Finite Group Databasebeta

← #323455

Subgroups — #323455

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic, complemented
2#29190C211yesmaximal, characteristic
3#29179C211yesmaximal, characteristic, complemented
4#29141C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic
6#21646C211yesmaximal, characteristic
7#10679C211yesmaximal, characteristic, complemented
8#10678C211yesmaximal, characteristic
9#10675C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic
11#9838C211yesmaximal, characteristic
12#9822C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8816C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#474—22no—
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2093—22no—
23#2087—22no—
24#2065—22no—
25#2024—22no—
26#1993—22no—
27#1992—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1936—22no—
39#1921—22no—
40#1864—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1803—22no—
55C4×C4⋊C8C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57#1762—22no—
58C4×#97C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73#486—22no—
74#486Q814yescomplemented
75#474—48no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D412yes—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#97D412yes—
91C2×#97D412yes—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#382—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C43C2311yescharacteristic
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#336D412yes—
134#336D412yes—
135#322—22no—
136C82—22no—
137C22×Q8—21no—
138C22×Q8—41no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C23×C4—41no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C4⋊Q8—82no—
155C4⋊Q8C2×Q812yes—
156#127—21no—
157#127—21no—
158#127—21no—
159#127C2×Q812yes—
160#125—44no—
161C4×Q8—44no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—82no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2411yescharacteristic
202#107—44no—
203#107—44no—
204#107—44no—
205#106—82no—
206C4⋊C8—24no—
207C4⋊C8—42no—
208#103—82no—
209#103—82no—
210#102—28no—
211#102—44no—
212#102—44no—
213#102—44no—
214#102—44no—
215#102—82no—
216#97—24no—
217#97—42no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8C2×Q812yes—
229C4×C8#4112yes—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234C24—41no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#12711yescharacteristic
265C22×C4#12711yescharacteristic
266C22×C4#12711yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8C22⋊Q814yes—
288C2×C8AGL(1,ℤ/8)14yes—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—82no—
326C42#13514yes—
327C42C22×Q811yescharacteristic
328C42C2×#4111yescharacteristic
329C422−1+411yescharacteristic
330C422−1+411yescharacteristic
331C23—42no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C2×#12711yescharacteristic
339Q8—44no—
340Q8—82no—
341Q8—164no—
342D4—44no—
343D4—44no—
344D4—44no—
345D4—82no—
346D4—82no—
347D4—82no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—164no—
380C2×C4#49512yes—
381C2×C4#49512yes—
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4#53611yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4#54711yescharacteristic
386C2×C4D4×Q811yescharacteristic
387C2×C4D4×Q811yescharacteristic
388C2×C4C2×#13512yes—
389C2×C4C2×AGL(1,ℤ/8)12yes—
390C2×C4#57512yes—
391C2×C4#57512yes—
392C2×C4#57612yes—
393C2×C4#57812yes—
394C8—24no—
395C8—24no—
396C8—42no—
397C8—42no—
398C8—42no—
399C8—42no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22#254211yescharacteristic
408C22C2×#49511yescharacteristic
409C22#315111yescharacteristic
410C22#315311yescharacteristic
411C22#315311yescharacteristic
412C22#315511yescharacteristic
413C22#315611yescharacteristic
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#307612yes—
426C4#323312yes—
427C4#334312yes—
428C4#334712yes—
429C4#338612yes—
430C4#338912yes—
431C2—162no—
432C2#1962811yescharacteristic
433C2#1963511yescharacteristic
434C2#1973011yescharacteristic
435C2#1975011yescharacteristic
436C2#2851011yescharacteristic
437C2#2854411yescharacteristic
438C2#2854511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33297C2semidirect
3. #29179C2semidirect
4. #29141C2semidirect
7. #10679C2semidirect
9. #10675C2semidirect
13. #8821C2semidirect
14. #8816C2semidirect
15. #8262C2semidirect
74. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#291902
#291792
#291412
#285082
#216462
#106792
#106782
#106752
#106712
#98382
#98222
#88212
#88162
#82622