4116yes—
175C4⋊Q8C2×D414yes—
176#127—22no—
177#127C2×D414yes—
178#124—24no—
179#124C2×D418yes—
180#121—216no—
181C4×Q8—28no—
182C4×Q8—28no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—28no—
187C4×Q8—48no—
188C4×Q8—48no—
189C4×D4—216no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—416no—
193#117—24no—
194#117—24no—
195#117—48no—
196C2×C4⋊C4—216no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—48no—
220C2×C42—48no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D414yes—
225C2×C42C2×D414yes—
226C2×C42#4114yes—
227C2×C42#4114yes—
228C2×C42C2412yes—
229C2×C42C2412yes—
230C2×C42C2411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—28no—
233C4⋊C8—216no—
234C4⋊C8—28no—
235C4⋊C8—48no—
236#103—216no—
237#103—232no—
238#102—216no—
239#95—28no—
240#95—216no—
241#95—28no—
242#95—24no—
243#95C2×D418yes—
244#95C2×D418yes—
245C24—22no—
246C24C22×D411yescharacteristic
247#41—216no—
248#41—216no—
249#41—48no—
250#41—48no—
251C2×Q8—28no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—216no—
255C2×Q8—216no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8C22×D414yes—
269C2×Q8C22×D418yes—
270C2×D4—28no—
271C2×D4—28no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—84no—
280C2×D4C22×D414yes—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—48no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—88no—
299C22×C4#12118yes—
300C22×C4#12118yes—
301C22×C4C22×D414yes—
302C22×C4C22×D412yes—
303C22×C4C22×D414yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4H(5,𝔽2)12yes—
311C22×C42−1+412yes—
312C22×C4C2511yescharacteristic
313Q16—432no—
314SD16—432no—
315M16—416no—
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—28no—
319C4⋊C4—232no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—84no—
330C22⋊C4—216no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—88no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—44no—
338C42—48no—
339C42—48no—
340C42—48no—
341C42#12118yes—
342C42#12118yes—
343C42C22×D414yes—
344C42C22×D412yes—
345C42C22×D412yes—
346C42C2×#4114yes—
347C42C2×#4112yes—
348C42C2×#4112yes—
349C23—24no—
350C23—24no—
351C23—48no—
352C23—42no—
353C23—42no—
354C23C2×#12112yes—
355C23C2×#12112yes—
356C23#53712yes—
357C23C23×D411yescharacteristic
358Q8—28no—
359Q8—24no—
360Q8—28no—
361Q8—24no—
362Q8—44no—
363Q8—42no—
364Q8—44no—
365Q8—42no—
366Q8—416no—
367D4—24no—
368D4—24no—
369D4—416no—
370D4—42no—
371D4—42no—
372D4—88no—
373C2×C4—24no—
374C2×C4—28no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—24no—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—416no—
387C2×C4—416no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—816no—
393C2×C4—84no—
394C2×C4—88no—
395C2×C4C2×#12114yes—
396C2×C4C2×#12114yes—
397C2×C4C2×#12118yes—
398C2×C4C2×#12112yes—
399C2×C4C2×#12112yes—
400C2×C4#53712yes—
401C2×C4D4214yes—
402C2×C4#54714yes—
403C2×C4#54714yes—
404C2×C4#55514yes—
405C2×C4C23×D411yescharacteristic
406C2×C4C23×D411yescharacteristic
407C2×C4C22×#4111yescharacteristic
408C2×C4C22×#4111yescharacteristic
409C2×C4C2×H(5,𝔽2)11yescharacteristic
410C2×C4C2×2−1+411yescharacteristic
411C8—48no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—48no—
416C22—44no—
417C22—84no—
418C22—84no—
419C22#253714yes—
420C22C22×#12111yescharacteristic
421C22C22×#12111yescharacteristic
422C22C2×#53711yescharacteristic
423C22#358614yes—
424C4—28no—
425C4—24no—
426C4—24no—
427C4—44no—
428C4—48no—
429C4—44no—
430C4—84no—
431C4—84no—
432C4#253714yes—
433C4C2×D4211yescharacteristic
434C4C2×#54711yescharacteristic
435C4C2×#54711yescharacteristic
436C4C2×#55511yescharacteristic
437C2—24no—
438C2—84no—
439C2C2×#253711yescharacteristic
440C2#6042412yes—
441C2#6042712yes—
442C2#6092112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3039C2semidirect
4. C2×#2327C2semidirect
5. #28261C2semidirect
6. #28260C2semidirect
7. #28258C2semidirect
8. #23547C2semidirect
9. #20792C2semidirect
10. #20649C2semidirect
11. #20647C2semidirect
64. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
C2×#30372
#340702
C2×#23272
#282612
#282602
#282582
#235472
#207922
#206492
#206472