Finite Group Databasebeta

← #161111

Subgroups — #161111

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34331C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7649C212yesmaximal, complemented
5#7642C212yesmaximal, complemented
6#7618C212yesmaximal, complemented
7#7608C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13#2367—21no—
14#2367C412yes—
15#2366C411yescharacteristic
16#2359—22no—
17#2355C411yescharacteristic
18#2354—21no—
19Q8×C4⋊C4C411yescharacteristic
20#2349C412yes—
21#2335—22no—
22#2326—21no—
23C4×#127C2211yescharacteristic
24C4×#124C2212yes—
25#2295C2211yescharacteristic
26#2293C2212yes—
27#2275—21no—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1770—21no—
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—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×#97—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#391—21no—
81#391C2×C411yescharacteristic
82#391C2311yescharacteristic
83#390—21no—
84#390—21no—
85#390—21no—
86#390C2×C412yes—
87#390C2312yes—
88#389—22no—
89#389C2×C411yescharacteristic
90#389C2×C412yes—
91#389C2311yescharacteristic
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—21no—
100#387—21no—
101#386—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#384—21no—
106#384C2×C411yescharacteristic
107#382C2×C412yes—
108#380—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124#376—21no—
125C2×#95—22no—
126C43C2311yescharacteristic
127C43C2312yes—
128#336—22no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156#127—22no—
157#127M1614yes—
158#124—28no—
159#124—24no—
160#124—24no—
161#123—216no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—42no—
168C4×Q8—82no—
169C4×Q8M1614yes—
170C4×D4—28no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—44no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4—81no—
178C4×D4—82no—
179C4×D4—82no—
180#117—82no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—22no—
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×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—82no—
229#97—44no—
230C4×C8—42no—
231C4×C8—42no—
232C4×C8—81no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—22no—
248C2×Q8—21no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8C2×M1612yes—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4C2×M1614yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—28no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4C2×M1612yes—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—22no—
344C42—22no—
345C42—21no—
346C42—24no—
347C42—22no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—41no—
354C42—42no—
355C42—82no—
356C42—82no—
357C42C2×M1612yes—
358C42C2×M1612yes—
359C42#13512yes—
360C42#13512yes—
361C23—24no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23C22×C22⋊C411yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—44no—
372D4—44no—
373D4—44no—
374D4—48no—
375D4—84no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—88no—
417C2×C4—164no—
418C2×C4—164no—
419C2×C4#40712yes—
420C2×C4#40712yes—
421C2×C4#41612yes—
422C2×C4#41612yes—
423C2×C4#42112yes—
424C2×C4C22×M1611yescharacteristic
425C2×C4C22×M1611yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C8—42no—
431C8—81no—
432C8—82no—
433C8—81no—
434C8—81no—
435C8—82no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22C2×#40711yescharacteristic
444C22#288411yescharacteristic
445C22#288811yescharacteristic
446C22C2×#41611yescharacteristic
447C22#289711yescharacteristic
448C22C2×#42111yescharacteristic
449C22#290111yescharacteristic
450C4—22no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4—82no—
467C4—81no—
468C4#148814yes—
469C2—84no—
470C2#1737211yescharacteristic
471C2C2×#148811yescharacteristic
472C2#1745311yescharacteristic
473C2#1760211yescharacteristic
474C2#1761611yescharacteristic
475C2#1768111yescharacteristic
476C2#1772411yescharacteristic
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. #22832C2semidirect
3. #22018C2semidirect
4. #7649C2semidirect
5. #7642C2semidirect
6. #7618C2semidirect
7. #7608C2semidirect
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
#228322
#220182
#76492
#76422
#76182
#76082
#75872
#75772
#75692
#75622