Finite Group Databasebeta

← #162473

Subgroups — #162473

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35185C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7706C212yesmaximal
5#7697C212yesmaximal, complemented
6#7686C212yesmaximal
7#7677C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2864C2212yes—
15#2860C2212yes—
16#2650—22no—
17#2571C411yescharacteristic
18#2567C411yescharacteristic
19#2563C411yescharacteristic
20#2538C411yescharacteristic
21#2513C412yes—
22#2509C412yes—
23#2496—21no—
24#2469—22no—
25#2421—21no—
26#2323—21no—
27#2314—21no—
28C4×C4⋊Q8C2211yescharacteristic
29#2293C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1939—21no—
32#1939—22no—
33#1831C2212yes—
34#1831C2211yescharacteristic
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2211yescharacteristic
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—28no—
63C8×Q8—44no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×#97—22no—
71C2×#97—22no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#400—22no—
77#400—22no—
78#400—22no—
79#397—22no—
80#397—22no—
81#396—22no—
82#395—22no—
83#393—22no—
84#391—21no—
85#391—21no—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C411yescharacteristic
89#390C2311yescharacteristic
90#389—21no—
91#389—22no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C411yescharacteristic
98#389C2312yes—
99#389C2311yescharacteristic
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—21no—
105#388—21no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#384—21no—
113#384C2×C411yescharacteristic
114#382—21no—
115#382C2×C412yes—
116#382C2×C412yes—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—22no—
133#336—21no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C4⋊Q8—24no—
153#126—82no—
154#126—82no—
155C22⋊Q8—44no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—81no—
161C4×Q8—81no—
162C4×D4—24no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—24no—
211C4⋊C8—28no—
212C4⋊C8—44no—
213C4⋊C8—42no—
214C4⋊C8—44no—
215C4⋊C8—42no—
216#97—28no—
217#97—82no—
218C4×C8—24no—
219C4×C8—44no—
220C4×C8—42no—
221C4×C8—81no—
222#95—21no—
223#95—21no—
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#95—22no—
233#95—21no—
234#95—22no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C24—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×Q8#10414yes—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—81no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—24no—
328C42—41no—
329C42—42no—
330C42—41no—
331C42—42no—
332C42—41no—
333C42—42no—
334C42—82no—
335C42#13114yes—
336C42AGL(1,ℤ/8)12yes—
337C42#13712yes—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23C22×C22⋊C411yescharacteristic
346Q8—28no—
347Q8—84no—
348D4—82no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—84no—
389C2×C4#40812yes—
390C2×C4#40812yes—
391C2×C4#41712yes—
392C2×C4#41712yes—
393C2×C4C2×#10414yes—
394C2×C4C2×#10412yes—
395C2×C4C2×#13112yes—
396C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57611yescharacteristic
399C2×C4#57711yescharacteristic
400C8—42no—
401C8—44no—
402C8—42no—
403C8—42no—
404C8—82no—
405C8—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C2×#40811yescharacteristic
414C22#288411yescharacteristic
415C22#288911yescharacteristic
416C22C2×#41711yescharacteristic
417C22#289711yescharacteristic
418C22C22×#10411yescharacteristic
419C22#290111yescharacteristic
420C4—24no—
421C4—22no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—22no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—81no—
433C4—82no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4#148314yes—
438C2—161no—
439C2#1739111yescharacteristic
440C2C2×#148311yescharacteristic
441C2#1748311yescharacteristic
442C2#1755011yescharacteristic
443C2#1762811yescharacteristic
444C2#1765411yescharacteristic
445C2#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. #22734C2semidirect
3. #20570C2semidirect
5. #7697C2semidirect
7. #7677C2semidirect
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
#227342
#205702
#77062
#76972
#76862
#76772
#75912
#75742
#75702
#75602