Finite Group Databasebeta

← #323622

Subgroups — #323622

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30305C211yesmaximal, characteristic, complemented
2#29207C211yesmaximal, characteristic, complemented
3#29206C211yesmaximal, characteristic, complemented
4#29190C211yesmaximal, characteristic
5#28508C211yesmaximal, characteristic
6#22268C211yesmaximal, characteristic
7#10683C211yesmaximal, characteristic
8#10680C211yesmaximal, characteristic, complemented
9#10675C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic
11#9829C211yesmaximal, characteristic
12#9820C211yesmaximal, characteristic
13#8819C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#474—22no—
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2093—22no—
23#2084—22no—
24#2065—22no—
25#2059—22no—
26#2058—22no—
27#2058—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1956C2211yescharacteristic
38#1949—22no—
39#1948—22no—
40#1880—22no—
41#1877—22no—
42#1864—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#474—48no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D412yes—
88C2×#103—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#390—21no—
102#390—21no—
103#390—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C43C2311yescharacteristic
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#336D412yes—
132#336D412yes—
133#336D412yes—
134#336D412yes—
135#322—22no—
136C82—22no—
137C22×Q8—21no—
138C22×Q8—41no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C23×C4—41no—
143C2×SD16—28no—
144C2×SD16—82no—
145C2×SD16—82no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—44no—
156C4⋊Q8—82no—
157C4⋊Q8—82no—
158C4⋊Q8C2×Q812yes—
159#127—21no—
160#127—21no—
161#127—21no—
162#127C2×Q812yes—
163#125—82no—
164C4×Q8—82no—
165C4×D4—44no—
166C4×D4—82no—
167C4×D4—82no—
168C2×C4⋊C4—21no—
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—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—44no—
206#107—44no—
207#107—82no—
208#106—82no—
209#106—82no—
210#106—82no—
211#106—82no—
212#106—82no—
213#106—82no—
214C4⋊C8—24no—
215C4⋊C8SD1618yescomplemented
216#103—82no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—82no—
221#102—82no—
222C4×C8—21no—
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×C8C2×Q812yes—
233C4×C8#4112yes—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24—41no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4#12711yescharacteristic
269C22×C4#12711yescharacteristic
270C22×C4#12711yescharacteristic
271C22×C4#12711yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275SD16—88no—
276C2×C8—22no—
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×C8C22⋊Q814yes—
291C2×C8C2×SD1614yes—
292C2×C8C2×SD1614yes—
293C2×C8#13714yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
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—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—82no—
331C42C2×SD1614yes—
332C42C22×Q811yescharacteristic
333C42C2×#4111yescharacteristic
334C422−1+411yescharacteristic
335C422−1+411yescharacteristic
336C23—42no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23C2×#12711yescharacteristic
344Q8—44no—
345Q8—82no—
346Q8—164no—
347D4—44no—
348D4—44no—
349D4—44no—
350D4—82no—
351D4—82no—
352D4—82no—
353C2×C4—21no—
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—42no—
364C2×C4—81no—
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—164no—
385C2×C4#49212yes—
386C2×C4#49212yes—
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4#53611yescharacteristic
389C2×C4#54711yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4D4×Q811yescharacteristic
392C2×C4D4×Q811yescharacteristic
393C2×C4C22×SD1612yes—
394C2×C4C2×#13712yes—
395C2×C4#57612yes—
396C2×C4#57612yes—
397C2×C4#57712yes—
398C2×C4#57712yes—
399C8—24no—
400C8—24no—
401C8—24no—
402C8—24no—
403C8—42no—
404C8—42no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22#254211yescharacteristic
413C22C2×#49211yescharacteristic
414C22#315211yescharacteristic
415C22#315411yescharacteristic
416C22#315411yescharacteristic
417C22#315511yescharacteristic
418C22#315511yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#308112yes—
431C4#324112yes—
432C4#333512yes—
433C4#334612yes—
434C4Q8×SD1612yes—
435C4#338512yes—
436C2—162no—
437C2#1962211yescharacteristic
438C2#1965211yescharacteristic
439C2#1972611yescharacteristic
440C2#1974211yescharacteristic
441C2#2852511yescharacteristic
442C2#2853011yescharacteristic
443C2#2854311yescharacteristic
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. #29207C2semidirect
3. #29206C2semidirect
8. #10680C2semidirect
9. #10675C2semidirect
13. #8819C2semidirect
14. #8814C2semidirect
15. #8264C2semidirect
75. #492Q8semidirect
215. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#303052
#292072
#292062
#291902
#285082
#222682
#106832
#106802
#106752
#106722
#98292
#98202
#88192
#88142
#82642