Finite Group Databasebeta

← #6480103

Subgroups — #6480103

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#33129C214yesmaximal, complemented
3#30291C211yesmaximal, characteristic, complemented
4#30239C211yesmaximal, characteristic
5#30185C212yesmaximal, complemented
6#29572C214yesmaximal, complemented
7#29552C214yesmaximal, complemented
8#29514C214yesmaximal, complemented
9#29493C214yesmaximal, complemented
10#22181C214yesmaximal, complemented
11C2×#1744C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×#466C2214yes—
14C22×C4×C8C2212yes—
15#2861—22no—
16#2860—22no—
17#2860C2214yes—
18#2859C2212yes—
19#2858C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2854—22no—
23#2854C2214yes—
24#2850—22no—
25C4×C4⋊Q8C2212yes—
26C4×#127C2212yes—
27C4×#124—22no—
28C4×#121—22no—
29C4×#121C2214yes—
30#2293—22no—
31#2293C2214yes—
32C2×C43C2211yescharacteristic
33#2099C2214yes—
34#2093C2214yes—
35#2083C2214yes—
36#2071C2214yes—
37#2057C2214yes—
38#2041C2214yes—
39#2023C2214yes—
40#2011C2214yes—
41#1970C2214yes—
42#1970C2214yes—
43#1942C2214yes—
44#1942C2214yes—
45#1934C2218yes—
46#1934C2218yes—
47#1925C2218yes—
48#1918C2218yes—
49#1893C2218yes—
50#1879C2218yes—
51#1862C2218yes—
52#1808C2214yes—
53#1744C2214yes—
54#1744C2214yes—
55C2×#336C2211yescharacteristic
56C2×#336C2212yes—
57C2×#336C2211yescharacteristic
58C22×D8C2314yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×#127C2312yes—
63#529—416no—
64C2×#126—24no—
65C2×#125—22no—
66C2×#124—22no—
67C2×#124—22no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2314yes—
73C2×C4×Q8—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2314yes—
78C2×#117—24no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#466—216no—
87C2×#107C2314yes—
88C2×#107C2314yes—
89C2×#107C2314yes—
90C2×#102C2318yes—
91C2×#102C2318yes—
92C2×#102C2318yes—
93C2×C4×C8—28no—
94C2×C4×C8C2314yes—
95C2×C4×C8C2314yes—
96#391—22no—
97#391—22no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390C2314yes—
105#390C2314yes—
106#390C2314yes—
107#390C2314yes—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#388—24no—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#388—24no—
124#386—24no—
125#386—24no—
126#384C2312yes—
127#384C2312yes—
128#382—22no—
129#380—22no—
130#380C2314yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C43D412yes—
136C43D412yes—
137C43C2312yes—
138C43C2312yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144#336C2318yes—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2414yes—
150C23×C4—24no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×D8—28no—
159C2×D8—44no—
160C2×D8—44no—
161C22×C8—24no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8#4118yes—
167C22×C8C2414yes—
168C22×C8C2414yes—
169C4⋊Q8—44no—
170#127—44no—
171#126—416no—
172#125—48no—
173#124—48no—
174#124—48no—
175#123—416no—
176C22⋊Q8—48no—
177#121—216no—
178#121—48no—
179#121—48no—
180C4×Q8—48no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—48no—
184C4×D4—416no—
185C4×D4—48no—
186#117—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C4⋊C4C2414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—42no—
218C2×C42D818yescomplemented
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2412yes—
232C2×C42C2412yes—
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235#107—28no—
236#107—44no—
237#107—44no—
238#102—216no—
239#102—48no—
240#102—48no—
241C4×C8—28no—
242C4×C8—28no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249C24—24no—
250C24C22×D411yescharacteristic
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×D4—28no—
255C2×D4—24no—
256C2×D4—28no—
257C2×D4—24no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—44no—
266C2×D4—44no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4C2×D814yes—
291C22×C4C2×D814yes—
292C22×C4C2×D814yes—
293C22×C4#13514yes—
294C22×C4C22×D412yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4114yes—
304C22×C4C2511yescharacteristic
305D8—88no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—48no—
312C2×C8—48no—
313C2×C8C2×#4118yes—
314C4⋊C4—28no—
315C4⋊C4—24no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—48no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—24no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—48no—
354C42C2×D818yes—
355C42C2×D818yes—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23#53212yes—
368C23C22×D812yes—
369C23C2×#13512yes—
370C23C23×D411yescharacteristic
371Q8—88no—
372D4—48no—
373D4—48no—
374D4—88no—
375D4—88no—
376D4—816no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—816no—
414C2×C4#53212yes—
415C2×C4C22×D814yes—
416C2×C4C22×D812yes—
417C2×C4C22×D812yes—
418C2×C4C22×D814yes—
419C2×C4C2×#13514yes—
420C2×C4#57512yes—
421C2×C4#57512yes—
422C2×C4C22×#4112yes—
423C2×C4#58511yescharacteristic
424C2×C4#58512yes—
425C2×C4#58511yescharacteristic
426C8—28no—
427C8—44no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22—84no—
438C22—84no—
439C22—84no—
440C22—84no—
441C22#316914yes—
442C22C2×#53211yescharacteristic
443C22#344911yescharacteristic
444C22#344911yescharacteristic
445C22C23×D811yescharacteristic
446C22C22×#13511yescharacteristic
447C22C2×#57511yescharacteristic
448C22C2×#57511yescharacteristic
449C4—24no—
450C4—24no—
451C4—22no—
452C4—22no—
453C4—24no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—42no—
458C4—84no—
459C4—84no—
460C4—84no—
461C4#349112yes—
462C4#349312yes—
463C2—24no—
464C2—164no—
465C2#3511911yescharacteristic
466C2C2×#316911yescharacteristic
467C2#6068811yescharacteristic
468C2#6094011yescharacteristic
469C2#6094011yescharacteristic
470C2#6094511yescharacteristic
471C2#6094811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#30239C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. #33129C2semidirect
3. #30291C2semidirect
5. #30185C2semidirect
6. #29572C2semidirect
7. #29552C2semidirect
8. #29514C2semidirect
9. #29493C2semidirect
10. #22181C2semidirect
11. C2×#1744C2semidirect
218. C2×C42D8semidirect

Maximal subgroups

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

subgroupindex
#366312
#331292
#302912
#302392
#301852
#295722
#295522
#295142
#294932
#221812
C2×#17442