Finite Group Databasebeta

← #161859

Subgroups — #161859

Order 512.

705 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#35185C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal
5#7717C212yesmaximal, complemented
6#7712C212yesmaximal
7#7697C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#446—21no—
13#2864C2212yes—
14#2860C2212yes—
15#2650—22no—
16#2571C411yescharacteristic
17#2567C411yescharacteristic
18#2563C411yescharacteristic
19#2538C411yescharacteristic
20#2513C412yes—
21#2509C412yes—
22#2496—21no—
23#2469—22no—
24#2421—21no—
25#2323—21no—
26#2314—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2293C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2212yes—
38#1831C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#126—22no—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—21no—
62#446—44no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#400—22no—
76#397—22no—
77#397—22no—
78#396—22no—
79#395—22no—
80#393—22no—
81#391—21no—
82#391—21no—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2×C411yescharacteristic
86#390C2311yescharacteristic
87#389—21no—
88#389—22no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C411yescharacteristic
95#389C2312yes—
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—21no—
102#388—21no—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C4⋊Q8—24no—
152#126—82no—
153#126—82no—
154C22⋊Q8—44no—
155C4×Q8—24no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—42no—
159C4×Q8—81no—
160C4×Q8—81no—
161C4×D4—24no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—44no—
211C4⋊C8—44no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—82no—
215#97—44no—
216C4×C8—42no—
217C4×C8—42no—
218C4×C8—81no—
219#95—21no—
220#95—21no—
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—21no—
231#95—22no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C24—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—81no—
241C2×Q8—81no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—44no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—44no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—81no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—21no—
322C42—22no—
323C42—22no—
324C42—41no—
325C42—42no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—42no—
331C42—82no—
332C42#13114yes—
333C42AGL(1,ℤ/8)12yes—
334C42#13712yes—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23C22×C22⋊C411yescharacteristic
343Q8—44no—
344Q8—84no—
345D4—82no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—84no—
387C2×C4#40812yes—
388C2×C4#40812yes—
389C2×C4#41712yes—
390C2×C4#41712yes—
391C2×C4#42112yes—
392C2×C4C2×#13112yes—
393C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4#57611yescharacteristic
396C2×C4#57711yescharacteristic
397C8—42no—
398C8—81no—
399C8—82no—
400C8—81no—
401C8—81no—
402C8—82no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C2×#40811yescharacteristic
411C22#288411yescharacteristic
412C22#288911yescharacteristic
413C22C2×#41711yescharacteristic
414C22#289711yescharacteristic
415C22C2×#42111yescharacteristic
416C22#290111yescharacteristic
417C4—22no—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4#149514yes—
435C2—161no—
436C2#1739111yescharacteristic
437C2#1746811yescharacteristic
438C2C2×#149511yescharacteristic
439C2#1760211yescharacteristic
440C2#1762811yescharacteristic
441C2#1768311yescharacteristic
442C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22736C2semidirect
3. #22018C2semidirect
5. #7717C2semidirect
7. #7697C2semidirect
8. #7591C2semidirect
10. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#351852
#227362
#220182
#77322
#77172
#77122
#76972
#75912
#75742
#75702
#75602