Finite Group Databasebeta

← #215193

Subgroups — #215193

Order 512.

1316 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#36644C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20778C211yesmaximal, characteristic, complemented
4#9689C214yesmaximal, complemented
5#9689C214yesmaximal, complemented
6#8337C214yesmaximal, complemented
7C2×#532C412yes—
8C2×#532C412yes—
9C2×#532C2212yes—
10#2868C411yescharacteristic
11#2867C411yescharacteristic
12#2858C412yes—
13#2858C2212yes—
14#2858C2212yes—
15#2856—21no—
16#2856—21no—
17#2856—21no—
18#2856—21no—
19C4×#127—21no—
20C4×#127—21no—
21C4×#127—21no—
22C4×#127—21no—
23C2×C43C2211yescharacteristic
24#1839—21no—
25#1839—21no—
26#1831—21no—
27#1831—21no—
28#1817—21no—
29#1817—21no—
30#1797—21no—
31#1797—21no—
32#1767C2212yes—
33#1767C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1745C2212yes—
37#1745C2212yes—
38C2×#329C22116yes—
39C22×#41C2×C414yes—
40#532—48no—
41#532—48no—
42#532—48no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#127—21no—
47C2×#127—21no—
48C2×#127—21no—
49C2×#127—21no—
50C2×#127C2×C412yes—
51C2×#127C2×C412yes—
52C2×#127C2×C412yes—
53C2×#124C2×C414yes—
54C2×#124C2×C414yes—
55C2×#124C2314yes—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97C2314yes—
80C2×#97C2314yes—
81#390—22no—
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390—22no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97C43C2×C411yescharacteristic
98C43C2×C411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43D411yescharacteristic
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#329—416no—
114C2×#41—48no—
115C2×#41—48no—
116C2×#41—48no—
117C22×Q8C22×C414yes—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C22×C414yes—
126C23×C4—24no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—24no—
135C22×C8—24no—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139#127—44no—
140#127—44no—
141#127—44no—
142#127—44no—
143#127—44no—
144#127—44no—
145#127—44no—
146#124—216no—
147#124—48no—
148#124—48no—
149#121—48no—
150#121—48no—
151#121—48no—
152#121—48no—
153#121—48no—
154#121—48no—
155C4×D4—48no—
156C4×D4—48no—
157C4×D4—48no—
158C4×D4—48no—
159C4×D4—48no—
160C4×D4—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—44no—
209C4⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—44no—
212#97—28no—
213#97—28no—
214C24—24no—
215C24C2×C22⋊C411yescharacteristic
216#41—832no—
217C2×Q8—28no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×D4—28no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—48no—
236C2×D4—48no—
237C2×D4—48no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4#10014yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—48no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—48no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42#10014yes—
328C42H(5,𝔽2)14yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23C2×#10012yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—88no—
345D4—88no—
346D4—88no—
347D4—88no—
348D4—88no—
349D4—88no—
350D4—88no—
351D4—88no—
352D4—816no—
353C2×C4—28no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—816no—
394C2×C4C2×#10012yes—
395C2×C4C2×#10012yes—
396C2×C4C2×#10012yes—
397C2×C4C2×#10012yes—
398C2×C4C2×#10012yes—
399C2×C4#42112yes—
400C2×C4#42112yes—
401C2×C4#42112yes—
402C2×C4#42112yes—
403C2×C4#51812yes—
404C2×C4#53512yes—
405C2×C4#53512yes—
406C8—84no—
407C8—84no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—84no—
420C22—84no—
421C22—84no—
422C22#230911yescharacteristic
423C22C22×#10011yescharacteristic
424C22C22×#10011yescharacteristic
425C22C2×#42111yescharacteristic
426C22C2×#42111yescharacteristic
427C22C2×#42111yescharacteristic
428C22C2×#42111yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—84no—
444C2—42no—
445C2—164no—
446C2#1766311yescharacteristic
447C2#1767111yescharacteristic
448C2#1767111yescharacteristic
449C2#1767511yescharacteristic
450C2#2179711yescharacteristic
451C2#2181611yescharacteristic
452C2#2181611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36644C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20778C2semidirect
4. #9689C2semidirect
5. #9689C2semidirect
6. #8337C2semidirect

Maximal subgroups

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

subgroupindex
#366442
#207782
#96892
#83372