Finite Group Databasebeta

← #323614

Subgroups — #323614

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30305C211yesmaximal, characteristic, complemented
2#29208C211yesmaximal, characteristic, complemented
3#29177C211yesmaximal, characteristic, complemented
4#28514C211yesmaximal, characteristic
5#28509C211yesmaximal, characteristic
6#22268C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10680C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic
10#10672C211yesmaximal, characteristic
11#9829C211yesmaximal, characteristic
12#9820C211yesmaximal, characteristic
13#8819C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#478—22no—
18C2×#476—22no—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2094—22no—
24#2092—22no—
25#2088—22no—
26#2084—22no—
27#2058—22no—
28#2057—22no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1956C2211yescharacteristic
39#1947—22no—
40#1919—22no—
41#1880—22no—
42#1877—22no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#492—22no—
75#492Q814yescomplemented
76#478—48no—
77#476—48no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×C4⋊C8D412yes—
88C2×C4⋊C8D412yes—
89C2×#103—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C43C2311yescharacteristic
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#336D412yes—
133#336D412yes—
134#336D412yes—
135#336D412yes—
136#322—22no—
137C82—22no—
138C22×Q8—21no—
139C22×Q8—41no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C23×C4—41no—
144C2×SD16—28no—
145C2×SD16—82no—
146C2×SD16—82no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—44no—
157C4⋊Q8—82no—
158C4⋊Q8—82no—
159C4⋊Q8C2×Q812yes—
160#127—21no—
161#127—21no—
162#127—21no—
163#127C2×Q812yes—
164#125—44no—
165C4×Q8—82no—
166C4×D4—44no—
167C4×D4—44no—
168C4×D4—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
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—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206#107—44no—
207#107—82no—
208#107—82no—
209#106—44no—
210#106—44no—
211#106—44no—
212#106—44no—
213#106—82no—
214#106—82no—
215C4⋊C8—24no—
216C4⋊C8SD1618yescomplemented
217#103—82no—
218#102—44no—
219#102—44no—
220#102—44no—
221#102—44no—
222#102—82no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8C2×Q812yes—
234C4×C8#4112yes—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239C24—41no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4#12711yescharacteristic
270C22×C4#12711yescharacteristic
271C22×C4#12711yescharacteristic
272C22×C4#12711yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276SD16—88no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8C22⋊Q814yes—
292C2×C8C2×SD1614yes—
293C2×C8C2×SD1614yes—
294C2×C8AGL(1,ℤ/8)14yes—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—82no—
332C42C2×SD1614yes—
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C422−1+411yescharacteristic
336C422−1+411yescharacteristic
337C23—42no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23C2×#12711yescharacteristic
345Q8—44no—
346Q8—82no—
347Q8—164no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—82no—
352D4—82no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—164no—
386C2×C4#49212yes—
387C2×C4#49212yes—
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4#53611yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4D4×Q811yescharacteristic
393C2×C4D4×Q811yescharacteristic
394C2×C4C22×SD1612yes—
395C2×C4C2×AGL(1,ℤ/8)12yes—
396C2×C4#57612yes—
397C2×C4#57712yes—
398C2×C4#57712yes—
399C2×C4#57812yes—
400C8—24no—
401C8—24no—
402C8—24no—
403C8—24no—
404C8—42no—
405C8—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22#254211yescharacteristic
414C22C2×#49211yescharacteristic
415C22#315111yescharacteristic
416C22#315411yescharacteristic
417C22#315411yescharacteristic
418C22#315511yescharacteristic
419C22#315611yescharacteristic
420C4—22no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#308112yes—
432C4#324112yes—
433C4#333512yes—
434C4#334312yes—
435C4Q8×SD1612yes—
436C4#338612yes—
437C2—162no—
438C2#1962111yescharacteristic
439C2#1965511yescharacteristic
440C2#1971911yescharacteristic
441C2#1973911yescharacteristic
442C2#2852511yescharacteristic
443C2#2853011yescharacteristic
444C2#2854511yescharacteristic
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. #30305C2semidirect
2. #29208C2semidirect
3. #29177C2semidirect
7. #10685C2semidirect
8. #10680C2semidirect
13. #8819C2semidirect
14. #8814C2semidirect
15. #8264C2semidirect
75. #492Q8semidirect
216. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#303052
#292082
#291772
#285142
#285092
#222682
#106852
#106802
#106722
#98292
#98202
#88192
#88142
#82642