Finite Group Databasebeta

← #6483130

Subgroups — #6483130

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic, complemented
2C2×#2931C211yesmaximal, characteristic, complemented
3#37639C212yesmaximal, complemented
4#32933C214yesmaximal, complemented
5#29738C214yesmaximal, complemented
6#28705C212yesmaximal, complemented
7#28605C214yesmaximal, complemented
8#28485C212yesmaximal, complemented
9#28227C214yesmaximal, complemented
10#28149C214yesmaximal, complemented
11#21893C212yesmaximal, complemented
12#20817C211yesmaximal, characteristic, complemented
13C22×#131C2212yes—
14C2×#532C2212yes—
15C2×#532C2211yescharacteristic
16#3169—28no—
17C2×#500C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#486C2212yes—
20C2×#466C2212yes—
21C2×#466C2212yes—
22C2×#466C2218yes—
23C2×C4×D8C2212yes—
24C2×C4×D8C2212yes—
25#2931—28no—
26C2×#436C2212yes—
27C2×#436C2212yes—
28C2×C8×D4C2212yes—
29C2×C8×D4C2212yes—
30C2×#423C2211yescharacteristic
31C2×#408C2212yes—
32C22×C4×C8C2211yescharacteristic
33#2819—24no—
34#2619—24no—
35#2615—24no—
36#2615C2218yes—
37#2563—22no—
38#2563C2214yes—
39#2541—22no—
40#2541C2214yes—
41#2537C2218yes—
42#2285C2212yes—
43#2076C2218yes—
44#2033C22116yes—
45#1999C2218yes—
46#1987C2214yes—
47#1982C2214yes—
48#1895C2218yes—
49#1895C22116yes—
50#1845C2212yes—
51#1834C2212yes—
52#1808C2218yes—
53#1808C2212yes—
54#1808C2212yes—
55#1802C2218yes—
56C4×#107C2212yes—
57C4×#107C2212yes—
58C22×#41—22no—
59C22×#41C2312yes—
60C22×D8C2312yes—
61C22×D8C2312yes—
62C2×#131—28no—
63C2×#131—48no—
64C22×M16C2312yes—
65C22×M16C2314yes—
66C23×C8C2314yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69#532—24no—
70#532—28no—
71#532—24no—
72#532—28no—
73C2×C4⋊Q8D411yescharacteristic
74C2×C4⋊Q8C2312yes—
75C2×#127—24no—
76C2×#127D411yescharacteristic
77C2×#127C2312yes—
78C2×#124—22no—
79C2×#124D412yes—
80C2×#121—22no—
81C2×#121—28no—
82C2×#121D412yes—
83C2×#121D412yes—
84C2×#121C2314yes—
85C2×#121C23116yes—
86C2×C4×D4—22no—
87C2×C4×D4C2314yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C2×#117—22no—
91C22×C42C2311yescharacteristic
92#500—24no—
93#493—24no—
94#486—28no—
95#466—28no—
96#466—28no—
97#466—416no—
98C4×D8—28no—
99C4×D8—28no—
100#436—28no—
101#436—28no—
102C8×D4—28no—
103C8×D4—28no—
104C2×#107C2312yes—
105C2×#107C2314yes—
106C2×#107C2312yes—
107C2×#107C2314yes—
108#423—24no—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C23116yes—
114#408—28no—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C4×C8D418yescomplemented
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#394—216no—
123#393—28no—
124#393—28no—
125#392—28no—
126#392—216no—
127#392—28no—
128#390—24no—
129#390—24no—
130#384C2314yes—
131#382—24no—
132#382—22no—
133#380—22no—
134#380—28no—
135#380C23116yes—
136#380C2314yes—
137C4×C4⋊C4—22no—
138C4×C4⋊C4C2314yes—
139#340C23116yes—
140#340C2314yes—
141#340C2314yes—
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—44no—
145C2×#41—44no—
146C2×#41—48no—
147C2×#41—88no—
148C22×Q8—22no—
149C22×Q8C2×D412yes—
150C22×D4—24no—
151C22×D4—216no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4C2×D412yes—
155C22×D4C2×D412yes—
156C22×D4C2×D414yes—
157C22×D4C2×D412yes—
158C22×D4C2414yes—
159C23×C4—24no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2414yes—
163C23×C4C2411yescharacteristic
164C2×D8—24no—
165C2×D8—24no—
166C2×D8—28no—
167C2×D8—44no—
168#131—432no—
169C2×M16—24no—
170C2×M16—28no—
171C2×M16—48no—
172C2×M16—44no—
173C22×C8—24no—
174C22×C8—28no—
175C22×C8—216no—
176C22×C8—42no—
177C22×C8C2×D414yes—
178C22×C8C2×D414yes—
179C22×C8C2414yes—
180C22×C8C2411yescharacteristic
181C22×C8C2411yescharacteristic
182C22×C8C2411yescharacteristic
183C22×C8C2411yescharacteristic
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187#127—24no—
188#127—24no—
189#127—24no—
190#127—88no—
191#124—28no—
192#124—48no—
193#121—28no—
194#121—28no—
195#121—28no—
196#121—28no—
197#121—48no—
198#121—416no—
199#121—416no—
200#121—816no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—28no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—28no—
207C4×D4—48no—
208#117—84no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2414yes—
215C2×C4⋊C4C2418yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—216no—
220C2×C22⋊C4—216no—
221C2×C22⋊C4C2×D414yes—
222C2×C22⋊C4C2×D414yes—
223C2×C42—24no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D414yes—
227C2×C42#4114yes—
228C2×C42C2411yescharacteristic
229C2×C42C2411yescharacteristic
230#107—24no—
231#107—24no—
232#107—28no—
233#107—44no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236#102—28no—
237#102—28no—
238#102—416no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241C4×C8C2×D414yes—
242C4×C8C2×D414yes—
243C4×C8C2×D414yes—
244C4×C8C2×D414yes—
245#95—24no—
246#95—24no—
247#95—216no—
248C24—24no—
249C24C22×D414yes—
250C24C22×D411yescharacteristic
251#41—48no—
252#41—48no—
253#41—416no—
254#41—832no—
255C2×Q8—24no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—84no—
259C2×D4—28no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—28no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—84no—
278C2×D4—88no—
279C2×D4—816no—
280C2×D4—816no—
281C2×D4—816no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—48no—
294C22×C4—416no—
295C22×C4#12114yes—
296C22×C4#12114yes—
297C22×C4C22×D414yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2511yescharacteristic
306D8—48no—
307D8—48no—
308M16—416no—
309M16—48no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—28no—
314C2×C8—24no—
315C2×C8—28no—
316C2×C8—44no—
317C2×C8—416no—
318C2×C8C22×D412yes—
319C2×C8C22×D412yes—
320C2×C8C22×D412yes—
321C2×C8C22×D412yes—
322C2×C8H(5,𝔽2)14yes—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—84no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—816no—
337C22⋊C4—84no—
338C22⋊C4—816no—
339C42—48no—
340C42#12114yes—
341C42#12114yes—
342C42C22×D412yes—
343C42C22×D412yes—
344C42C2×#4112yes—
345C42C2×#4112yes—
346C23—22no—
347C23—21no—
348C23—22no—
349C23—21no—
350C23—24no—
351C23—28no—
352C23—28no—
353C23—24no—
354C23—24no—
355C23—44no—
356C23—48no—
357C23—48no—
358C23—44no—
359C23—44no—
360C23C2×#12112yes—
361C23C23×D411yescharacteristic
362Q8—48no—
363Q8—88no—
364D4—48no—
365D4—416no—
366D4—44no—
367D4—44no—
368D4—48no—
369D4—48no—
370D4—48no—
371D4—88no—
372D4—816no—
373D4—88no—
374D4—1632no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—816no—
395C2×C4C2×#12112yes—
396C2×C4C2×#12112yes—
397C2×C4C2×#12112yes—
398C2×C4C2×#12112yes—
399C2×C4C2×#12112yes—
400C2×C4#57614yes—
401C2×C4#57612yes—
402C2×C4#57812yes—
403C2×C4C23×D411yescharacteristic
404C2×C4C22×#4111yescharacteristic
405C2×C4C2×H(5,𝔽2)12yes—
406C2×C4#58512yes—
407C8—22no—
408C8—22no—
409C8—22no—
410C8—44no—
411C8#53514yes—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—48no—
421C22—48no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—88no—
426C22—88no—
427C22C22×#12111yescharacteristic
428C22#349612yes—
429C22C2×#57612yes—
430C22C2×#57611yescharacteristic
431C22C2×#57811yescharacteristic
432C4—22no—
433C4—22no—
434C4—22no—
435C4—44no—
436C4—84no—
437C4—88no—
438C4#305814yes—
439C4#306114yes—
440C4C2×#53511yescharacteristic
441C4#344911yescharacteristic
442C4#350612yes—
443C2—42no—
444C2—84no—
445C2—164no—
446C2#3516511yescharacteristic
447C2C2×#305811yescharacteristic
448C2C2×#306111yescharacteristic
449C2#6232812yes—
450C2#6233212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3169C2semidirect
2. C2×#2931C2semidirect
3. #37639C2semidirect
4. #32933C2semidirect
5. #29738C2semidirect
6. #28705C2semidirect
7. #28605C2semidirect
8. #28485C2semidirect
9. #28227C2semidirect
10. #28149C2semidirect
11. #21893C2semidirect
12. #20817C2semidirect
117. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#29312
#376392
#329332
#297382
#287052
#286052
#284852
#282272
#281492
#218932
#208172