Finite Group Databasebeta

← #204560

Subgroups — #204560

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic
3#20470C211yesmaximal, characteristic
4C2×#1497C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic
8#9677C214yesmaximal, complemented
9#9673C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2584—22no—
17#2583—22no—
18#2567C412yes—
19#2546C412yes—
20#2502C412yes—
21#2434C412yes—
22#2399—21no—
23#2399—21no—
24C4×C4⋊Q8C2214yes—
25#2296—22no—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1851—22no—
29#1767C2212yes—
30C4×C4⋊C8C2212yes—
31C4×C4⋊C8C2214yes—
32#1750—22no—
33#1746—21no—
34#1726—21no—
35#1497—24no—
36#1491—24no—
37#1485—24no—
38#1479—24no—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2214yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2214yes—
45C23×Q8C2×C411yescharacteristic
46C22×M16—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C414yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2312yes—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432D414yes—
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yes—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8C2312yes—
95C2×C4×C8C2312yes—
96#398—22no—
97#398—22no—
98#398—22no—
99#398—22no—
100#395—22no—
101#395—22no—
102#395—22no—
103#395—22no—
104#391C2×C414yes—
105#391C2×C414yes—
106#389—24no—
107#389—24no—
108#389—24no—
109#389—24no—
110#385—24no—
111#385—24no—
112#384C2×C414yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4D412yes—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119#376D411yescharacteristic
120#376D411yescharacteristic
121C43C2312yes—
122#336—24no—
123#336—22no—
124#336—22no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128Q8⋊C8—24no—
129Q8⋊C8—28no—
130Q8⋊C8—28no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8C22×C412yes—
138C22×Q8C22×C412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—44no—
145C2×M16—42no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—28no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—44no—
157#125—42no—
158#125—42no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179#117—24no—
180#117—24no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C414yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C414yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22×C414yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D414yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—22no—
224C4⋊C8—48no—
225C4⋊C8—42no—
226C4⋊C8C2×D412yes—
227C4⋊C8C2×D412yes—
228C4⋊C8C2×D414yes—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—42no—
232#97—24no—
233#97—42no—
234#97—44no—
235C4×C8—24no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24C2×C22⋊C411yescharacteristic
250C2×Q8—22no—
251C2×Q8—28no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—44no—
261C2×Q8—48no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—82no—
266C2×Q8#13114yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C414yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12412yes—
290C22×C4#12712yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—48no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—44no—
306C2×C8—42no—
307C4⋊C4—28no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C42—24no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42C4×D412yes—
339C42C4×D412yes—
340C42#12112yes—
341C42#12112yes—
342C42#12412yes—
343C42#12712yes—
344C42#13714yes—
345C42#13714yes—
346C42C22×D411yescharacteristic
347C42C22×D411yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—22no—
351C23—22no—
352C23—22no—
353C23#39012yes—
354C23C22×C22⋊C411yescharacteristic
355Q8—24no—
356Q8—24no—
357Q8—48no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—44no—
363Q8—84no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—48no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4#39012yes—
394C2×C4#39012yes—
395C2×C4#39012yes—
396C2×C4#41918yes—
397C2×C4C2×C4×D411yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12411yescharacteristic
402C2×C4C2×#12711yescharacteristic
403C2×C4C2×#13112yes—
404C2×C4#56812yes—
405C2×C4C2×#13712yes—
406C2×C4C2×#13712yes—
407C8—44no—
408C8—42no—
409C8—42no—
410C8—44no—
411C8—82no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—44no—
416C22C2×#39011yescharacteristic
417C22#288012yes—
418C22C2×#41912yes—
419C22#290112yes—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—22no—
424C4—28no—
425C4—22no—
426C4—44no—
427C4—44no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4#292911yescharacteristic
432C4#293011yescharacteristic
433C4#293312yes—
434C4#303911yescharacteristic
435C4#303911yescharacteristic
436C4#303912yes—
437C2—42no—
438C2#1751414yes—
439C2#2119111yescharacteristic
440C2#2125311yescharacteristic
441C2#2125911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊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
8. #9677C2semidirect
9. #9673C2semidirect

Maximal subgroups

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

subgroupindex
#349532
#213502
#204702
C2×#14972
C2×#14852
C2×#14912
C2×#14792
#96772
#96732