Finite Group Databasebeta

← #6435865

Subgroups — #6435865

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35341C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#30217C214yesmaximal, complemented
4#30216C214yesmaximal, complemented
5#30210C214yesmaximal, complemented
6#30209C214yesmaximal, complemented
7#23634C214yesmaximal, complemented
8#23458C214yesmaximal, complemented
9#21325C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11#21200C211yesmaximal, characteristic
12#21199C211yesmaximal, characteristic
13#20367C211yesmaximal, characteristic, complemented
14C2×#544C2212yes—
15C2×#532C2211yescharacteristic
16C22×#125C2211yescharacteristic
17C2×#417C2214yes—
18C2×#416C2214yes—
19C2×C4×M16C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2868C2212yes—
22#2861C2214yes—
23#2859C2212yes—
24#2660C2214yes—
25#2642C2212yes—
26#2641C2212yes—
27#2591—24no—
28#2562—22no—
29#2562—22no—
30#2552—22no—
31#2551—22no—
32#2440—24no—
33#2317C2212yes—
34#2311C2212yes—
35#2311C2212yes—
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#2074C2218yes—
40#2073C2218yes—
41#2072C2218yes—
42#2071C2218yes—
43#2026C2218yes—
44#2025C2218yes—
45#2024C2218yes—
46#2023C2218yes—
47#1992C2214yes—
48#1992C2212yes—
49#1992C2214yes—
50#1992C2212yes—
51#1991C2214yes—
52#1961C2212yes—
53#1960C2212yes—
54#1959C2212yes—
55#1959C2214yes—
56#1959C2212yes—
57#1957C2212yes—
58#1956C2212yes—
59#1746C2214yes—
60#1745C2214yes—
61#1736C2214yes—
62C2×#340C2214yes—
63#1729C2212yes—
64#1729C2212yes—
65C22×#41C2312yes—
66C22×M16C2312yes—
67C23×C8C2312yes—
68#544—48no—
69#532—24no—
70#532—24no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2314yes—
75C2×#127C2311yescharacteristic
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125C2312yes—
81C2×#125C2312yes—
82C2×#125C2314yes—
83C2×#124C2312yes—
84C2×C22⋊Q8—24no—
85C2×#121—22no—
86C2×C4×D4—22no—
87C2×#117C2312yes—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C42C2311yescharacteristic
95#417—48no—
96#416—48no—
97C2×#103C2314yes—
98C2×#103C2318yes—
99C2×#103C2314yes—
100C2×#102C2314yes—
101C2×#102C2318yes—
102C2×#102C2314yes—
103C4×M16—24no—
104C2×#97—24no—
105C2×#97C2312yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108C2×C4×C8C2312yes—
109#400—24no—
110#400—24no—
111#398—28no—
112#395—28no—
113#394—24no—
114#394—24no—
115#391C2314yes—
116#391C2314yes—
117#390C2314yes—
118#390C2314yes—
119#389—24no—
120#389—24no—
121#389—24no—
122#388—24no—
123#388—24no—
124#388—24no—
125#388—24no—
126#386—24no—
127#386—24no—
128#385—24no—
129#384—24no—
130#384—22no—
131#384C2314yes—
132#384C2311yescharacteristic
133#384C2312yes—
134#384C2311yescharacteristic
135#384C2312yes—
136#382C2314yes—
137#382C2312yes—
138#380—22no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D412yes—
146C4×C22⋊C4D414yes—
147#376C2312yes—
148C43C2312yes—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2318yes—
154#336C2318yes—
155C2×#41—24no—
156C2×#41—24no—
157C2×#41—28no—
158C2×#41—28no—
159C22×Q8—24no—
160C22×Q8C2412yes—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C2412yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C2×D412yes—
168C23×C4C2411yescharacteristic
169C2×M16—24no—
170C2×M16—24no—
171C2×M16—24no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—24no—
175C22×C8C2414yes—
176C22×C8C2414yes—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—48no—
180C4⋊Q8—48no—
181C4⋊Q8—88no—
182#127—22no—
183#127—22no—
184#125—24no—
185#125—24no—
186#125—24no—
187#125—24no—
188#125—48no—
189#125—48no—
190#125—48no—
191#124—24no—
192#124—24no—
193C22⋊Q8—416no—
194#121—48no—
195C4×D4—48no—
196#117—24no—
197#117—24no—
198#117—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2412yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—22no—
234C2×C42C2×D414yes—
235C2×C42C2×D414yes—
236C2×C42C2×D412yes—
237C2×C42#4114yes—
238C2×C42#4114yes—
239C2×C42C2412yes—
240C2×C42C2412yes—
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243#103—44no—
244#103—44no—
245#103—48no—
246#102—44no—
247#102—44no—
248#102—48no—
249#97—24no—
250#97—24no—
251#97—24no—
252C4×C8—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—28no—
259#95C2×D414yes—
260#95C2×D414yes—
261C24—22no—
262C24C22×D411yescharacteristic
263#41—416no—
264#41—416no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—48no—
270C2×Q8—88no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—82no—
280C2×D4—82no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—84no—
296C22×C4#12418yes—
297C22×C4C22×D414yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4112yes—
305C22×C4H(5,𝔽2)14yes—
306C22×C42−1+414yes—
307C22×C4C2511yescharacteristic
308M16—48no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—48no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—84no—
336C4⋊C4—88no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—24no—
347C42—24no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42#12418yes—
352C42C2×#4112yes—
353C42C2×#4112yes—
354C42C2×#4112yes—
355C42C2×#4112yes—
356C23—22no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23C2×#12412yes—
366C23#53512yes—
367C23#53612yes—
368C23C23×D411yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—168no—
372D4—44no—
373D4—44no—
374D4—88no—
375D4—88no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—88no—
401C2×C4—88no—
402C2×C4—164no—
403C2×C4C2×#12414yes—
404C2×C4C2×#12414yes—
405C2×C4C2×#12412yes—
406C2×C4#53512yes—
407C2×C4#53612yes—
408C2×C4#54218yes—
409C2×C4#54218yes—
410C2×C4C22×#4111yescharacteristic
411C2×C4C22×#4111yescharacteristic
412C2×C4C2×H(5,𝔽2)12yes—
413C2×C4C2×2−1+412yes—
414C8—44no—
415C8—44no—
416C22—22no—
417C22—22no—
418C22—24no—
419C22—24no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22#264014yes—
428C22C22×#12411yescharacteristic
429C22C2×#53511yescharacteristic
430C22C2×#53611yescharacteristic
431C22#358512yes—
432C22#358612yes—
433C4—24no—
434C4—24no—
435C4—42no—
436C4—42no—
437C4—44no—
438C4—82no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4#264014yes—
444C4C2×#54212yes—
445C4C2×#54212yes—
446C2—24no—
447C2—162no—
448C2C2×#264011yescharacteristic
449C2#6062112yes—
450C2#6083011yescharacteristic
451C2#6083111yescharacteristic
452C2#6088712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35275C2semidirect
3. #30217C2semidirect
4. #30216C2semidirect
5. #30210C2semidirect
6. #30209C2semidirect
7. #23634C2semidirect
8. #23458C2semidirect
9. #21325C2semidirect
10. C2×#1992C2semidirect
13. #20367C2semidirect

Maximal subgroups

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

subgroupindex
#353412
#352752
#302172
#302162
#302102
#302092
#236342
#234582
#213252
C2×#19922
#212002
#211992
#203672