Finite Group Databasebeta

← #320910

Subgroups — #320910

Order 512.

730 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#30205C211yesmaximal, characteristic, complemented
2#29218C211yesmaximal, characteristic, complemented
3#28718C211yesmaximal, characteristic
4#28616C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic
6#22219C211yesmaximal, characteristic
7#10684C211yesmaximal, characteristic, complemented
8#10675C211yesmaximal, characteristic, complemented
9#10311C211yesmaximal, characteristic
10#10306C211yesmaximal, characteristic
11#9838C211yesmaximal, characteristic
12#9833C211yesmaximal, characteristic, complemented
13#9825C211yesmaximal, characteristic, complemented
14#9822C211yesmaximal, characteristic
15#8264C211yesmaximal, characteristic, complemented
16C2×#494C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#486C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#474—22no—
22#2868C2211yescharacteristic
23#2862C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26#2100—22no—
27#2093—22no—
28#2073—22no—
29#2072—22no—
30#2056—22no—
31#2024—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1987—22no—
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1926—22no—
48#1878—22no—
49#1864—22no—
50#1845C2211yescharacteristic
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1804—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×Q16—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#494—22no—
76#494—41no—
77#487—22no—
78#487—41no—
79#486—22no—
80#486—22no—
81#486—22no—
82#486—22no—
83#486—41no—
84#486—41no—
85#474—48no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8D412yes—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#382—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C43C2311yescharacteristic
129#340—22no—
130#340—22no—
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#336D412yes—
142#336D412yes—
143#322—41no—
144C82—41no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—41no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C23×C4—41no—
153C2×Q16—44no—
154C2×Q16—44no—
155C2×Q16—82no—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
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—44no—
174C4⋊Q8—44no—
175C4⋊Q8—82no—
176C4⋊Q8#4112yes—
177#127—21no—
178#127—21no—
179#127—21no—
180#127#4112yes—
181C4×Q8—82no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—82no—
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×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2411yescharacteristic
218#107—44no—
219#107—44no—
220#106—82no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223#103—44no—
224#103—82no—
225#103—82no—
226#103—82no—
227#102—28no—
228#102—44no—
229#102—44no—
230#102—44no—
231#102—44no—
232#102—82no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4#12711yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296Q16—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×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8AGL(1,ℤ/8)14yes—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
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—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—82no—
352C42#13514yes—
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C42H(5,𝔽2)11yescharacteristic
356C422−1+411yescharacteristic
357C23—42no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23C2×#12711yescharacteristic
365Q8—44no—
366Q8—82no—
367Q8—82no—
368Q8—82no—
369Q8—82no—
370Q8—82no—
371Q8—164no—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—82no—
376D4—82no—
377D4—82no—
378C2×C4—21no—
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—42no—
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×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—164no—
410C2×C4#49512yes—
411C2×C4#49512yes—
412C2×C4#52511yescharacteristic
413C2×C4#53711yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#54611yescharacteristic
416C2×C4#54811yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4C2×#13512yes—
419C2×C4C2×AGL(1,ℤ/8)12yes—
420C2×C4#57612yes—
421C2×C4#57712yes—
422C2×C4#57812yes—
423C2×C4#57812yes—
424C8—24no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8—42no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22#254011yescharacteristic
438C22C2×#49511yescharacteristic
439C22#315111yescharacteristic
440C22#315411yescharacteristic
441C22#315511yescharacteristic
442C22#315611yescharacteristic
443C22#315611yescharacteristic
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#312412yes—
456C4#326212yes—
457C4#330212yes—
458C4#330612yes—
459C4#336412yes—
460C4#337312yes—
461C2—162no—
462C2#1963511yescharacteristic
463C2#1965911yescharacteristic
464C2#1973111yescharacteristic
465C2#1974111yescharacteristic
466C2#2839411yescharacteristic
467C2#2840311yescharacteristic
468C2#2843011yescharacteristic
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. #30205C2semidirect
2. #29218C2semidirect
4. #28616C2semidirect
7. #10684C2semidirect
8. #10675C2semidirect
12. #9833C2semidirect
13. #9825C2semidirect
15. #8264C2semidirect

Maximal subgroups

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

subgroupindex
#302052
#292182
#287182
#286162
#285082
#222192
#106842
#106752
#103112
#103062
#98382
#98332
#98252
#98222
#82642