Finite Group Databasebeta

← #6493940

Subgroups — #6493940

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3147C211yesmaximal, characteristic, complemented
2C2×#2965C211yesmaximal, characteristic, complemented
3#35292C212yesmaximal, complemented
4#33176C212yesmaximal, complemented
5#33175C212yesmaximal, complemented
6#29807C214yesmaximal, complemented
7#28742C214yesmaximal, complemented
8#28487C214yesmaximal, complemented
9#24271C218yesmaximal, complemented
10#21805C212yesmaximal, complemented
11#20816C211yesmaximal, characteristic, complemented
12C22×#135C2214yes—
13C2×#532C2212yes—
14C2×C4×#41C2211yescharacteristic
15#3147—28no—
16C2×#495C2214yes—
17C2×#494C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#492C2212yes—
20C2×#470C2218yes—
21C2×#469C2218yes—
22#2965—28no—
23C2×#446C2212yes—
24C2×#436C2212yes—
25C2×#435C2214yes—
26C2×#431C2211yescharacteristic
27C2×#431C2212yes—
28C2×C4×M16C2212yes—
29C22×C4×C8C2211yescharacteristic
30#2857C2214yes—
31#2856C2214yes—
32#2624—22no—
33#2615—22no—
34#2592C2214yes—
35#2580—22no—
36#2540C2214yes—
37#2535—22no—
38#2414—24no—
39#2407C2218yes—
40#2324—24no—
41#2285C2212yes—
42#2055C22116yes—
43#2000C22116yes—
44#1983C2218yes—
45#1972C2218yes—
46#1971C2218yes—
47#1957C2214yes—
48#1956C2214yes—
49#1844C2212yes—
50#1834C2212yes—
51#1807C2214yes—
52#1788C2218yes—
53C4×#106C2214yes—
54C22×#41D411yescharacteristic
55C22×#41C2314yes—
56C2×#135—216no—
57C2×#135—416no—
58C22×Q16C2314yes—
59C22×SD16C2318yes—
60C22×D8C2314yes—
61C22×M16C2314yes—
62C23×C8C2312yes—
63#532—28no—
64#532—28no—
65C2×C4⋊Q8C2312yes—
66C2×#127C2312yes—
67C2×#124C2314yes—
68C2×C22⋊Q8C2318yes—
69C2×#121—24no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—24no—
73C2×#121C2318yes—
74C4×#41—24no—
75C4×#41—24no—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4D412yes—
81C2×C4×D4D411yescharacteristic
82C2×C4×D4C2312yes—
83C2×C4×D4C2311yescharacteristic
84C2×#117—22no—
85C2×#117D412yes—
86C2×#117D411yescharacteristic
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#495—216no—
90#494—24no—
91#493—24no—
92#492—28no—
93#470—416no—
94#469—416no—
95#446—28no—
96#436—28no—
97#435—216no—
98#431—24no—
99#431—28no—
100C2×#106C2318yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2312yes—
103C2×#103C23116yes—
104C2×#102C23116yes—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2314yes—
107C4×M16—28no—
108C2×#97C2314yes—
109C2×C4×C8D418yescomplemented
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114#399—24no—
115#397—24no—
116#394—28no—
117#393—24no—
118#392—28no—
119#392—24no—
120#391C2318yes—
121#390—28no—
122#390—24no—
123#390—24no—
124#390C2318yes—
125#389—24no—
126#389—24no—
127#388—28no—
128#386—28no—
129#385—28no—
130#384C2314yes—
131#382—22no—
132#380—24no—
133#380—24no—
134#380—22no—
135#380C2314yes—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2314yes—
138#340C2318yes—
139C2×#41—24no—
140C2×#41—28no—
141C2×#41—28no—
142C2×#41—44no—
143C2×#41—416no—
144C22×Q8C2×D411yescharacteristic
145C22×Q8C2414yes—
146C22×D4—28no—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C2×D412yes—
151C22×D4C2×D411yescharacteristic
152C22×D4C2414yes—
153C23×C4—24no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2412yes—
159C23×C4C2411yescharacteristic
160#135—464no—
161C2×Q16—28no—
162C2×Q16—216no—
163C2×SD16—216no—
164C2×SD16—416no—
165C2×D8—28no—
166C2×D8—216no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C22×C8—28no—
171C22×C8C2×D418yes—
172C22×C8C2414yes—
173C22×C8C2412yes—
174C22×C8C2412yes—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177#127—24no—
178#127—24no—
179#124—28no—
180#124—28no—
181C22⋊Q8—416no—
182#121—416no—
183#121—48no—
184#121—48no—
185#121—416no—
186#121—88no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×D4—28no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—48no—
197C4×D4—48no—
198#117—24no—
199#117—28no—
200#117—84no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2418yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4C2×D412yes—
219C2×C22⋊C4C2×D412yes—
220C2×C22⋊C4C2×D412yes—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D414yes—
229C2×C42#4114yes—
230C2×C42C2412yes—
231C2×C42C2411yescharacteristic
232C2×C42C2411yescharacteristic
233#106—216no—
234C4⋊C8—28no—
235C4⋊C8—24no—
236#103—416no—
237#102—416no—
238C22⋊C8—24no—
239C22⋊C8—28no—
240#97—28no—
241C4×C8—24no—
242C4×C8C2×D414yes—
243C4×C8C2×D414yes—
244C4×C8C2×D418yes—
245#95—24no—
246#95—24no—
247#95—28no—
248#95—28no—
249C24—24no—
250C24C22×D412yes—
251C24C22×D411yescharacteristic
252#41—416no—
253#41—432no—
254#41—816no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—216no—
259C2×Q8—42no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—216no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4—416no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—88no—
275C2×D4—88no—
276C2×D4—88no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—48no—
291C22×C4—88no—
292C22×C4#12114yes—
293C22×C4#12114yes—
294C22×C4C22×D412yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2511yescharacteristic
305Q16—416no—
306SD16—432no—
307D8—416no—
308M16—416no—
309C2×C8—24no—
310C2×C8—216no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8C22×D414yes—
315C2×C8C22×D414yes—
316C2×C8H(5,𝔽2)18yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—216no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—88no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—416no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—88no—
334C22⋊C4—88no—
335C22⋊C4—84no—
336C22⋊C4—88no—
337C42—24no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—84no—
342C42#12114yes—
343C42#12114yes—
344C42C22×D412yes—
345C42C22×D412yes—
346C42C2×#4112yes—
347C42C2×#4112yes—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—44no—
357C23—48no—
358C23—48no—
359C23—44no—
360C23—44no—
361C23C2×#12112yes—
362C23C23×D411yescharacteristic
363Q8—44no—
364Q8—48no—
365Q8—48no—
366D4—44no—
367D4—48no—
368D4—48no—
369D4—48no—
370D4—816no—
371D4—88no—
372D4—88no—
373D4—1616no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—816no—
398C2×C4C2×#12112yes—
399C2×C4C2×#12112yes—
400C2×C4C2×#12112yes—
401C2×C4C2×#12112yes—
402C2×C4C2×#12112yes—
403C2×C4#57514yes—
404C2×C4#57714yes—
405C2×C4C23×D411yescharacteristic
406C2×C4C22×#4111yescharacteristic
407C2×C4C2×H(5,𝔽2)14yes—
408C8—24no—
409C8—44no—
410C8#53518yes—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—44no—
419C22—44no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—88no—
424C22—88no—
425C22C22×#12111yescharacteristic
426C22#349812yes—
427C22C2×#57512yes—
428C22C2×#57712yes—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—88no—
437C4#305414yes—
438C4#306014yes—
439C4C2×#53512yes—
440C4#350812yes—
441C2—42no—
442C2—82no—
443C2—164no—
444C2#3527211yescharacteristic
445C2C2×#305411yescharacteristic
446C2C2×#306011yescharacteristic
447C2#6242714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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×#3147C2semidirect
2. C2×#2965C2semidirect
3. #35292C2semidirect
4. #33176C2semidirect
5. #33175C2semidirect
6. #29807C2semidirect
7. #28742C2semidirect
8. #28487C2semidirect
9. #24271C2semidirect
10. #21805C2semidirect
11. #20816C2semidirect
109. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31472
C2×#29652
#352922
#331762
#331752
#298072
#287422
#284872
#242712
#218052
#208162