Finite Group Databasebeta

← #213726

Subgroups — #213726

Order 512.

1269 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34265C211yesmaximal, characteristic
2#17351C212yesmaximal, complemented
3#9107C214yesmaximal, complemented
4#8965C214yesmaximal, complemented
5#8735C212yesmaximal, complemented
6#8301C212yesmaximal, complemented
7C2×#535C411yescharacteristic
8C2×#520C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2972C2214yes—
11#2367—22no—
12#2366—22no—
13#2363—22no—
14#2361—22no—
15#2321C412yes—
16#2321C2212yes—
17#2309C411yescharacteristic
18C4×#120C412yes—
19C4×#120C2212yes—
20#2297C411yescharacteristic
21#2296C411yescharacteristic
22#2274C2211yescharacteristic
23#1779C2214yes—
24#1461—22no—
25#1459—22no—
26#1459C2214yes—
27#1458—22no—
28C2×#324—22no—
29C2×#324C2214yes—
30C2×#323C2214yes—
31#1282C2218yes—
32C22×#41C2×C411yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36#535—24no—
37#535C818yescomplemented
38C2×#127—21no—
39C2×#124—21no—
40C2×#121—22no—
41C2×#120C2×C412yes—
42C2×#120C2×C412yes—
43#520—44no—
44C2×C4×Q8D412yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—44no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2312yes—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117D412yes—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C411yescharacteristic
62C8×D4—48no—
63#431—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—44no—
66C2×C22⋊C8C2318yes—
67#391—21no—
68#390—21no—
69#390—21no—
70#390—21no—
71#390—22no—
72#390—22no—
73#390—22no—
74#389—21no—
75#389—21no—
76#388—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#386—22no—
84#386C2×C412yes—
85#386C2×C412yes—
86#385—22no—
87#385—22no—
88#385C2×C412yes—
89#385C2×C412yes—
90#379C2×C412yes—
91#379C2×C412yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2312yes—
100#324—24no—
101#324—24no—
102#324—24no—
103#323—24no—
104#323—44no—
105C25C22×C411yescharacteristic
106C25C22×C411yescharacteristic
107C2×#41—42no—
108C2×#41—42no—
109C2×#41C2×C814yes—
110C2×#41M1614yes—
111C22×Q8C22⋊C411yescharacteristic
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—42no—
118C22×D4—42no—
119C22×D4C22⋊C411yescharacteristic
120C22×D4C2×C814yescomplemented
121C22×D4C22×C412yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—42no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C22×C8—28no—
137C22×C8—44no—
138#127—28no—
139#124—28no—
140#121—216no—
141#120—24no—
142#120—24no—
143#120C2×C818yescomplemented
144#120C2×C818yescomplemented
145C4×Q8—44no—
146C4×D4—28no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—48no—
150C4×D4—48no—
151C4×D4—84no—
152C4×D4—84no—
153#117—22no—
154#117—44no—
155#117—42no—
156#117—42no—
157#117C2×C814yescomplemented
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22×C412yes—
184C2×C22⋊C4C22×C412yes—
185C2×C22⋊C4C22×C412yes—
186C2×C22⋊C4C2×D412yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C2×D412yes—
197C4⋊C8—44no—
198C22⋊C8—28no—
199C22⋊C8—28no—
200C22⋊C8—44no—
201C22⋊C8—88no—
202C4×C8—44no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—81no—
218C24—81no—
219C24—81no—
220C24—81no—
221C24C2×C22⋊C412yes—
222C24C22×C812yes—
223C24C22×C812yes—
224C24C23×C411yescharacteristic
225#41—48no—
226#41—48no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8C22⋊C814yescomplemented
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—41no—
237C2×D4—48no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—48no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4C22⋊C814yescomplemented
249C2×D4C22×C814yes—
250C2×D4C2×M1614yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—84no—
271C22×C4—81no—
272C22×C4—82no—
273C22×C4—84no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—82no—
279C22×C4—84no—
280C22×C4C22⋊C818yescomplemented
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C22×C812yes—
288C22×C4C2×M1612yes—
289C22×C4C22×D412yes—
290C22×C4H(5,𝔽2)14yes—
291C2×C8—48no—
292C2×C8—88no—
293C2×C8—164no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—82no—
304C22⋊C4—24no—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4C22×C814yes—
320C42—24no—
321C42—24no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—82no—
328C42—82no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—24no—
335C23—41no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—84no—
358C23—84no—
359C23C2×C22⋊C814yes—
360C23C22×C22⋊C411yescharacteristic
361C23#51812yes—
362C23C23×C811yescharacteristic
363C23C22×M1611yescharacteristic
364C23#56812yes—
365Q8—44no—
366D4—28no—
367D4—28no—
368D4—44no—
369D4—48no—
370D4—48no—
371D4—88no—
372D4—88no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—84no—
405C2×C4—88no—
406C2×C4—82no—
407C2×C4—88no—
408C2×C4—82no—
409C2×C4—162no—
410C2×C4C2×C22⋊C812yes—
411C2×C4C2×C22⋊C814yes—
412C2×C4C2×C22⋊C812yes—
413C2×C4#53414yes—
414C8—88no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—22no—
419C22—24no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—44no—
424C22—44no—
425C22—44no—
426C22—48no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—82no—
432C22—84no—
433C22—82no—
434C22—82no—
435C22—84no—
436C22—84no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#230811yescharacteristic
442C22C22×C22⋊C811yescharacteristic
443C22#287911yescharacteristic
444C22#297212yes—
445C22#297312yes—
446C4—24no—
447C4—22no—
448C4—22no—
449C4—24no—
450C4—44no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—81no—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—22no—
461C2—44no—
462C2—82no—
463C2—82no—
464C2#2173811yescharacteristic
465C2#2177611yescharacteristic
466C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC2×#535C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17351C2semidirect
3. #9107C2semidirect
4. #8965C2semidirect
5. #8735C2semidirect
6. #8301C2semidirect
37. #535C8semidirect
120. C22×D4C2×C8semidirect
143. #120C2×C8semidirect
144. #120C2×C8semidirect
157. #117C2×C8semidirect
229. C2×Q8C22⋊C8semidirect
248. C2×D4C22⋊C8semidirect
280. C22×C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
#342652
#173512
#91072
#89652
#87352
#83012