Finite Group Databasebeta

← #508324

Subgroups — #508324

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C212yesmaximal, complemented
2#33171C211yesmaximal, characteristic
3#21151C211yesmaximal, characteristic, complemented
4#13506C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13476C212yesmaximal, complemented
7#13474C212yesmaximal, complemented
8#13407C211yesmaximal, characteristic, complemented
9#13407C211yesmaximal, characteristic, complemented
10#12992C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2856—21no—
19#2296—21no—
20#2234—28no—
21C2×#367C2212yes—
22C2×#366C2212yes—
23C2×C4⋊C16C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×C22⋊C16C2212yes—
29#1970—21no—
30#1970C2212yes—
31#1957—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1941—21no—
37#1941—21no—
38#1845C2211yescharacteristic
39#1735—21no—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44C22×D8—21no—
45C22×D8C2312yes—
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#127—21no—
49C2×#121—22no—
50C2×#121—41no—
51C2×#121—41no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42D411yescharacteristic
55C22×C16C2312yes—
56C22×C16C2312yes—
57#493—42no—
58#466—24no—
59#466—24no—
60#466—44no—
61#427—22no—
62#427—42no—
63#427D414yescomplemented
64C2×#107—21no—
65C2×#107C2312yes—
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#106—21no—
71C2×#106—21no—
72C2×#102—22no—
73C2×#102—22no—
74C2×C4×C8—42no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—22no—
80#390—41no—
81#390—41no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#385—41no—
87#385—41no—
88#384—21no—
89C4×C4⋊C4—41no—
90C4×C4⋊C4—41no—
91#367—24no—
92#366—24no—
93C4⋊C16—42no—
94#357—24no—
95#357—24no—
96#357—42no—
97#357—42no—
98C22⋊C16—24no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103C22×D4—22no—
104C22×D4—42no—
105C22×D4—41no—
106C22×D4—41no—
107C23×C4D811yescharacteristic
108C23×C4D811yescharacteristic
109C23×C4C2×D411yescharacteristic
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—22no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8—42no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D412yes—
127C22×C8C2×D411yescharacteristic
128C22×C8#4112yes—
129C22×C8C2411yescharacteristic
130#127—42no—
131#127—42no—
132#121—44no—
133#121—44no—
134#121—84no—
135#121—84no—
136#117—24no—
137#117—42no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42D811yescharacteristic
161C2×C42D811yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16—22no—
170#107—21no—
171#107—21no—
172#107—22no—
173#107—22no—
174#107—22no—
175#107—22no—
176#107—42no—
177#107C2×D412yes—
178#107C2×D412yes—
179#106—24no—
180#106—42no—
181#102—44no—
182#102—44no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191C24—42no—
192C24C2×D811yescharacteristic
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—82no—
203C2×D4—84no—
204C2×D4—84no—
205C2×D4—84no—
206C2×D4—82no—
207C2×D4—82no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4#12112yes—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4AGL(1,ℤ/8)12yes—
230C22×C4C22×D411yescharacteristic
231D8—42no—
232D8—42no—
233D8—42no—
234D8—42no—
235D8—84no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—44no—
246C2×C8#12112yes—
247C2×C8#13712yes—
248C2×C8C22×D411yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C8H(5,𝔽2)12yes—
251C2×C82−1+412yes—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C22⋊C4—44no—
269C22⋊C4—84no—
270C22⋊C4—82no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—41no—
278C42—41no—
279C42—44no—
280C42—82no—
281C16—42no—
282C16—42no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23#45912yes—
297C23C22×D811yescharacteristic
298D4—82no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—164no—
303D4—164no—
304D4—168no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4#45912yes—
338C2×C4#50812yes—
339C2×C4#50912yes—
340C2×C4#50912yes—
341C2×C4C2×#12111yescharacteristic
342C2×C4#53411yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4C2×#13711yescharacteristic
346C2×C4#57511yescharacteristic
347C8—22no—
348C8—22no—
349C8—41no—
350C8#53712yes—
351C8#53912yes—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—162no—
360C22—162no—
361C22—162no—
362C22—162no—
363C22—162no—
364C22—162no—
365C22—162no—
366C22C2×#45911yescharacteristic
367C22#318611yescharacteristic
368C22#322611yescharacteristic
369C22C2×#50811yescharacteristic
370C22C2×#50911yescharacteristic
371C22C2×#50911yescharacteristic
372C22#341411yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—82no—
379C4—82no—
380C4—162no—
381C4—161no—
382C4—161no—
383C4#261811yescharacteristic
384C4#303711yescharacteristic
385C4#319311yescharacteristic
386C4#323411yescharacteristic
387C2—42no—
388C2—322no—
389C2#2975411yescharacteristic
390C2#3357911yescharacteristic
391C2#3358211yescharacteristic
392C2#3375411yescharacteristic
393C2#3376011yescharacteristic
394C2#3379411yescharacteristic
395C2#3380111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33171C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2234C2semidirect
3. #21151C2semidirect
4. #13506C2semidirect
5. #13505C2semidirect
6. #13476C2semidirect
7. #13474C2semidirect
8. #13407C2semidirect
9. #13407C2semidirect
10. #12992C2semidirect
11. #12783C2semidirect
63. #427D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
#331712
#211512
#135062
#135052
#134762
#134742
#134072
#129922
#127832