Finite Group Databasebeta

← #482108

Subgroups — #482108

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36018C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11882C214yesmaximal, complemented
5#11879C214yesmaximal, complemented
6#11699C212yesmaximal, complemented
7#11697C212yesmaximal, complemented
8C2×#550C411yescharacteristic
9C2×#548—21no—
10C2×#519C2211yescharacteristic
11#2982—21no—
12#2905—21no—
13C22×#101—21no—
14#2619C412yes—
15#2619C2211yescharacteristic
16#2548C411yescharacteristic
17#2548C2212yes—
18#2536C411yescharacteristic
19#2536C2212yes—
20#2426—22no—
21#2426—21no—
22#2414—22no—
23#2414—21no—
24#2392C412yes—
25#2392C2211yescharacteristic
26#2351C411yescharacteristic
27#2341—21no—
28#1801—22no—
29#1801C2214yes—
30#1742C2212yes—
31#1461—21no—
32#1461C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1346C2214yes—
36#1345C2218yes—
37#1345C2214yes—
38C22×#41—21no—
39C23×Q8D411yescharacteristic
40C22×M16—22no—
41#550—44no—
42#548—44no—
43#548—44no—
44C2×#127C2×C412yes—
45C2×#127C2×C411yescharacteristic
46C2×#124—21no—
47C2×#124—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×#121—22no—
51C2×#121—21no—
52C2×#121—22no—
53C2×#121—21no—
54#519—44no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×#117C2×C412yes—
67C2×#117D412yes—
68C2×#117D411yescharacteristic
69C2×#117C2311yescharacteristic
70C8×Q8—44no—
71#431—24no—
72#431—22no—
73#424—44no—
74#423—24no—
75C2×#101—24no—
76C2×#101—44no—
77#398C2×C412yes—
78#398C2311yescharacteristic
79#397C2×C411yescharacteristic
80#396—22no—
81#396—21no—
82#394—22no—
83#394—21no—
84#394—22no—
85#394—21no—
86#394C2×C412yes—
87#394C2×C412yes—
88#394C2×C412yes—
89#394C2312yes—
90#394C2311yescharacteristic
91#392—22no—
92#392—21no—
93#392C2×C411yescharacteristic
94#392C2×C412yes—
95#390—21no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390—21no—
100#390—22no—
101#390—22no—
102#389—22no—
103#389—21no—
104#388—22no—
105#388—22no—
106#388—22no—
107#382C2×C412yes—
108#382C2×C411yescharacteristic
109#380—21no—
110#380—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115#343—22no—
116#343C2314yes—
117#324—24no—
118#324C2314yes—
119#323—22no—
120C2×#41—44no—
121C2×#41—44no—
122C2×#41—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8D814yes—
128C22×Q8SD1614yes—
129C22×Q8C2×D412yes—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4C22×C412yes—
137C22×D4C22×C411yescharacteristic
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C22⋊C412yes—
141C23×C4C2×D411yescharacteristic
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—42no—
147#127—44no—
148#127—42no—
149#124—44no—
150#124—44no—
151#124—44no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155#121—48no—
156#121—48no—
157#121—42no—
158#121—42no—
159#121—44no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—44no—
163C4×Q8—44no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—48no—
170C4×D4—42no—
171C4×D4—42no—
172#117—24no—
173#117—42no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22×C411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C42—22no—
210C2×C42—21no—
211C2×C42C22⋊C412yes—
212C2×C42C22⋊C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C4⋊C8—48no—
219C4⋊C8—44no—
220C4⋊C8—42no—
221#101—48no—
222C22⋊C8—24no—
223C4×C8—44no—
224C4×C8—42no—
225#95—21no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95C22×C411yescharacteristic
231#95C22×C411yescharacteristic
232#95C22×C412yes—
233C24—22no—
234C24—21no—
235C24C2×C22⋊C411yescharacteristic
236#41—816no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8#10214yes—
246C2×Q8#10214yes—
247C2×Q8C2×D814yes—
248C2×Q8C2×SD1614yes—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C2×D4—81no—
263C2×D4—82no—
264C2×D4—81no—
265C2×D4—82no—
266C2×D4—81no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#10218yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×D812yes—
293C22×C4C2×SD1612yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—44no—
298M16—44no—
299M16—84no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—84no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4#9914yes—
319C22⋊C4—24no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4—82no—
339C42—24no—
340C42—22no—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42—44no—
347C42—42no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23—42no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23C2×#10212yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—28no—
369Q8—28no—
370Q8—44no—
371Q8—44no—
372D4—88no—
373D4—82no—
374D4—84no—
375D4—88no—
376D4—84no—
377D4—84no—
378D4—164no—
379D4—162no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4C2×#9914yes—
415C2×C4C2×#9912yes—
416C2×C4C2×#10214yes—
417C2×C4C2×#10214yes—
418C2×C4C2×#10212yes—
419C2×C4#42112yes—
420C2×C4C22×D811yescharacteristic
421C2×C4C22×SD1611yescharacteristic
422C8—84no—
423C8—84no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22—81no—
432C22—82no—
433C22—81no—
434C22—82no—
435C22—82no—
436C22—81no—
437C22—82no—
438C22—81no—
439C22—81no—
440C22—82no—
441C22—81no—
442C22—82no—
443C22—82no—
444C22—81no—
445C22C22×#9911yescharacteristic
446C22C22×#10211yescharacteristic
447C22C2×#42111yescharacteristic
448C4—24no—
449C4—21no—
450C4—21no—
451C4—24no—
452C4—24no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4#151914yes—
463C2—21no—
464C2—21no—
465C2—42no—
466C2—161no—
467C2—162no—
468C2C2×#151911yescharacteristic
469C2#3335111yescharacteristic
470C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36018C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20639C2semidirect
3. #17352C2semidirect
4. #11882C2semidirect
5. #11879C2semidirect
6. #11699C2semidirect
7. #11697C2semidirect

Maximal subgroups

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

subgroupindex
#360182
#206392
#173522
#118822
#118792
#116992
#116972