Finite Group Databasebeta

← #481905

Subgroups — #481905

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35704C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11824C214yesmaximal, complemented
5#11820C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11693C212yesmaximal, complemented
8C2×#550C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2983—21no—
11C2×#410—21no—
12#2635C2212yes—
13#2619C2211yescharacteristic
14#2591C412yescomplemented
15#2581C411yescharacteristic, complemented
16#2548C411yescharacteristic, complemented
17#2476—22no—
18#2456—22no—
19#2445—21no—
20#2431—21no—
21#2406C412yescomplemented
22#2396C2211yescharacteristic
23#2388C2212yes—
24#2367—21no—
25#2361—21no—
26#2357C411yescharacteristic, complemented
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1466C2214yes—
32#1464—22no—
33#1460—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1343C2218yes—
38C22×#41C2×C412yes—
39C22×#41D411yescharacteristic
40#550—44no—
41C2×#127—21no—
42C2×#127C2×C411yescharacteristic
43C2×#124—21no—
44C2×#123—22no—
45C2×#121—21no—
46C2×#121—22no—
47C2×#121C2×C412yes—
48#520—44no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117D412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63#446—44no—
64#432—24no—
65#431—22no—
66#410—44no—
67C2×#99—24no—
68#400C2×C412yes—
69#400C2×C411yescharacteristic
70#399C2311yescharacteristic
71#398—21no—
72#396—21no—
73#395C2×C412yes—
74#395C2×C411yescharacteristic
75#394—21no—
76#394—22no—
77#394—22no—
78#394C2×C412yes—
79#394C2×C412yes—
80#394C2×C412yes—
81#393C2311yescharacteristic
82#392—22no—
83#392—22no—
84#392—21no—
85#392C2312yes—
86#391—21no—
87#390—21no—
88#390—21no—
89#390—22no—
90#390—21no—
91#389—21no—
92#389—44no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#385—22no—
103#385—22no—
104#382C2×C411yescharacteristic
105#381C2×C412yes—
106#380—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—44no—
123C2×#41—84no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C22⋊C411yescharacteristic
132C22×D4C22×C412yes—
133C22×D4C2×D412yes—
134C23×C4—22no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140#127—42no—
141#127—44no—
142#124—44no—
143#123—84no—
144#121—48no—
145#121—42no—
146#121—42no—
147#121—44no—
148C4×Q8—24no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×D4—24no—
152C4×D4—42no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—44no—
156C4×D4—84no—
157#117—42no—
158#117—44no—
159#117—44no—
160#117—44no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C22⋊C4C2×D412yes—
195C2×C22⋊C4C2411yescharacteristic
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42C22⋊C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C4⋊C8—42no—
205C4⋊C8—44no—
206#99—48no—
207C22⋊C8—24no—
208#97—44no—
209C4×C8—42no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—44no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—22no—
224C24C2×C22⋊C412yes—
225C24C22×D411yescharacteristic
226#41—48no—
227#41—48no—
228#41—88no—
229#41—88no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4#13712yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—164no—
306C4⋊C4#9914yes—
307C22⋊C4—24no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—81no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—22no—
326C42—24no—
327C42—22no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—84no—
332C42—82no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23C22×C22⋊C411yescharacteristic
350C23C2×AGL(1,ℤ/8)11yescharacteristic
351C23C2×#13711yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—44no—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—88no—
359D4—84no—
360D4—84no—
361D4—84no—
362D4—84no—
363D4—162no—
364D4—164no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—48no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—88no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—162no—
406C2×C4C2×#9914yes—
407C2×C4C2×#9912yes—
408C2×C4#41712yes—
409C2×C4#41712yes—
410C2×C4#42112yes—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22C22×#9911yescharacteristic
434C22C2×#41711yescharacteristic
435C22C2×#42111yescharacteristic
436C4—24no—
437C4—22no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—164no—
450C4#151314yes—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2—162no—
457C2C2×#151311yescharacteristic
458C2#3334811yescharacteristic
459C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35704C2
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. #17372C2semidirect
4. #11824C2semidirect
5. #11820C2semidirect
6. #11715C2semidirect
7. #11693C2semidirect
8. C2×#550C4semidirect
14. #2591C4semidirect
15. #2581C4semidirect
16. #2548C4semidirect
21. #2406C4semidirect
26. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#357042
#206342
#173722
#118242
#118202
#117152
#116932