Finite Group Databasebeta

← #160915

Subgroups — #160915

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34331C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7635C212yesmaximal, complemented
5#7628C212yesmaximal, complemented
6#7618C212yesmaximal, complemented
7#7614C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2367—21no—
16#2367C412yes—
17#2366C411yescharacteristic
18#2359—22no—
19#2355C411yescharacteristic
20#2354—21no—
21Q8×C4⋊C4C411yescharacteristic
22#2349C412yes—
23#2335—22no—
24#2326—21no—
25C4×#127C2211yescharacteristic
26C4×#124C2212yes—
27#2295C2211yescharacteristic
28#2293C2212yes—
29#2275—21no—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1939—21no—
33#1831C2211yescharacteristic
34#1831C2212yes—
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38#1767C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53#559—24no—
54#559—24no—
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×#123—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C412yes—
71C2×#117—22no—
72#446—28no—
73#446—44no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#391—21no—
87#391C2×C411yescharacteristic
88#391C2311yescharacteristic
89#390—21no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2312yes—
94#389—22no—
95#389C2×C411yescharacteristic
96#389C2×C412yes—
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—21no—
106#387—21no—
107#386—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#384—21no—
112#384C2×C411yescharacteristic
113#382C2×C412yes—
114#380—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#376—21no—
131C2×#95—22no—
132C43C2311yescharacteristic
133C43C2312yes—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158#127—22no—
159#127M1614yes—
160#124—28no—
161#124—24no—
162#124—24no—
163#123—216no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—82no—
171C4×Q8M1614yes—
172C4×D4—28no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—44no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—81no—
179C4×D4—81no—
180C4×D4—82no—
181C4×D4—82no—
182#117—82no—
183#117—82no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C22×C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—24no—
228C4⋊C8—28no—
229C4⋊C8—44no—
230C4⋊C8—44no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#97—28no—
234#97—44no—
235#97—82no—
236C4×C8—24no—
237C4×C8—42no—
238C4×C8—81no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—21no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—22no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8C2×M1612yes—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4C2×M1614yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—28no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C4⋊C4—21no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—81no—
328C4⋊C4—82no—
329C4⋊C4—81no—
330C4⋊C4—82no—
331C4⋊C4#10414yes—
332C4⋊C4C2×M1612yes—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—28no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—22no—
350C42—22no—
351C42—21no—
352C42—24no—
353C42—22no—
354C42—24no—
355C42—42no—
356C42—41no—
357C42—41no—
358C42—42no—
359C42—41no—
360C42—42no—
361C42—82no—
362C42—82no—
363C42C2×M1612yes—
364C42C2×M1612yes—
365C42AGL(1,ℤ/8)12yes—
366C42#13712yes—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23C22×C22⋊C411yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—44no—
378D4—44no—
379D4—44no—
380D4—48no—
381D4—84no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—88no—
422C2×C4—164no—
423C2×C4—164no—
424C2×C4#40712yes—
425C2×C4#40712yes—
426C2×C4#41712yes—
427C2×C4#41712yes—
428C2×C4C2×#10414yes—
429C2×C4C2×#10412yes—
430C2×C4C22×M1611yescharacteristic
431C2×C4C22×M1611yescharacteristic
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57711yescharacteristic
436C8—42no—
437C8—44no—
438C8—42no—
439C8—42no—
440C8—81no—
441C8—82no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22C2×#40711yescharacteristic
450C22#288411yescharacteristic
451C22#288811yescharacteristic
452C22C2×#41711yescharacteristic
453C22#289711yescharacteristic
454C22C22×#10411yescharacteristic
455C22#290111yescharacteristic
456C4—22no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—24no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—41no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4#148914yes—
475C2—84no—
476C2#1737211yescharacteristic
477C2C2×#148911yescharacteristic
478C2#1745311yescharacteristic
479C2#1755011yescharacteristic
480C2#1762811yescharacteristic
481C2#1765211yescharacteristic
482C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #20570C2semidirect
4. #7635C2semidirect
5. #7628C2semidirect
6. #7618C2semidirect
7. #7614C2semidirect
9. #7577C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#343312
#227362
#205702
#76352
#76282
#76182
#76142
#75872
#75772
#75692
#75622