Finite Group Databasebeta

← #329415

Subgroups — #329415

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#32907C212yesmaximal
3#21373C211yesmaximal, characteristic
4#10617C211yesmaximal, characteristic, complemented
5#10615C211yesmaximal, characteristic, complemented
6#10478C212yesmaximal, complemented
7#10476C212yesmaximal, complemented
8#10265C211yesmaximal, characteristic, complemented
9#10264C211yesmaximal, characteristic, complemented
10#9130C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#462C2212yes—
15C2×#461C2212yes—
16C2×#452—22no—
17C2×#431C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2857C2212yes—
20#1990C2212yes—
21#1984C2211yescharacteristic
22#1984C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1942—21no—
26#1942—21no—
27#1878—22no—
28#1877—22no—
29#1873—22no—
30#1869—21no—
31#1869—21no—
32#1845C2211yescharacteristic
33#1808—21no—
34#1808—21no—
35#1803C2212yes—
36#1802C2212yes—
37#1797C2212yes—
38C4×C22⋊C8—21no—
39C4×C22⋊C8—21no—
40C2×#335C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×#41D411yescharacteristic
48C22×Q16—22no—
49C22×SD16—22no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×C4⋊Q8C2312yes—
53C2×C22⋊Q8—22no—
54C4×#41—42no—
55C4×#41—42no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#500—42no—
67#462—42no—
68#462—42no—
69#461—42no—
70#461—42no—
71#452—48no—
72#431—42no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—42no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—22no—
98#389—22no—
99#389—22no—
100#384C2312yes—
101#380—22no—
102#380—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—22no—
112#335—42no—
113#332—42no—
114#332—42no—
115C2×#41—22no—
116C2×#41—42no—
117C2×#41—42no—
118C2×#41D814yes—
119C22×Q8—22no—
120C22×Q8C2×D411yescharacteristic
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×Q16—44no—
130C2×Q16—44no—
131C2×Q16—44no—
132C2×SD16—44no—
133C2×SD16—44no—
134C2×SD16—44no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—42no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—42no—
161C4×D4—44no—
162C4×D4—42no—
163C4×D4—42no—
164#117—44no—
165#117—42no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—42no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191#107—42no—
192#107—42no—
193#107—42no—
194#107—42no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201#103—44no—
202#103—44no—
203#103—44no—
204#102—44no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207C22⋊C8—44no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—42no—
216#95—42no—
217#95—42no—
218C24—21no—
219C24#12111yescharacteristic
220#41—48no—
221#41—48no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8C2×D812yes—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4C2×D812yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4C2×D812yes—
271C22×C4#13512yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275Q16—88no—
276SD16—88no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8#13512yes—
301C2×C8#13712yes—
302C2×C8#13712yes—
303C4⋊C4—22no—
304C4⋊C4—22no—
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—82no—
316C4⋊C4—84no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—84no—
321C42—21no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—41no—
329C42H(5,𝔽2)12yes—
330C42H(5,𝔽2)12yes—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23#46612yes—
345C23C2×#12111yescharacteristic
346Q8—24no—
347Q8—42no—
348Q8—82no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352D4—24no—
353D4—48no—
354D4—42no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4#46612yes—
393C2×C4#47312yes—
394C2×C4#53412yes—
395C2×C4#53411yescharacteristic
396C2×C4#53511yescharacteristic
397C2×C4#53812yes—
398C2×C4C22×D811yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4#57512yes—
404C2×C4#57511yescharacteristic
405C8—22no—
406C8—22no—
407C8—22no—
408C8—41no—
409C8—82no—
410C8—82no—
411C8—81no—
412C8—81no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#261711yescharacteristic
428C22#303511yescharacteristic
429C22#303911yescharacteristic
430C22C2×#46611yescharacteristic
431C22#305411yescharacteristic
432C22C2×#47311yescharacteristic
433C22#306711yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—162no—
442C4—162no—
443C4#318612yes—
444C4#319011yescharacteristic
445C4#319112yes—
446C4#319611yescharacteristic
447C4#321011yescharacteristic
448C4#321611yescharacteristic
449C2—42no—
450C2—82no—
451C2#1912511yescharacteristic
452C2#1916111yescharacteristic
453C2#1927111yescharacteristic
454C2#1929811yescharacteristic
455C2#2971711yescharacteristic
456C2#2973011yescharacteristic
457C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33209C2semidirect
4. #10617C2semidirect
5. #10615C2semidirect
6. #10478C2semidirect
7. #10476C2semidirect
8. #10265C2semidirect
9. #10264C2semidirect
10. #9130C2semidirect
11. #8596C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#329072
#213732
#106172
#106152
#104782
#104762
#102652
#102642
#91302
#85962