Finite Group Databasebeta

← #207643

Subgroups — #207643

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20473C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9647C211yesmaximal, characteristic, complemented
7#9643C211yesmaximal, characteristic, complemented
8#9073C212yesmaximal
9#9071C212yesmaximal
10#8175C212yesmaximal, complemented
11#8173C212yesmaximal, complemented
12C2×#541C411yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2336—22no—
16#2334—22no—
17#2330C411yescharacteristic
18#2327C411yescharacteristic
19#2323C412yes—
20#2320C412yes—
21#2314C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125—22no—
24C4×#125C2211yescharacteristic
25C4×#126—22no—
26C4×C22⋊Q8C2212yes—
27C42×Q8C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1770—22no—
35#1767C2212yes—
36#1752—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C23×C8—21no—
50#541—28no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#126—22no—
53C2×#125—22no—
54C2×#125C2×C411yescharacteristic
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2312yes—
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391C2×C412yes—
81#391C2×C412yes—
82#389—22no—
83#389—22no—
84#389—22no—
85#389C2×C412yes—
86#389C2×C412yes—
87#388—22no—
88#388—22no—
89#388—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#383—22no—
102#382—22no—
103#382C2×C411yescharacteristic
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—22no—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43—22no—
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129Q8⋊C8—24no—
130Q8⋊C8—24no—
131Q8⋊C8—24no—
132Q8⋊C8—24no—
133C22×Q8C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—21no—
138C22×C8—24no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—44no—
145C4⋊Q8—24no—
146#126—216no—
147#125—216no—
148#125—24no—
149C22⋊Q8—24no—
150C22⋊Q8M1618yes—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—24no—
160#117—42no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—82no—
207C22⋊C8—24no—
208C22⋊C8—82no—
209#97—44no—
210#97—44no—
211#97—82no—
212C4×C8—24no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8#4112yes—
218C4×C8#4112yes—
219C4×C8#4112yes—
220C4×C8#4112yes—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236C24C2×C22⋊C411yescharacteristic
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C2×M1614yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×M1614yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—48no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—22no—
287C4⋊C4—28no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4C2×M1614yes—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—82no—
305C42—28no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—42no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—82no—
328C42—82no—
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C23—21no—
334C23—22no—
335C23—22no—
336C23—21no—
337C23#40712yes—
338C23C22×C22⋊C411yescharacteristic
339Q8—24no—
340Q8—24no—
341Q8—42no—
342Q8—42no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4#40712yes—
382C2×C4C4×#4111yescharacteristic
383C2×C4C4×#4111yescharacteristic
384C2×C4#52511yescharacteristic
385C2×C4#52511yescharacteristic
386C2×C4#53211yescharacteristic
387C2×C4#53211yescharacteristic
388C2×C4C22×M1612yes—
389C2×C4#56812yes—
390C2×C4#57712yes—
391C2×C4#57812yes—
392C8—24no—
393C8—42no—
394C8—42no—
395C8—42no—
396C8—82no—
397C8—82no—
398C22—24no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22#229711yescharacteristic
404C22C2×#40711yescharacteristic
405C22#287911yescharacteristic
406C22#289812yes—
407C22#290112yes—
408C4—22no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—42no—
413C4—44no—
414C4—42no—
415C4—42no—
416C4—44no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—82no—
421C4#297012yes—
422C4#297012yes—
423C4#317411yescharacteristic
424C4#317711yescharacteristic
425C4#317812yes—
426C2—42no—
427C2#1745412yes—
428C2#1750612yes—
429C2#2137911yescharacteristic
430C2#2141311yescharacteristic
431C2#2141911yescharacteristic
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
4. #9876C2semidirect
5. #9875C2semidirect
6. #9647C2semidirect
7. #9643C2semidirect
10. #8175C2semidirect
11. #8173C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219262
#204732
#98762
#98752
#96472
#96432
#90732
#90712
#81752
#81732