Finite Group Databasebeta

← #482996

Subgroups — #482996

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62363C211yesmaximal, characteristic
2#33497C211yesmaximal, characteristic, complemented
3#22746C211yesmaximal, characteristic, complemented
4#11866C212yesmaximal, complemented
5#11864C212yesmaximal, complemented
6#11862C211yesmaximal, characteristic, complemented
7#11861C211yesmaximal, characteristic, complemented
8#11751C212yesmaximal, complemented
9#11750C212yesmaximal, complemented
10#11749C211yesmaximal, characteristic, complemented
11#11747C211yesmaximal, characteristic, complemented
12#3522—21no—
13#3396C412yes—
14#3396C411yescharacteristic
15#3391C411yescharacteristic
16Q8×Q16C411yescharacteristic
17#3380C411yescharacteristic
18#3380C412yes—
19#3362—21no—
20#3361—22no—
21#3361—21no—
22#3247—21no—
23#3247—22no—
24#3169C2211yescharacteristic
25#3169C2212yes—
26#3168C2211yescharacteristic
27#3168C2212yes—
28Q8×M16—21no—
29C2×C8×Q8C2211yescharacteristic
30#2183—22no—
31#2183—21no—
32#2183—21no—
33#2163C2212yes—
34#2163C2212yes—
35#2162C2211yescharacteristic
36#2162C2211yescharacteristic
37#1950—22no—
38#1950—21no—
39#1837C2211yescharacteristic
40#1837C2212yes—
41C4×#108C2211yescharacteristic
42C4×#108C2212yes—
43#1350C2212yes—
44#1350C2211yescharacteristic
45#1350C2211yescharacteristic
46#1349C2212yes—
47#1349C2212yes—
48#1349C2212yes—
49#1349C2211yescharacteristic
50#1349C2212yes—
51#1349C2212yes—
52#1349C2211yescharacteristic
53#559—21no—
54#559—22no—
55Q82—21no—
56#554—21no—
57#554—22no—
58#550—21no—
59#548—41no—
60#547—41no—
61#547—42no—
62#542—41no—
63#542—42no—
64#532—21no—
65#532—22no—
66#529—21no—
67#529—22no—
68#520—41no—
69C2×C4×Q8D411yescharacteristic
70#500C2×C411yescharacteristic
71#500C2×C412yes—
72#499C2×C411yescharacteristic
73#499C2×C412yes—
74#495C2×C411yescharacteristic
75#495C2×C412yes—
76#493C2×C411yescharacteristic
77#493C2×C412yes—
78#488—21no—
79#488—22no—
80#486—21no—
81#486—22no—
82#478—22no—
83#478—22no—
84#478—22no—
85#477—22no—
86#477—21no—
87#476—22no—
88#476—21no—
89#467C2311yescharacteristic
90#466—21no—
91#466—22no—
92#466C2311yescharacteristic
93#466C2312yes—
94#465—21no—
95#465—22no—
96#463—22no—
97#463—22no—
98#463—22no—
99#460—22no—
100#460—21no—
101#459—22no—
102#459—21no—
103#446—41no—
104#446—42no—
105C8×Q8—41no—
106C8×Q8C2×C412yes—
107C8×Q8C2×C411yescharacteristic
108C8×Q8C2×C411yescharacteristic
109C8×Q8D412yes—
110C8×Q8D411yescharacteristic
111C8×Q8D411yescharacteristic
112C4×Q16C2×C411yescharacteristic
113C4×Q16C2×C412yes—
114C4×SD16—21no—
115C4×SD16—22no—
116C4×D8—21no—
117C4×D8—22no—
118C4×D8C2×C411yescharacteristic
119C4×D8C2×C412yes—
120C4×D8C2×C412yes—
121C4×D8C2×C412yes—
122C4×D8C2×C412yes—
123C2×#108—21no—
124C2×#108—22no—
125#423—21no—
126#423—22no—
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8D412yes—
129C4×M16—21no—
130C4×M16—22no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16—22no—
136C4⋊C16—22no—
137C22⋊C16C2312yes—
138C22⋊C16C2311yescharacteristic
139C22⋊C16C2311yescharacteristic
140#346—22no—
141#346—22no—
142C4×C16—21no—
143C4×C16—21no—
144#341—21no—
145#341—22no—
146#341C2312yes—
147#341C2311yescharacteristic
148#341C2311yescharacteristic
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—42no—
152C22×Q8SD1611yescharacteristic
153C2×Q16C22×C411yescharacteristic
154C2×SD16—21no—
155C2×SD16—22no—
156C2×D8—21no—
157C2×D8—22no—
158C2×D8C22×C411yescharacteristic
159C2×D8C22×C412yes—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D412yes—
165C22×C8C2411yescharacteristic
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—42no—
169C4⋊Q8—41no—
170#127—22no—
171#127—21no—
172#126—42no—
173#126—41no—
174#125—21no—
175#125—22no—
176#124—21no—
177#124—22no—
178#124—41no—
179#124—42no—
180#124—42no—
181#124—41no—
182#123—41no—
183#123—42no—
184#123—42no—
185#123—42no—
186#123—42no—
187C22⋊Q8—21no—
188C22⋊Q8—41no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193#121—21no—
194#121—22no—
195#121—41no—
196#121—42no—
197#121—42no—
198#121—41no—
199C4×Q8—22no—
200C4×Q8—21no—
201C4×Q8—41no—
202C4×Q8C22⋊C411yescharacteristic
203C4×Q8C22⋊C411yescharacteristic
204C4×Q8C22⋊C412yes—
205C4×Q8D811yescharacteristic
206C4×Q8D811yescharacteristic
207C4×Q8D812yes—
208C4×D4—22no—
209C4×D4—21no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—42no—
219#117—41no—
220#117—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4SD1611yescharacteristic
224C2×C4⋊C4SD1612yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C16—22no—
228C2×C16—21no—
229C2×C16—21no—
230#108—42no—
231#108—44no—
232#107—21no—
233#107—22no—
234#107C22×C411yescharacteristic
235#107C22×C412yes—
236#107C22×C411yescharacteristic
237#106—21no—
238#106—22no—
239C4⋊C8—41no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—41no—
243C4⋊C8C22⋊C412yes—
244C4⋊C8C22⋊C411yescharacteristic
245C4⋊C8C22⋊C412yes—
246C4⋊C8C22⋊C411yescharacteristic
247C4⋊C8C22⋊C412yes—
248C4⋊C8C22⋊C412yes—
249C4⋊C8C22⋊C412yes—
250#103—21no—
251#103—22no—
252#103—21no—
253#102—21no—
254#102—22no—
255#102—22no—
256#102—22no—
257#102—22no—
258#102—22no—
259#102—21no—
260#97—42no—
261#97—41no—
262C4×C8—41no—
263C4×C8—42no—
264C4×C8C22×C411yescharacteristic
265C4×C8C22×C412yes—
266C4×C8C22×C411yescharacteristic
267C4×C8C22×C412yes—
268C4×C8C2×D411yescharacteristic
269C4×C8C2×D411yescharacteristic
270C4×C8C2×D412yes—
271C4×C8C2×D412yes—
272#41—84no—
273#41—162no—
274#41—164no—
275C2×Q8—21no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8#10211yescharacteristic
280C2×Q8#10211yescharacteristic
281C2×Q8#10212yes—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4C2×SD1611yescharacteristic
296C22×C4C2×SD1611yescharacteristic
297C22×C4C2×SD1612yes—
298C22×C4C2×SD1612yes—
299C22×C4C22×D411yescharacteristic
300Q16—41no—
301SD16—81no—
302SD16—82no—
303D8—41no—
304D8—42no—
305D8—81no—
306D8—82no—
307M16—22no—
308M16—41no—
309M16—41no—
310M16—42no—
311M16—42no—
312M16—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8C2×C22⋊C411yescharacteristic
320C2×C8C2×C22⋊C412yes—
321C2×C8C2×C22⋊C411yescharacteristic
322C2×C8C2×C22⋊C412yes—
323C2×C8C23×C411yescharacteristic
324C2×C8C22×D411yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4#10212yes—
338C4⋊C4#10211yescharacteristic
339C4⋊C4#10212yes—
340C4⋊C4#10211yescharacteristic
341C4⋊C4#10212yes—
342C4⋊C4#10212yes—
343C4⋊C4#10212yes—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—42no—
347C22⋊C4—41no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C42—42no—
357C42—41no—
358C42C2×C22⋊C411yescharacteristic
359C42C2×C22⋊C412yes—
360C42C2×C22⋊C412yes—
361C42C2×C22⋊C411yescharacteristic
362C42C2×D811yescharacteristic
363C42C2×D812yes—
364C42C2×D812yes—
365C42C2×D811yescharacteristic
366C16—42no—
367C16—41no—
368C16—41no—
369C23—42no—
370C23—41no—
371C23C22×SD1611yescharacteristic
372Q8—22no—
373Q8—22no—
374Q8—42no—
375Q8—41no—
376Q8—41no—
377Q8—162no—
378Q8—161no—
379D4—42no—
380D4—41no—
381D4—42no—
382D4—41no—
383D4—84no—
384D4—82no—
385D4—84no—
386D4—82no—
387D4—162no—
388D4—161no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4—162no—
404C2×C4C2×#10212yes—
405C2×C4C2×#10211yescharacteristic
406C2×C4C2×#10212yes—
407C2×C4C2×#10211yescharacteristic
408C2×C4C2×#10212yes—
409C2×C4C2×#10212yes—
410C2×C4C2×#10211yescharacteristic
411C2×C4C2×#10211yescharacteristic
412C2×C4C22×C22⋊C411yescharacteristic
413C2×C4C22×D811yescharacteristic
414C8—21no—
415C8—21no—
416C8—21no—
417C8—22no—
418C8—21no—
419C8—22no—
420C8—42no—
421C8—81no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—82no—
427C22—81no—
428C22—82no—
429C22—81no—
430C22—82no—
431C22—81no—
432C22C22×#10211yescharacteristic
433C4—21no—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4#214312yes—
446C4#288411yescharacteristic
447C4#290111yescharacteristic
448C2—41no—
449C2—162no—
450C2—161no—
451C2#1761811yescharacteristic
452C2C2×#214311yescharacteristic
453C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62363C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33497C2semidirect
3. #22746C2semidirect
4. #11866C2semidirect
5. #11864C2semidirect
6. #11862C2semidirect
7. #11861C2semidirect
8. #11751C2semidirect
9. #11750C2semidirect
10. #11749C2semidirect
11. #11747C2semidirect

Maximal subgroups

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

subgroupindex
#623632
#334972
#227462
#118662
#118642
#118622
#118612
#117512
#117502
#117492
#117472