Finite Group Databasebeta

← #201192

Subgroups — #201192

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3#17734C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9454C214yesmaximal, complemented
6#9452C214yesmaximal, complemented
7#7949C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983—22no—
12#2983C2214yes—
13C2×#446—24no—
14#2920C2212yes—
15C2×#432—21no—
16C2×#432C2212yes—
17C2×#431—21no—
18#2584—22no—
19#2583—21no—
20#2583—21no—
21#2567C2212yes—
22#2546C2212yes—
23#2502C412yes—
24#2434C412yes—
25#2399—22no—
26C4×C4⋊Q8C414yes—
27#2296—21no—
28#2296—21no—
29C4×#117C2211yescharacteristic
30#1559—24no—
31#1558C2214yes—
32#1556C2214yes—
33#1542—24no—
34#1467C2214yes—
35#1463—21no—
36#1463—21no—
37#1461—22no—
38#1285C2218yes—
39C23×Q8C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2×C414yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8C2312yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#446—216no—
69#446—216no—
70#432—24no—
71#432—24no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432D414yes—
76#432C2314yes—
77#432C2314yes—
78#431—24no—
79#431—24no—
80#431—22no—
81#431—22no—
82#423—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8D414yes—
86#407—22no—
87C2×C22⋊C8—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×C4×C8—22no—
92#398—24no—
93#398—22no—
94#398—22no—
95#395—24no—
96#395—22no—
97#395—22no—
98#391C2×C414yes—
99#391C2×C414yes—
100#389—24no—
101#389—24no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#384C2×C414yes—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4C2×C412yes—
117#376—21no—
118C43C2×C412yes—
119#333—28no—
120#332—28no—
121#324—24no—
122#324—24no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—44no—
127C22×Q8C22⋊C412yes—
128C22×Q8C22×C412yes—
129C22×Q8C22×C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—42no—
138C22×C8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—48no—
144#125—42no—
145#125—42no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162#117—28no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42C22×C414yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—28no—
217C4⋊C8—28no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—42no—
221C4⋊C8C2×D418yes—
222C22⋊C8—24no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226#97—28no—
227#97—24no—
228#97—24no—
229#97—24no—
230C4×C8—28no—
231C4×C8—42no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—48no—
247C2×Q8—48no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—82no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—84no—
288C2×C8—28no—
289C2×C8—48no—
290C2×C8—44no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—48no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—28no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—28no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—82no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12312yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—22no—
335C23—22no—
336C23—41no—
337C23#38611yescharacteristic
338C23#38611yescharacteristic
339C23C22×C22⋊C411yescharacteristic
340Q8—48no—
341Q8—48no—
342Q8—84no—
343Q8—84no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#38611yescharacteristic
382C2×C4#38612yes—
383C2×C4#38612yes—
384C2×C4#38611yescharacteristic
385C2×C4#41314yes—
386C2×C4#45614yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12011yescharacteristic
389C2×C4C2×#12112yes—
390C2×C4C2×#12311yescharacteristic
391C8—48no—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22#188612yes—
402C22C2×#38611yescharacteristic
403C22C2×#41312yes—
404C4—22no—
405C4—22no—
406C4—21no—
407C4—21no—
408C4—24no—
409C4—22no—
410C4—22no—
411C4—48no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—82no—
417C4#188612yes—
418C4C2×#45612yes—
419C2—22no—
420C2—22no—
421C2—41no—
422C2C2×#188611yescharacteristic
423C2#2098911yescharacteristic
424C2#2100911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9454C2semidirect
6. #9452C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#349532
#177342
#173652
#94542
#94522
#79492