Finite Group Databasebeta

← #482099

Subgroups — #482099

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35737C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11878C214yesmaximal, complemented
5#11875C214yesmaximal, complemented
6#11694C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic, complemented
9C2×#548—21no—
10C2×#520C2211yescharacteristic
11#2972—21no—
12C2×#410—21no—
13#2621C412yescomplemented
14#2621C2211yescharacteristic
15#2547C411yescharacteristic, complemented
16#2547C2212yes—
17#2535C411yescharacteristic, complemented
18#2535C2212yes—
19#2427—22no—
20#2427—21no—
21#2417—21no—
22#2417—22no—
23#2392C412yescomplemented
24#2392C2211yescharacteristic
25#2351C411yescharacteristic, complemented
26Q8×C22⋊C4—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1344C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39C23×Q8—21no—
40D4×Q8—44no—
41#548—44no—
42#548—44no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124—21no—
46C2×#124—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×#121—21no—
52C2×#121—22no—
53#520—44no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2312yes—
65C2×#117—22no—
66C2×#117C2×C412yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C8×D4—44no—
71#431—22no—
72#410—44no—
73C2×#99—24no—
74C2×C22⋊C8—24no—
75#398—22no—
76#398—21no—
77#398C2×C411yescharacteristic
78#398C2×C412yes—
79#397—22no—
80#397—21no—
81#397C2×C412yes—
82#397C2×C412yes—
83#397C2×C412yes—
84#397C2312yes—
85#397C2311yescharacteristic
86#396—22no—
87#396—21no—
88#394—22no—
89#394—21no—
90#394C2×C411yescharacteristic
91#392C2×C412yes—
92#392C2311yescharacteristic
93#391—21no—
94#391—22no—
95#391—22no—
96#391—21no—
97#389—22no—
98#389—21no—
99#389—44no—
100#386—22no—
101#386—22no—
102#386—22no—
103#382C2×C412yes—
104#382C2×C411yescharacteristic
105#380—22no—
106#380—22no—
107#380—22no—
108#380—21no—
109#380—22no—
110#380—24no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115#342—22no—
116#342C2314yes—
117#324—22no—
118#324—24no—
119#323C2314yes—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41D814yes—
123C2×#41SD1614yes—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8C22⋊C411yescharacteristic
130C22×D4—22no—
131C22×D4—21no—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22×C411yescharacteristic
134C22×D4C2×D412yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C22×C8—44no—
143C4⋊Q8—42no—
144C4⋊Q8—44no—
145#124—44no—
146#124—44no—
147#124—44no—
148C22⋊Q8—48no—
149C22⋊Q8—48no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C22⋊Q8—44no—
153#121—44no—
154#121—44no—
155#121—44no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—44no—
159C4×D4—24no—
160C4×D4—42no—
161C4×D4—44no—
162C4×D4—42no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—48no—
166C4×D4—42no—
167C4×D4—42no—
168#117—42no—
169#117—44no—
170#117—44no—
171#117—44no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22×C412yes—
207C2×C22⋊C4C2×D411yescharacteristic
208C2×C22⋊C4C2411yescharacteristic
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—42no—
212C2×C42C22⋊C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C4⋊C8—42no—
217#99—48no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220C4×C8—42no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—44no—
231#95C22×C411yescharacteristic
232#95C22×C412yes—
233#95C22×C411yescharacteristic
234C24—21no—
235C24C2×C22⋊C412yes—
236C24C22×D411yescharacteristic
237#41—48no—
238#41—48no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—816no—
248C2×Q8#10214yes—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—44no—
260C2×D4—82no—
261C2×D4—81no—
262C2×D4#10214yes—
263C2×D4C2×D814yes—
264C2×D4C2×SD1614yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#10218yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×D812yes—
296C22×C4C2×SD1612yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—44no—
300C2×C8—164no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—84no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4—164no—
319C4⋊C4#9914yes—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C22⋊C4—81no—
339C42—22no—
340C42—24no—
341C42—22no—
342C42—42no—
343C42—44no—
344C42—42no—
345C42—84no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×#10214yes—
362C23C22×C22⋊C411yescharacteristic
363C23C22×D811yescharacteristic
364C23C22×SD1611yescharacteristic
365Q8—28no—
366Q8—88no—
367Q8—84no—
368D4—28no—
369D4—44no—
370D4—44no—
371D4—42no—
372D4—42no—
373D4—82no—
374D4—82no—
375D4—88no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—88no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—162no—
417C2×C4C2×#9914yes—
418C2×C4C2×#9912yes—
419C2×C4C2×#10214yes—
420C2×C4C2×#10212yes—
421C2×C4C2×#10212yes—
422C2×C4#42112yes—
423C8—84no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—41no—
430C22—41no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—41no—
435C22—42no—
436C22—41no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22C22×#9911yescharacteristic
446C22C22×#10211yescharacteristic
447C22C2×#42111yescharacteristic
448C4—24no—
449C4—22no—
450C4—24no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4—164no—
462C4#151914yes—
463C2—21no—
464C2—21no—
465C2—41no—
466C2—41no—
467C2—44no—
468C2—82no—
469C2C2×#151911yescharacteristic
470C2#3334911yescharacteristic
471C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35737C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17351C2semidirect
4. #11878C2semidirect
5. #11875C2semidirect
6. #11694C2semidirect
7. #11691C2semidirect
8. C2×D4×Q8C4semidirect
13. #2621C4semidirect
15. #2547C4semidirect
17. #2535C4semidirect
23. #2392C4semidirect
25. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#357372
#206342
#173512
#118782
#118752
#116942
#116912