Finite Group Databasebeta

← #501747

Subgroups — #501747

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×#2150C211yesmaximal, characteristic, complemented
3#33209C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic
5#13515C212yesmaximal, complemented
6#13476C212yesmaximal, complemented
7#13275C211yesmaximal, characteristic, complemented
8#13177C211yesmaximal, characteristic, complemented
9#13061C212yesmaximal
10#12836C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#500C2211yescharacteristic
14C2×#466C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18C4×#121—21no—
19#2224—24no—
20C2×#367C2212yes—
21C2×#366C2212yes—
22#2150—24no—
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2212yes—
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2212yes—
27C2×#357C2212yes—
28#1990—21no—
29#1970C2211yescharacteristic
30#1942—21no—
31#1942C2211yescharacteristic
32#1942C2212yes—
33#1934—21no—
34#1893—21no—
35#1845C2211yescharacteristic
36#1808—21no—
37#1808C2212yes—
38#1808C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#102—21no—
41#1380C2212yes—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45C22×D8C2311yescharacteristic
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—21no—
49C2×C22⋊Q8—41no—
50C2×#121—21no—
51C2×C4×D4—21no—
52C2×C4×D4—41no—
53C22×C42D411yescharacteristic
54C2×M32—24no—
55C2×M32—24no—
56C2×M32—42no—
57C22×C16C2312yes—
58C22×C16C2312yes—
59#500—22no—
60#500—22no—
61#466—22no—
62#466Q814yescomplemented
63C4×D8—22no—
64C4×D8—22no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107C2312yes—
68C2×#107C2312yes—
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#102—21no—
73C2×#102—21no—
74C2×#102—21no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—41no—
82#390—21no—
83#389—22no—
84#389—21no—
85#389—41no—
86#388—41no—
87#386—41no—
88#384—21no—
89#380—22no—
90#380—21no—
91#380—41no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4—41no—
94#367—42no—
95#366—24no—
96C4⋊C16—24no—
97C4⋊C16—22no—
98C4⋊C16D414yescomplemented
99#357—24no—
100#357—42no—
101#340—22no—
102#340—21no—
103#340—22no—
104#340—21no—
105#340—21no—
106C22×Q8—41no—
107C22×D4—21no—
108C22×D4—41no—
109C23×C4—41no—
110C23×C4D811yescharacteristic
111C23×C4D811yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×D8—21no—
114C2×D8—21no—
115C2×D8—21no—
116C2×D8—21no—
117C2×D8—21no—
118C2×D8C2×Q812yes—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—41no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×Q812yes—
130C22×C8#4112yes—
131C22×C8C2411yescharacteristic
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C22⋊Q8—84no—
135#121—24no—
136#121—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—84no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42D811yescharacteristic
160C2×C42D811yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163M32—48no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16C2×D414yes—
170#107—21no—
171#107—22no—
172#107—21no—
173#107—21no—
174#107—22no—
175#107—22no—
176#107—21no—
177#107—22no—
178#107—21no—
179#107—42no—
180#107—42no—
181#107C2×Q812yes—
182#102—24no—
183#102—42no—
184#102—42no—
185C4×C8—22no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—22no—
189C4×C8C2×D412yes—
190C4×C8#4112yes—
191#95—41no—
192#95—41no—
193#95—41no—
194#95—42no—
195#95—42no—
196C24—41no—
197C24C2×D811yescharacteristic
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×D4—22no—
202C2×D4—44no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—82no—
209C2×D4—82no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4C22⋊Q812yes—
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C2×D811yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4#13712yes—
235C22×C4C22×D411yescharacteristic
236D8—22no—
237D8—22no—
238D8—41no—
239D8—41no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—44no—
252C2×C8#12112yes—
253C2×C8C22⋊Q812yes—
254C2×C8C22×D411yescharacteristic
255C2×C8C22×Q811yescharacteristic
256C2×C8C2×#4111yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8H(5,𝔽2)11yescharacteristic
259C2×C82−1+411yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—84no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C22⋊C4—44no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C22⋊C4—84no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42—21no—
282C42—21no—
283C42—22no—
284C42—22no—
285C42—82no—
286C42#12112yes—
287C42#13512yes—
288C16—24no—
289C16—42no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23#47412yes—
304C23C22×D811yescharacteristic
305Q8—164no—
306D4—42no—
307D4—42no—
308D4—88no—
309D4—81no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—164no—
342C2×C4#46612yes—
343C2×C4#46612yes—
344C2×C4#47412yes—
345C2×C4#50912yes—
346C2×C4#50912yes—
347C2×C4#51012yes—
348C2×C4C2×#12111yescharacteristic
349C2×C4C2×C22⋊Q811yescharacteristic
350C2×C4#53711yescharacteristic
351C2×C4C2×#13511yescharacteristic
352C2×C4C2×#13711yescharacteristic
353C2×C4#57511yescharacteristic
354C8—22no—
355C8—22no—
356C8—41no—
357C8#54611yescharacteristic
358C8D4×Q811yescharacteristic
359C8#55211yescharacteristic
360C8#55411yescharacteristic
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22C2×#46611yescharacteristic
376C22C2×#47411yescharacteristic
377C22#324511yescharacteristic
378C22C2×#50911yescharacteristic
379C22C2×#50911yescharacteristic
380C22C2×#51011yescharacteristic
381C22#341411yescharacteristic
382C4—22no—
383C4—22no—
384C4—22no—
385C4—41no—
386C4—82no—
387C4—162no—
388C4—161no—
389C4—161no—
390C4—162no—
391C4#221814yes—
392C4#241711yescharacteristic
393C4#305611yescharacteristic
394C4#307611yescharacteristic
395C4#324811yescharacteristic
396C2—42no—
397C2—162no—
398C2#2483511yescharacteristic
399C2C2×#221811yescharacteristic
400C2#3360611yescharacteristic
401C2#3362711yescharacteristic
402C2#3362911yescharacteristic
403C2#3381711yescharacteristic
404C2#3382011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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×#2224C2semidirect
2. C2×#2150C2semidirect
3. #33209C2semidirect
5. #13515C2semidirect
6. #13476C2semidirect
7. #13275C2semidirect
8. #13177C2semidirect
10. #12836C2semidirect
11. #12811C2semidirect
62. #466Q8semidirect
98. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×#21502
#332092
C4×#4662
#135152
#134762
#132752
#131772
#130612
#128362
#128112