Finite Group Databasebeta

← #160653

Subgroups — #160653

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34331C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7601C212yesmaximal, complemented
5#7594C212yesmaximal, complemented
6#7587C211yesmaximal, characteristic
7#7584C212yesmaximal, complemented
8#7577C211yesmaximal, characteristic, complemented
9#7574C212yesmaximal, complemented
10#7569C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×C8×Q8—21no—
14#2367—21no—
15#2367C412yes—
16#2366C411yescharacteristic
17#2359—22no—
18#2355C411yescharacteristic
19#2354—21no—
20Q8×C4⋊C4C411yescharacteristic
21#2349C412yes—
22#2335—22no—
23#2326—21no—
24C4×#127C2211yescharacteristic
25C4×#124C2212yes—
26#2295C2211yescharacteristic
27#2293C2212yes—
28#2275—21no—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—22no—
33#1939—21no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
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#559C818yescomplemented
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—
72C8×Q8—44no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—21no—
84#391C2×C411yescharacteristic
85#391C2311yescharacteristic
86#390—21no—
87#390—21no—
88#390—21no—
89#390C2×C412yes—
90#390C2312yes—
91#389—22no—
92#389C2×C411yescharacteristic
93#389C2×C412yes—
94#389C2311yescharacteristic
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—21no—
103#387—21no—
104#386—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#384C2×C411yescharacteristic
110#382C2×C412yes—
111#380—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127#376—21no—
128C2×#95—22no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—22no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157#127—22no—
158#127C2×C814yes—
159#124—28no—
160#124—24no—
161#124C2×C818yes—
162#123—216no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—42no—
167C4×Q8—82no—
168C4×Q8C2×C814yes—
169C4×Q8C2×C814yes—
170C4×Q8M1614yes—
171C4×D4—28no—
172C4×D4—24no—
173C4×D4—44no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4—81no—
178C4×D4—82no—
179C4×D4—82no—
180C4×D4C2×C818yes—
181#117—82no—
182#117—82no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—22no—
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×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—42no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—82no—
232C4×C8—42no—
233C4×C8—44no—
234C4×C8—42no—
235C4×C8—81no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250C24—22no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8C22⋊C814yes—
257C2×Q8C22×C812yes—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4C22×C814yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—28no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—81no—
326C4⋊C4—82no—
327C4⋊C4C22⋊C814yes—
328C4⋊C4C22⋊C818yes—
329C4⋊C4C22×C812yes—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—21no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—24no—
348C42—22no—
349C42—42no—
350C42—41no—
351C42—41no—
352C42—42no—
353C42—42no—
354C42—41no—
355C42—42no—
356C42—82no—
357C42—82no—
358C42C22×C814yes—
359C42C22×C812yes—
360C42C2×M1614yes—
361C42C2×M1612yes—
362C42AGL(1,ℤ/8)12yes—
363C42#13712yes—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23C22×C22⋊C411yescharacteristic
372Q8—28no—
373Q8—42no—
374Q8—42no—
375D4—44no—
376D4—44no—
377D4—48no—
378D4—84no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—88no—
418C2×C4—164no—
419C2×C4—164no—
420C2×C4C2×C22⋊C814yes—
421C2×C4C2×C22⋊C814yes—
422C2×C4C2×C22⋊C812yes—
423C2×C4C2×C22⋊C812yes—
424C2×C4#41712yes—
425C2×C4#41712yes—
426C2×C4#42112yes—
427C2×C4C23×C811yescharacteristic
428C2×C4C22×M1611yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4C2×#13711yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57711yescharacteristic
433C8—42no—
434C8—81no—
435C8—82no—
436C8—81no—
437C8—82no—
438C8—81no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22C22×C22⋊C811yescharacteristic
447C22#288411yescharacteristic
448C22#288811yescharacteristic
449C22C2×#41711yescharacteristic
450C22#289711yescharacteristic
451C22C2×#42111yescharacteristic
452C22#290111yescharacteristic
453C4—24no—
454C4—24no—
455C4—22no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—82no—
470C4—81no—
471C4#147714yes—
472C2—84no—
473C2#1735311yescharacteristic
474C2C2×#147711yescharacteristic
475C2#1743811yescharacteristic
476C2#1760211yescharacteristic
477C2#1762811yescharacteristic
478C2#1768111yescharacteristic
479C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C2×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. #22018C2semidirect
4. #7601C2semidirect
5. #7594C2semidirect
7. #7584C2semidirect
8. #7577C2semidirect
9. #7574C2semidirect
11. #7562C2semidirect
54. #559C8semidirect

Maximal subgroups

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

subgroupindex
#343312
#227342
#220182
#76012
#75942
#75872
#75842
#75772
#75742
#75692
#75622