Finite Group Databasebeta

← #320135

Subgroups — #320135

Order 512.

771 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33277C211yesmaximal, characteristic
2#29178C211yesmaximal, characteristic, complemented
3#28655C211yesmaximal, characteristic
4#28515C211yesmaximal, characteristic, complemented
5#28444C211yesmaximal, characteristic
6C4×#500C211yesmaximal, characteristic, complemented
7#10675C211yesmaximal, characteristic, complemented
8#10673C211yesmaximal, characteristic
9#10309C211yesmaximal, characteristic, complemented
10#10308C211yesmaximal, characteristic
11#9825C211yesmaximal, characteristic, complemented
12#9823C211yesmaximal, characteristic, complemented
13#9822C211yesmaximal, characteristic
14#9820C211yesmaximal, characteristic
15C8×C4⋊C8C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#494C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#479—22no—
22C2×#464—22no—
23#2859C2211yescharacteristic
24#2858C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27#2097—22no—
28#2086—22no—
29#2067—22no—
30#2015—22no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1956C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1942—22no—
46#1934—22no—
47#1934—22no—
48#1925—22no—
49#1918—22no—
50#1845C2211yescharacteristic
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55C4×#107—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C2×C82C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×Q16—22no—
64C22×SD16—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×#124—21no—
71C2×#124—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74#500—22no—
75#500—22no—
76#500—22no—
77#500—41no—
78#494—22no—
79#494—41no—
80#492—22no—
81#492—41no—
82#486—22no—
83#486—41no—
84#479—48no—
85#464—48no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#106—22no—
92C2×C4⋊C8D412yes—
93C2×C4⋊C8D412yes—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×C4×C8D412yes—
103C2×C4×C8D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C43C2311yescharacteristic
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#336D412yes—
143#336D412yes—
144C82—41no—
145C82—41no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C2×Q16—44no—
153C2×Q16—44no—
154C2×Q16—82no—
155C2×SD16—44no—
156C2×SD16—44no—
157C2×SD16—82no—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8#4112yes—
175C4⋊Q8#4112yes—
176#127—21no—
177#127—21no—
178#127—21no—
179#127—21no—
180#125—44no—
181#124—44no—
182#124—82no—
183C4×Q8—44no—
184C4×Q8—44no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—28no—
221#107—44no—
222#107—44no—
223#107—44no—
224#107—44no—
225#106—44no—
226C4⋊C8—24no—
227C4⋊C8D818yes—
228#103—44no—
229#103—44no—
230#103—44no—
231#102—44no—
232#102—44no—
233#102—44no—
234#102—44no—
235#102—44no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—41no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4#12711yescharacteristic
289C22×C4#12711yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4#12711yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295Q16—88no—
296SD16—88no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×D814yes—
311C2×C8C2×D814yes—
312C2×C8#13514yes—
313C2×C8#13514yes—
314C2×C8#13714yes—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—82no—
353C42—82no—
354C42C2×D814yes—
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C42H(5,𝔽2)11yescharacteristic
358C422−1+411yescharacteristic
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23C2×#12711yescharacteristic
367Q8—44no—
368Q8—44no—
369Q8—82no—
370Q8—82no—
371Q8—82no—
372Q8—82no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—82no—
377D4—82no—
378D4—82no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
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×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4#49312yes—
411C2×C4#49312yes—
412C2×C4#52511yescharacteristic
413C2×C4#53711yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#54611yescharacteristic
416C2×C4#54811yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4C22×D812yes—
419C2×C4C2×#13512yes—
420C2×C4C2×#13512yes—
421C2×C4C2×#13712yes—
422C2×C4#57712yes—
423C2×C4#57812yes—
424C8—24no—
425C8—24no—
426C8—24no—
427C8—24no—
428C8—24no—
429C8—42no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22#254011yescharacteristic
438C22C2×#49311yescharacteristic
439C22#314711yescharacteristic
440C22#314711yescharacteristic
441C22#315211yescharacteristic
442C22#315411yescharacteristic
443C22#315611yescharacteristic
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#310812yes—
456C4#324812yes—
457C4#329412yes—
458C4#330012yes—
459C4#335812yes—
460C4#336412yes—
461C2—321no—
462C2#1960311yescharacteristic
463C2#1963011yescharacteristic
464C2#1964211yescharacteristic
465C2#1966911yescharacteristic
466C2#2837711yescharacteristic
467C2#2839711yescharacteristic
468C2#2840711yescharacteristic
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
2. #29178C2semidirect
4. #28515C2semidirect
6. C4×#500C2semidirect
7. #10675C2semidirect
9. #10309C2semidirect
11. #9825C2semidirect
12. #9823C2semidirect
15. C8×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#332772
#291782
#286552
#285152
#284442
C4×#5002
#106752
#106732
#103092
#103082
#98252
#98232
#98222
#98202
C8×C4⋊C82