Finite Group Databasebeta

← #352672

Subgroups — #352672

Order 512.

701 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33287C211yesmaximal, characteristic, complemented
2#33286C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic, complemented
4#33226C211yesmaximal, characteristic
5#30307C211yesmaximal, characteristic, complemented
6#30278C211yesmaximal, characteristic
7#9834C211yesmaximal, characteristic, complemented
8#9830C211yesmaximal, characteristic
9#9828C211yesmaximal, characteristic, complemented
10#9827C211yesmaximal, characteristic
11#9823C211yesmaximal, characteristic, complemented
12#9822C211yesmaximal, characteristic
13#9822C211yesmaximal, characteristic
14#9819C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#488—22no—
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2858C2211yescharacteristic
25#2092—22no—
26#2092—22no—
27#2085—22no—
28#2059—22no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1958—22no—
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1956C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1936—22no—
47#1936—22no—
48#1935—22no—
49#1935—22no—
50#1933—22no—
51#1933—22no—
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×C82C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×#125—21no—
68C2×#125—21no—
69C2×#124—21no—
70C2×#124—21no—
71#500—22no—
72#500—22no—
73#498—22no—
74#498—22no—
75#488—48no—
76#486—22no—
77#486—22no—
78#486—22no—
79#486—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#97D412yes—
97C2×#97D412yes—
98C2×C4×C8D412yes—
99C2×C4×C8D412yes—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#391—21no—
108#391—21no—
109#391—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#382—21no—
129#382—21no—
130C43C2311yescharacteristic
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#336D412yes—
140#336D412yes—
141#322—22no—
142#322—22no—
143C82—22no—
144C82—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169#127—21no—
170#127—21no—
171#127—21no—
172#127—21no—
173#125—44no—
174#125—82no—
175#124—44no—
176#124—44no—
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—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210#107—44no—
211#107—82no—
212#106—44no—
213#106—82no—
214#103—44no—
215#103—44no—
216#103—44no—
217#103—44no—
218#103—44no—
219#103—44no—
220#102—44no—
221#102—44no—
222#102—44no—
223#102—44no—
224#102—44no—
225#102—44no—
226#97—24no—
227#97—24no—
228C4×C8—24no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—42no—
238C4×C8#4112yes—
239C4×C8#4112yes—
240C4×C8#4112yes—
241C4×C8#4112yes—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4#12711yescharacteristic
283C22×C4#12711yescharacteristic
284C22×C4#12711yescharacteristic
285C22×C4#12711yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8#12414yes—
303C2×C8#12414yes—
304C2×C8#13514yes—
305C2×C8#13514yes—
306C2×C8AGL(1,ℤ/8)14yes—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23C2×#12711yescharacteristic
357Q8—82no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362Q8—82no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12411yescharacteristic
404C2×C4#53211yescharacteristic
405C2×C4#53211yescharacteristic
406C2×C4#53211yescharacteristic
407C2×C4#53211yescharacteristic
408C2×C4C2×#13512yes—
409C2×C4C2×#13512yes—
410C2×C4C2×AGL(1,ℤ/8)12yes—
411C2×C4#57512yes—
412C2×C4#57712yes—
413C2×C4#57812yes—
414C8—24no—
415C8—24no—
416C8—24no—
417C8—24no—
418C8—24no—
419C8—42no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22#285811yescharacteristic
428C22#314711yescharacteristic
429C22#314711yescharacteristic
430C22#315111yescharacteristic
431C22#315311yescharacteristic
432C22#315411yescharacteristic
433C22#315611yescharacteristic
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4#313712yes—
446C4#313712yes—
447C4#316812yes—
448C4#316812yes—
449C4#317112yes—
450C4#318012yes—
451C2—321no—
452C2#1964111yescharacteristic
453C2#1964311yescharacteristic
454C2#1964511yescharacteristic
455C2#1965011yescharacteristic
456C2#3322911yescharacteristic
457C2#3324011yescharacteristic
458C2#3325111yescharacteristic
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. #33287C2semidirect
3. #33278C2semidirect
5. #30307C2semidirect
7. #9834C2semidirect
9. #9828C2semidirect
11. #9823C2semidirect
14. #9819C2semidirect
15. C8×#97C2semidirect

Maximal subgroups

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

subgroupindex
#332872
#332862
#332782
#332262
#303072
#302782
#98342
#98302
#98282
#98272
#98232
#98222
#98192
C8×#972