Finite Group Databasebeta

← #190841

Subgroups — #190841

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34058C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#21431C211yesmaximal, characteristic, complemented
4#8708C212yesmaximal, complemented
5#8569C212yesmaximal, complemented
6#8567C212yesmaximal, complemented
7#8310C212yesmaximal
8#8183C212yesmaximal, complemented
9#8181C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2319—22no—
12#2319C412yes—
13#2319C412yes—
14#2316C411yescharacteristic
15#2316C411yescharacteristic
16#2312—22no—
17#2309C411yescharacteristic
18#2309C411yescharacteristic
19#2293—22no—
20#2293C2212yes—
21#2293C2212yes—
22#2280—22no—
23C4×#117C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1917—22no—
28#1916—22no—
29#1842C2211yescharacteristic, complemented
30#1831C2211yescharacteristic, complemented
31#1797C2212yes—
32C4×C4⋊C8C2212yescomplemented
33C4×C4⋊C8C2212yescomplemented
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic, complemented
36C4×#97C2211yescharacteristic, complemented
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yescomplemented
42#1279C2212yescomplemented
43#1279C2212yescomplemented
44#1279C2212yescomplemented
45C23×D4C2×C411yescharacteristic
46C24×C4—21no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72#390—22no—
73#390—22no—
74#390C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#389—22no—
79#389—22no—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#381—22no—
93#381—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#379—22no—
99#379—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110#376—21no—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43—21no—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122D4⋊C8—24no—
123D4⋊C8—24no—
124D4⋊C8—24no—
125D4⋊C8—24no—
126C25—21no—
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—24no—
130C22×D4—42no—
131C22×D4C22×C412yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—28no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4#4114yes—
154C4×D4#4114yes—
155#117—24no—
156#117—24no—
157#117—42no—
158#117—42no—
159#117—82no—
160#117—82no—
161#117—82no—
162#117—82no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—44no—
212C4⋊C8—82no—
213C22⋊C8—44no—
214C22⋊C8—82no—
215#97—44no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—82no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231C24—24no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24—82no—
236C24—82no—
237C24C2×C22⋊C411yescharacteristic
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—416no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4#11714yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—88no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—82no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4#13114yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4#11714yes—
305C22⋊C4—28no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—22no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—82no—
339C42C2×#4112yes—
340C42C2×#4112yes—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—24no—
346C23—48no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—88no—
354C23—84no—
355C23—84no—
356C23—88no—
357C23#40812yes—
358C23C22×C22⋊C411yescharacteristic
359D4—28no—
360D4—48no—
361D4—44no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—28no—
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—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—88no—
400C2×C4—84no—
401C2×C4#40812yes—
402C2×C4C2×#11712yes—
403C2×C4#52512yes—
404C2×C4#52511yescharacteristic
405C2×C4#52511yescharacteristic
406C2×C4C2×#13112yes—
407C2×C4#56812yes—
408C2×C4#57612yes—
409C2×C4#57712yes—
410C8—44no—
411C8—82no—
412C8—82no—
413C8—82no—
414C22—24no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—816no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22#228011yescharacteristic
428C22C2×#40811yescharacteristic
429C22#287911yescharacteristic
430C22#289712yes—
431C22#290112yes—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—44no—
436C4—41no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—82no—
443C4#292414yes—
444C4#311812yes—
445C4#312211yescharacteristic
446C4#312311yescharacteristic
447C2—42no—
448C2—84no—
449C2#1746812yes—
450C2#1750512yes—
451C2#2071711yescharacteristic
452C2#2074411yescharacteristic
453C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21922C2semidirect
3. #21431C2semidirect
4. #8708C2semidirect
5. #8569C2semidirect
6. #8567C2semidirect
8. #8183C2semidirect
9. #8181C2semidirect
29. #1842C22semidirect
30. #1831C22semidirect
32. C4×C4⋊C8C22semidirect
33. C4×C4⋊C8C22semidirect
35. #1744C22semidirect
36. C4×#97C22semidirect
41. #1279C22semidirect
42. #1279C22semidirect
43. #1279C22semidirect
44. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#340582
#219222
#214312
#87082
#85692
#85672
#83102
#81832
#81812