Finite Group Databasebeta

← #482970

Subgroups — #482970

Order 512.

754 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic
2#33486C211yesmaximal, characteristic, complemented
3#22717C211yesmaximal, characteristic, complemented
4#11893C212yesmaximal, complemented
5#11892C212yesmaximal, complemented
6#11863C211yesmaximal, characteristic, complemented
7#11861C211yesmaximal, characteristic, complemented
8#11749C211yesmaximal, characteristic, complemented
9#11748C211yesmaximal, characteristic, complemented
10#11709C212yesmaximal, complemented
11C2×#1350C212yesmaximal, complemented
12#3520—21no—
13#3342C411yescharacteristic
14#3336C412yes—
15#3334C411yescharacteristic
16#3302—21no—
17#3300—22no—
18#3298—21no—
19D4×Q16C411yescharacteristic
20#3285C412yes—
21#3283C411yescharacteristic
22#3248—21no—
23#3190—22no—
24#3170C2211yescharacteristic
25#3168C2211yescharacteristic
26#3050C2212yes—
27C2×#467C2212yes—
28D4×M16—21no—
29C2×C8×D4C2211yescharacteristic
30D4×C16—21no—
31D4×C16—22no—
32D4×C16—21no—
33#2162C2211yescharacteristic
34#2162C2211yescharacteristic
35C2×C22⋊C16C2212yes—
36C2×C22⋊C16C2212yes—
37#1950—21no—
38#1855—22no—
39#1837C2211yescharacteristic
40#1810C2212yes—
41C4×#108C2211yescharacteristic
42C2×#341C2212yes—
43#1350C2212yes—
44#1350C2212yes—
45#1350C2212yes—
46#1350C2211yescharacteristic
47#1350C2212yes—
48#1350C2211yescharacteristic
49#1350C2212yes—
50#1349C2212yes—
51#1349C2211yescharacteristic
52#1349C2211yescharacteristic
53C2×#135C2×C412yes—
54C22×Q16C2×C412yes—
55C22×M16—22no—
56C23×C8C2312yes—
57#556—21no—
58#554—41no—
59#552—22no—
60#551—42no—
61D4×Q8—21no—
62#548—22no—
63#547—21no—
64#547—21no—
65#546—41no—
66#542—41no—
67#539—42no—
68#533—21no—
69#529—21no—
70#525—22no—
71C2×C22⋊Q8—22no—
72#520—41no—
73C2×C4×D4D411yescharacteristic
74C22×C16—22no—
75C22×C16—22no—
76#500C2×C411yescharacteristic
77#499C2×C411yescharacteristic
78#495C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#488—21no—
81#487—21no—
82#485—22no—
83#485—22no—
84#483—22no—
85#480—22no—
86#479—22no—
87#475—21no—
88#474—21no—
89#467—21no—
90#467—22no—
91#467—22no—
92#467C2×C412yes—
93#467C2×C412yes—
94#467C2×C412yes—
95#467C2311yescharacteristic
96#467C2312yes—
97#466C2×C412yes—
98#466C2311yescharacteristic
99#465—21no—
100#465—22no—
101#465—22no—
102#464—22no—
103#462—21no—
104#461—21no—
105#452—22no—
106#452—22no—
107#451—22no—
108#448—22no—
109C4×Q16—21no—
110C4×Q16C2×C412yes—
111C4×Q16C2×C411yescharacteristic
112C4×SD16—21no—
113C4×D8C2×C411yescharacteristic
114#436—41no—
115#435—42no—
116C8×D4—41no—
117C8×D4C2×C411yescharacteristic
118C8×D4C2×C412yes—
119C8×D4C2×C411yescharacteristic
120C8×D4D411yescharacteristic
121C8×D4D412yes—
122C8×D4D411yescharacteristic
123C2×#108—21no—
124C2×#108—22no—
125#427C2×C412yes—
126C2×#107C2×C412yes—
127#423—21no—
128C2×C4⋊C8D411yescharacteristic
129#416—22no—
130C2×#103—22no—
131#407—22no—
132C2×C22⋊C8D412yes—
133C4×M16—21no—
134C2×C4×C8C2311yescharacteristic
135C4⋊C16—21no—
136C4⋊C16—21no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139C22⋊C16C2311yescharacteristic
140C22⋊C16C2312yes—
141C22⋊C16C2311yescharacteristic
142C4×C16—21no—
143C4×C16—21no—
144#341—21no—
145#341—22no—
146#341C2311yescharacteristic
147#341C2312yes—
148#341C2311yescharacteristic
149C2×#41—22no—
150C2×#41—41no—
151C22×Q8—22no—
152C22×D4SD1611yescharacteristic
153C23×C4C2×D412yes—
154#135—82no—
155C2×Q16—22no—
156C2×Q16—21no—
157C2×Q16—82no—
158C2×Q16C22×C411yescharacteristic
159C2×Q16C22×C412yes—
160C2×SD16—22no—
161C2×SD16—21no—
162C2×D8C22×C411yescharacteristic
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—42no—
167C2×M16—44no—
168C2×M16—82no—
169C22×C8—42no—
170C22×C8—82no—
171C22×C8C22×C412yes—
172C22×C8C22×C412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—41no—
181#126—42no—
182#126—42no—
183#126—41no—
184#125—21no—
185#125—42no—
186#125—41no—
187#124—21no—
188#124—41no—
189#124—42no—
190#123—42no—
191#123—42no—
192#123—41no—
193#123—42no—
194#123—42no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—21no—
199C22⋊Q8—22no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—41no—
203C22⋊Q8—42no—
204C22⋊Q8—41no—
205C22⋊Q8—42no—
206#121—22no—
207#121—21no—
208#121—42no—
209#121—41no—
210#120—42no—
211C4×Q8—21no—
212C4×Q8—22no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×D4—21no—
216C4×D4—41no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—41no—
220C4×D4—42no—
221C4×D4C22⋊C411yescharacteristic
222C4×D4C22⋊C411yescharacteristic
223C4×D4C22⋊C412yes—
224C4×D4D811yescharacteristic
225C4×D4D812yes—
226C4×D4D811yescharacteristic
227#117—22no—
228#117—41no—
229#117—42no—
230#117—41no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4SD1611yescharacteristic
235C2×C22⋊C4—42no—
236C2×C22⋊C4SD1612yes—
237C2×C42C2×D411yescharacteristic
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—22no—
241C2×C16—82no—
242C2×C16—82no—
243#108—42no—
244#108—82no—
245#107—22no—
246#107—21no—
247#107C22×C411yescharacteristic
248#107C22×C412yes—
249#107C22×C411yescharacteristic
250#106—22no—
251#106—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8C22⋊C411yescharacteristic
255C4⋊C8C22⋊C411yescharacteristic
256C4⋊C8C22⋊C412yes—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—22no—
261#103—21no—
262#103—21no—
263#103—22no—
264#102—22no—
265#102—21no—
266#102—21no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271C22⋊C8C22⋊C412yes—
272C22⋊C8C22⋊C412yes—
273#97—41no—
274C4×C8—41no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D411yescharacteristic
279C24C2×SD1612yes—
280#41—82no—
281#41—82no—
282#41—84no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×D4—24no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4#10212yes—
295C2×D4#10211yescharacteristic
296C2×D4#10211yescharacteristic
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×D812yes—
306C22×C4C2×D812yes—
307C22×C4C2×SD1611yescharacteristic
308C22×C4C2×SD1611yescharacteristic
309C22×C4C2×SD1612yes—
310C22×C4C22×D411yescharacteristic
311Q16—41no—
312Q16—42no—
313Q16—81no—
314Q16—82no—
315SD16—81no—
316SD16—82no—
317D8—41no—
318M16—22no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—42no—
323M16—42no—
324M16—84no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—84no—
336C2×C8C2×C22⋊C411yescharacteristic
337C2×C8C2×C22⋊C411yescharacteristic
338C2×C8C2×C22⋊C412yes—
339C2×C8C2×C22⋊C412yes—
340C2×C8C23×C411yescharacteristic
341C2×C8C22×D411yescharacteristic
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4#10211yescharacteristic
359C4⋊C4#10212yes—
360C4⋊C4#10211yescharacteristic
361C22⋊C4—41no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4#10212yes—
373C22⋊C4#10212yes—
374C22⋊C4#10212yes—
375C22⋊C4#10212yes—
376C42—42no—
377C42—42no—
378C42—41no—
379C42C2×C22⋊C411yescharacteristic
380C42C2×C22⋊C411yescharacteristic
381C42C2×D811yescharacteristic
382C42C2×D811yescharacteristic
383C16—41no—
384C16—41no—
385C16—42no—
386C23—24no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23C2×#10212yes—
391C23C2×#10212yes—
392C23C22×SD1611yescharacteristic
393Q8—42no—
394Q8—41no—
395Q8—41no—
396Q8—42no—
397Q8—82no—
398Q8—162no—
399D4—24no—
400D4—22no—
401D4—41no—
402D4—41no—
403D4—82no—
404D4—84no—
405D4—162no—
406D4#35914yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—162no—
426C2×C4—162no—
427C2×C4C2×#10211yescharacteristic
428C2×C4C2×#10212yes—
429C2×C4C2×#10211yescharacteristic
430C2×C4C2×#10211yescharacteristic
431C2×C4C2×#10212yes—
432C2×C4C2×#10211yescharacteristic
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C22×D811yescharacteristic
435C8—21no—
436C8—21no—
437C8—22no—
438C8—22no—
439C8—21no—
440C8—21no—
441C8—42no—
442C8—82no—
443C8—81no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—22no—
450C22—21no—
451C22—22no—
452C22—84no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C2×#35914yes—
457C22C22×#10211yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—22no—
464C4—81no—
465C4—82no—
466C4—81no—
467C4—82no—
468C4—81no—
469C4C2×#35912yes—
470C4#288311yescharacteristic
471C4#290111yescharacteristic
472C2—24no—
473C2—42no—
474C2—41no—
475C2—161no—
476C2#1761311yescharacteristic
477C2C22×#35911yescharacteristic
478C2#3339311yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62252C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33486C2semidirect
3. #22717C2semidirect
4. #11893C2semidirect
5. #11892C2semidirect
6. #11863C2semidirect
7. #11861C2semidirect
8. #11749C2semidirect
9. #11748C2semidirect
10. #11709C2semidirect
11. C2×#1350C2semidirect

Maximal subgroups

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

subgroupindex
#622522
#334862
#227172
#118932
#118922
#118632
#118612
#117492
#117482
#117092
C2×#13502