Finite Group Databasebeta

← #320753

Subgroups — #320753

Order 512.

762 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#30228C211yesmaximal, characteristic, complemented
2#29178C211yesmaximal, characteristic, complemented
3#28716C211yesmaximal, characteristic
4#28514C211yesmaximal, characteristic
5#28440C211yesmaximal, characteristic, complemented
6#22220C211yesmaximal, characteristic
7#10675C211yesmaximal, characteristic, complemented
8#10674C211yesmaximal, characteristic, complemented
9#10306C211yesmaximal, characteristic
10#10306C211yesmaximal, characteristic
11#9839C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic
13#9825C211yesmaximal, characteristic, complemented
14#9822C211yesmaximal, characteristic
15#8264C211yesmaximal, characteristic, complemented
16C2×#494C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#486C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#478—22no—
22C2×#462—22no—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27#2099—22no—
28#2099—22no—
29#2092—22no—
30#2086—22no—
31#2071—22no—
32#2016—22no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1925—22no—
48#1918—22no—
49#1862—22no—
50#1845C2211yescharacteristic
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1802—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×Q16—22no—
64C22×Q16—22no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#125—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#494—22no—
77#494—22no—
78#494—41no—
79#494—41no—
80#486—22no—
81#486—22no—
82#486—22no—
83#486—22no—
84#486—41no—
85#486—41no—
86#478—48no—
87#462—48no—
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×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C43C2311yescharacteristic
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#340—22no—
140#336D412yes—
141#336D412yes—
142#336D412yes—
143#336D412yes—
144#322—41no—
145C82—41no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C23×C4—41no—
153C2×Q16—44no—
154C2×Q16—44no—
155C2×Q16—44no—
156C2×Q16—44no—
157C2×Q16—82no—
158C2×Q16—82no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
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—21no—
175C4⋊Q8—21no—
176C4⋊Q8—44no—
177C4⋊Q8—82no—
178C4⋊Q8#4112yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127#4112yes—
183#125—44no—
184C4×Q8—44no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—82no—
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—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—44no—
225#107—44no—
226#107—82no—
227#106—44no—
228C4⋊C8—42no—
229C4⋊C8D818yes—
230#103—44no—
231#103—44no—
232#102—28no—
233#102—44no—
234#102—44no—
235#102—44no—
236#102—44no—
237#102—82no—
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—
254C24—41no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
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—
297Q16—88no—
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×C8C2×D814yes—
315C2×C8C2×D814yes—
316C4⋊C4—22no—
317C4⋊C4—22no—
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—41no—
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—42no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—82no—
356C42C2×D814yes—
357C42C2×#4111yescharacteristic
358C42C2×#4111yescharacteristic
359C42H(5,𝔽2)11yescharacteristic
360C422−1+411yescharacteristic
361C23—42no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C2×#12711yescharacteristic
369Q8—44no—
370Q8—82no—
371Q8—82no—
372Q8—82no—
373Q8—82no—
374Q8—82no—
375D4—44no—
376D4—44no—
377D4—44no—
378D4—82no—
379D4—82no—
380D4—82no—
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—21no—
390C2×C4—21no—
391C2×C4—42no—
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—81no—
411C2×C4—81no—
412C2×C4—164no—
413C2×C4#49312yes—
414C2×C4#49312yes—
415C2×C4#52511yescharacteristic
416C2×C4#53711yescharacteristic
417C2×C4#54611yescharacteristic
418C2×C4#54611yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4#54811yescharacteristic
421C2×C4C22×D812yes—
422C2×C4#57512yes—
423C2×C4#57712yes—
424C2×C4#57712yes—
425C2×C4#57812yes—
426C2×C4#57812yes—
427C8—24no—
428C8—24no—
429C8—42no—
430C8—42no—
431C8—42no—
432C8—42no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22#254011yescharacteristic
441C22C2×#49311yescharacteristic
442C22#315311yescharacteristic
443C22#315411yescharacteristic
444C22#315411yescharacteristic
445C22#315611yescharacteristic
446C22#315611yescharacteristic
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—82no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#312012yes—
459C4#325812yes—
460C4#329412yes—
461C4#331512yes—
462C4#335812yes—
463C4#337712yes—
464C2—162no—
465C2#1963111yescharacteristic
466C2#1966211yescharacteristic
467C2#1971611yescharacteristic
468C2#1973911yescharacteristic
469C2#2839011yescharacteristic
470C2#2839711yescharacteristic
471C2#2842711yescharacteristic
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. #30228C2semidirect
2. #29178C2semidirect
5. #28440C2semidirect
7. #10675C2semidirect
8. #10674C2semidirect
11. #9839C2semidirect
13. #9825C2semidirect
15. #8264C2semidirect

Maximal subgroups

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

subgroupindex
#302282
#291782
#287162
#285142
#284402
#222202
#106752
#106742
#103062
#98392
#98302
#98252
#98222
#82642