Finite Group Databasebeta

← #163900

Subgroups — #163900

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#22591C212yesmaximal, complemented
3#7642C212yesmaximal, complemented
4#7628C212yesmaximal, complemented
5#7614C212yesmaximal, complemented
6#7608C212yesmaximal, complemented
7#7592C211yesmaximal, characteristic
8#7579C212yesmaximal, complemented
9C2×#1270C211yesmaximal, characteristic
10C2×#518C411yescharacteristic
11#2357—22no—
12#2354—22no—
13#2342—22no—
14#2342—22no—
15#2319C412yes—
16#2319C412yes—
17#2317C411yescharacteristic
18#2309C411yescharacteristic
19#2296C2211yescharacteristic
20#2293C2212yes—
21#2293C2212yes—
22C2×#390C2211yescharacteristic
23#2274C411yescharacteristic
24C4×#117C2211yescharacteristic
25#1917—22no—
26#1917—22no—
27#1850—22no—
28#1839C2212yescomplemented
29#1831C2212yescomplemented
30#1817C2212yes—
31#1767C2212yescomplemented
32#1752—22no—
33#1745C2212yescomplemented
34C2×#336C2212yes—
35#1271C2212yescomplemented
36#1271C2212yescomplemented
37#1271C2212yescomplemented
38#1271C2212yescomplemented
39#1270C2212yes—
40#1270C2211yescharacteristic
41#1270C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2211yescharacteristic
44#1270C2212yes—
45C23×D4C2×C411yescharacteristic
46C23×C8—22no—
47#518—24no—
48#518—24no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C412yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—22no—
67C2×#97—22no—
68#390—21no—
69#390—21no—
70#390—22no—
71#390C2×C412yes—
72#390C2×C412yes—
73#390C2×C411yescharacteristic
74#390C2×C411yescharacteristic
75#390C2311yescharacteristic
76#390C2311yescharacteristic
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2312yes—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#385—22no—
90#385—22no—
91#384—21no—
92#382—21no—
93#381—22no—
94#381—22no—
95#380—22no—
96#380—22no—
97#379—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110#376C2×C412yes—
111C43C2312yes—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C25—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4M1614yes—
128C22×D4C22×C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—44no—
140C22×C8—44no—
141C22×C8—82no—
142C4×D4—216no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—42no—
151#117—42no—
152#117—82no—
153#117—82no—
154#117—82no—
155#117—82no—
156#117M1614yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C22⋊C411yescharacteristic
187C2×C22⋊C4C22⋊C411yescharacteristic
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—44no—
204C4⋊C8—82no—
205C22⋊C8—44no—
206#97—44no—
207#97—82no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218C24—21no—
219C24—21no—
220C24—24no—
221C24—42no—
222C24—42no—
223C24—82no—
224C24—82no—
225C24C2×C22⋊C411yescharacteristic
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—24no—
229C2×D4—48no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—48no—
237C2×D4C2×M1612yes—
238C2×D4C2×M1612yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×M1612yes—
264C22×C4C2×M1612yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D412yes—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—88no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4C2×M1614yes—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—24no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—41no—
321C42—42no—
322C42—41no—
323C42—42no—
324C42—42no—
325C42—82no—
326C42—82no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—48no—
340C23—88no—
341C23—162no—
342C23—162no—
343C23—164no—
344C23#40712yes—
345C23#41012yes—
346C23#41312yes—
347C23C22×C22⋊C411yescharacteristic
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—48no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—88no—
389C2×C4—88no—
390C2×C4—162no—
391C2×C4#40712yes—
392C2×C4C22×M1611yescharacteristic
393C2×C4C22×M1611yescharacteristic
394C2×C4#57612yes—
395C2×C4#57712yes—
396C8—44no—
397C8—82no—
398C8—82no—
399C8—82no—
400C22—22no—
401C22—24no—
402C22—22no—
403C22—24no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—42no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—82no—
413C22—82no—
414C22—168no—
415C22#146414yes—
416C22C2×#40711yescharacteristic
417C22C2×#41011yescharacteristic
418C22C2×#41311yescharacteristic
419C22#289712yes—
420C22#290112yes—
421C4—22no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—81no—
434C4—82no—
435C2—24no—
436C2—82no—
437C2—82no—
438C2C2×#146411yescharacteristic
439C2#1745312yes—
440C2#1760412yes—
441C2#1768611yescharacteristic
442C2#1777111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22591C2semidirect
3. #7642C2semidirect
4. #7628C2semidirect
5. #7614C2semidirect
6. #7608C2semidirect
8. #7579C2semidirect
28. #1839C22semidirect
29. #1831C22semidirect
31. #1767C22semidirect
33. #1745C22semidirect
35. #1271C22semidirect
36. #1271C22semidirect
37. #1271C22semidirect
38. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#225912
#76422
#76282
#76142
#76082
#75922
#75792
C2×#12702