Finite Group Databasebeta

← #323483

Subgroups — #323483

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#29192C211yesmaximal, characteristic, complemented
3#29192C211yesmaximal, characteristic, complemented
4#29139C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic, complemented
6#21690C211yesmaximal, characteristic
7#10673C211yesmaximal, characteristic, complemented
8#10672C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic, complemented
11#9837C211yesmaximal, characteristic
12#9824C211yesmaximal, characteristic
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×#501—22no—
17C2×#493C2211yescharacteristic
18C2×#474—22no—
19C2×#442—22no—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#127C2211yescharacteristic
24#2093—22no—
25#2067—22no—
26#2067—22no—
27#2023—22no—
28#1972—22no—
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1862—22no—
40#1862—22no—
41#1847—22no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1802—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×D8—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×#127C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#501—48no—
77#493—22no—
78#493Q814yescomplemented
79#474—48no—
80#442—48no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×C4⋊C8D412yes—
91C2×C4⋊C8D412yes—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#97D412yes—
103C2×#97D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#390—21no—
111#390—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#384—21no—
121#384—21no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382—21no—
130C43C2311yescharacteristic
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#336D412yes—
136#336D412yes—
137#322—22no—
138C82—22no—
139C22×Q8—41no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C23×C4—41no—
147C23×C4—41no—
148C2×D8—28no—
149C2×D8—44no—
150C2×D8—44no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—82no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164#127—21no—
165#127—21no—
166#127C2×Q812yes—
167#127C2×Q812yes—
168#125—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—82no—
174C4×D4—82no—
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—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—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211#107—44no—
212#107—44no—
213#107—44no—
214#107—44no—
215#107—82no—
216#107—82no—
217#106—44no—
218#106—44no—
219#106—44no—
220C4⋊C8—24no—
221C4⋊C8—42no—
222#102—44no—
223#102—44no—
224#102—44no—
225#102—44no—
226#102—44no—
227#102—44no—
228#102—82no—
229#102—82no—
230#102—82no—
231#102—82no—
232#97—24no—
233#97D818yes—
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×C8C2×Q812yes—
245C4×C8#4112yes—
246#95—41no—
247#95—41no—
248C24—41no—
249C24—41no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
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
296D8—416no—
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×C8C22⋊Q814yes—
311C2×C8C2×D814yes—
312C2×C8C2×D814yes—
313C2×C8AGL(1,ℤ/8)14yes—
314C2×C8#13714yes—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
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—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42C2×D814yes—
350C42C22×Q811yescharacteristic
351C42C2×#4111yescharacteristic
352C422−1+411yescharacteristic
353C422−1+411yescharacteristic
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C2×#12711yescharacteristic
369Q8—164no—
370D4—44no—
371D4—44no—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—82no—
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—21no—
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—164no—
411C2×C4—164no—
412C2×C4#49312yes—
413C2×C4#49312yes—
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#54711yescharacteristic
417C2×C4#54711yescharacteristic
418C2×C4D4×Q811yescharacteristic
419C2×C4D4×Q811yescharacteristic
420C2×C4C22×D812yes—
421C2×C4C2×AGL(1,ℤ/8)12yes—
422C2×C4C2×#13712yes—
423C2×C4#57512yes—
424C2×C4#57612yes—
425C2×C4#57712yes—
426C8—24no—
427C8—24no—
428C8—42no—
429C8—42no—
430C8—42no—
431C8#47418yescomplemented
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22#254211yescharacteristic
447C22C2×#49311yescharacteristic
448C22#315111yescharacteristic
449C22#315211yescharacteristic
450C22#315311yescharacteristic
451C22#315411yescharacteristic
452C22#315511yescharacteristic
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4C2×#47412yes—
464C4#322612yes—
465C4#334512yes—
466C4#334712yes—
467C4#338412yes—
468C4#338912yes—
469C2—162no—
470C2—162no—
471C2#1962911yescharacteristic
472C2#1963111yescharacteristic
473C2#1972611yescharacteristic
474C2#1974811yescharacteristic
475C2#2850811yescharacteristic
476C2#2854411yescharacteristic
477C2#2854811yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #33316C2semidirect
2. #29192C2semidirect
3. #29192C2semidirect
4. #29139C2semidirect
5. #28508C2semidirect
7. #10673C2semidirect
8. #10672C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect
15. #8262C2semidirect
78. #493Q8semidirect
431. C8#474semidirect

Maximal subgroups

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

subgroupindex
#333162
#291922
#291392
#285082
#216902
#106732
#106722
#106712
#98372
#98242
#88202
#88142
#82622