Finite Group Databasebeta

← #512222

Subgroups — #512222

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33257C211yesmaximal, characteristic, complemented
2C42×D8C211yesmaximal, characteristic
3#13178C211yesmaximal, characteristic, complemented
4#13178C211yesmaximal, characteristic, complemented
5#13112C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#12937C211yesmaximal, characteristic
10#12937C211yesmaximal, characteristic
11#12935C211yesmaximal, characteristic
12#12935C211yesmaximal, characteristic
13#12879C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12813C211yesmaximal, characteristic
16C2×C4×D8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2864—21no—
20C42×D4—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1941—21no—
36#1941—21no—
37#1846C2211yescharacteristic
38#1846C2211yescharacteristic
39#1835—21no—
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#102—21no—
44C4×#102—21no—
45C4×#102—21no—
46C42×C8C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C22×C42—41no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C4×D8—22no—
69C4×D8—22no—
70C4×D8—22no—
71C4×D8—22no—
72C4×D8—22no—
73C4×D8Q814yes—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#387—41no—
97#387—41no—
98#387—41no—
99#387—41no—
100#382—21no—
101#382—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4—41no—
107C4×C22⋊C4—41no—
108C43D411yescharacteristic
109#367—22no—
110#367—22no—
111#366—22no—
112#366—22no—
113C4⋊C16—22no—
114C4⋊C16—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×D4—21no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—41no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8C2×Q812yes—
134C2×D8#4112yes—
135C2×D8#4112yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—84no—
164C2×C42D811yescharacteristic
165C2×C42D811yescharacteristic
166C2×C42D811yescharacteristic
167C2×C42D811yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107C2×Q812yes—
195#107#4112yes—
196#107#4112yes—
197#106—42no—
198#106—42no—
199#102—24no—
200#102—24no—
201#102—24no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8C2×Q812yes—
210C4×C8#4112yes—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222C24—41no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C22×D411yescharacteristic
252D8—22no—
253D8—22no—
254D8—22no—
255D8#12514yes—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8C22⋊Q812yes—
265C2×C8C22×Q811yescharacteristic
266C2×C8C2×#4111yescharacteristic
267C2×C8C2×#4111yescharacteristic
268C2×C8C2×#4111yescharacteristic
269C2×C8C2×#4111yescharacteristic
270C2×C8C2×#4111yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C42—82no—
301C42—82no—
302C42—82no—
303C42C22⋊Q812yes—
304C42#13712yes—
305C16—41no—
306C16—41no—
307C16—41no—
308C16—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23C22×D811yescharacteristic
317D4—24no—
318D4—42no—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—88no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#47412yes—
360C2×C4#47412yes—
361C2×C4#50812yes—
362C2×C4#50912yes—
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4#52511yescharacteristic
365C2×C4#52511yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4#57511yescharacteristic
368C2×C4#57511yescharacteristic
369C8—22no—
370C8—41no—
371C8—41no—
372C8C2×#12511yescharacteristic
373C8#52911yescharacteristic
374C8#52911yescharacteristic
375C8#53311yescharacteristic
376C22—44no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22C2×#47411yescharacteristic
384C22#310011yescharacteristic
385C22#310011yescharacteristic
386C22C2×#50811yescharacteristic
387C22C2×#50911yescharacteristic
388C22#341411yescharacteristic
389C22#341411yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4#285111yescharacteristic
404C4#308011yescharacteristic
405C4#311611yescharacteristic
406C4#311611yescharacteristic
407C2—84no—
408C2#3306611yescharacteristic
409C2#3362611yescharacteristic
410C2#3362711yescharacteristic
411C2#3366211yescharacteristic
412C2#3366511yescharacteristic
413C2#3367111yescharacteristic
414C2#3367411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33257C2semidirect
3. #13178C2semidirect
4. #13178C2semidirect
5. #13112C2semidirect
6. #13111C2semidirect
7. #13110C2semidirect
8. #13109C2semidirect
13. #12879C2semidirect

Maximal subgroups

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

subgroupindex
#332572
C42×D82
#131782
#131122
#131112
#131102
#131092
#129372
#129352
#128792
#128132