Finite Group Databasebeta

← #6498500

Subgroups — #6498500

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60862C211yesmaximal, characteristic
2#60800C211yesmaximal, characteristic
3C2×#3139C211yesmaximal, characteristic, complemented
4#60208C211yesmaximal, characteristic
5#33065C214yesmaximal, complemented
6#29862C214yesmaximal, complemented
7#28669C214yesmaximal, complemented
8#28631C214yesmaximal, complemented
9#28138C218yesmaximal, complemented
10#21831C211yesmaximal, characteristic, complemented
11#21799C211yesmaximal, characteristic, complemented
12#20827C211yesmaximal, characteristic, complemented
13#3444C2211yescharacteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16#3240—24no—
17#3231—22no—
18#3228—22no—
19#3218—24no—
20#3209—22no—
21#3206—22no—
22#3165C2212yes—
23#3152C2212yes—
24#3141C2212yes—
25#3139—24no—
26#3139—22no—
27#3139—22no—
28#3139C2212yes—
29C2×#491C2212yes—
30C2×#491C2212yes—
31C2×#487C2214yes—
32#3093C2214yes—
33#3080C2214yes—
34#3039C2214yes—
35#3004C2214yes—
36#2971—22no—
37#2970—22no—
38#2969—24no—
39#2960C2212yes—
40#2931C2212yes—
41#2930C2214yes—
42#2923C2212yes—
43#2920C2211yescharacteristic
44#2920C2211yescharacteristic
45#2920C2211yescharacteristic
46C2×#419C2214yes—
47#2875C2212yes—
48C2×C4×M16C2211yescharacteristic
49#2064C22116yes—
50#2052C22116yes—
51#1998C2214yes—
52#1997C2214yes—
53#1966C2214yes—
54#1966C2214yes—
55#1930C22116yes—
56#1908C22116yes—
57#1849C2212yes—
58#1838C2212yes—
59#1822C2214yes—
60#1811C2214yes—
61#1784C2214yes—
62C4×#108C2214yes—
63C23×Q8C2312yes—
64C2×#137C2318yes—
65C2×#131C2314yes—
66C22×M16C2312yes—
67#533—22no—
68#533—22no—
69#533—22no—
70#533—22no—
71#533D412yes—
72#533C2311yescharacteristic
73#532D412yes—
74#532C2311yescharacteristic
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83#529—22no—
84#529—22no—
85#529D414yes—
86C2×#125C2312yes—
87#525—22no—
88#525C2312yes—
89C2×C22⋊Q8—22no—
90C4×#41—22no—
91C4×#41C2312yes—
92C2×C4×Q8—22no—
93C2×#117C2312yes—
94C22×C4⋊C4—22no—
95C22×C42C2311yescharacteristic
96#501—24no—
97#500—22no—
98#498—22no—
99#497—24no—
100#494—22no—
101#492—22no—
102#491—24no—
103#491—22no—
104#491—22no—
105#491—24no—
106#491—24no—
107#491—24no—
108#489—24no—
109#487—24no—
110#487—22no—
111#487—24no—
112#487—24no—
113#487—24no—
114#487—24no—
115#486—22no—
116#485—28no—
117#482—28no—
118#479—28no—
119#478—28no—
120#462—28no—
121#461—28no—
122#451—28no—
123#448—28no—
124#446—24no—
125C8×Q8—24no—
126#436—28no—
127#436—24no—
128#435—28no—
129C8×D4—24no—
130#433—24no—
131#432—22no—
132#432—22no—
133#432—22no—
134#432—24no—
135#431—22no—
136#431—22no—
137#431—22no—
138C2×#108C2318yes—
139#428C2318yes—
140#424C2314yes—
141#423C2312yes—
142#423C2312yes—
143#423C2312yes—
144#419—28no—
145#419—28no—
146#419C2318yes—
147#417C2318yes—
148C2×#103C2318yes—
149C2×#103C2318yes—
150C2×#101C23116yes—
151#408C2314yes—
152#407C2312yes—
153#407C2312yes—
154#407C2312yes—
155#405—24no—
156#405C2314yes—
157C4×M16—22no—
158C4×M16D414yes—
159C4×M16D412yes—
160C4×M16D412yes—
161C4×M16C2312yes—
162C4×M16C2312yes—
163C4×M16C2312yes—
164C2×#97—22no—
165C2×#97C2311yescharacteristic
166C2×#97C2311yescharacteristic
167C2×C4×C8C2312yes—
168#343C23116yes—
169#341C2318yes—
170C2×#41—22no—
171C2×#41C2412yes—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—48no—
177C22×Q8C2414yes—
178C23×C4—22no—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181#137—416no—
182C2×Q16—28no—
183C2×SD16—28no—
184#131—48no—
185C2×M16—24no—
186C2×M16—44no—
187C2×M16C2×D414yes—
188C2×M16C2×D414yes—
189C2×M16C2414yes—
190C2×M16C2414yes—
191C2×M16C2414yes—
192C22×C8—24no—
193C22×C8C2412yes—
194C22×C8C2412yes—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8C2×D411yescharacteristic
210C4⋊Q8C2×D412yes—
211#127C2×D411yescharacteristic
212#126—24no—
213#125—22no—
214#125—24no—
215#125—22no—
216#125—22no—
217#125C2×D412yes—
218#124—22no—
219#124C2×D412yes—
220#123—24no—
221#123—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C22⋊Q8C2×D414yes—
229#121—22no—
230#121—22no—
231#121C2×D414yes—
232#120—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×Q8C2×D414yes—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C2×D414yes—
246#117—22no—
247#117—24no—
248#117—22no—
249#117—24no—
250#117—24no—
251#117C2412yes—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2414yes—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#108—48no—
272#107—28no—
273#106—28no—
274C4⋊C8—28no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277C4⋊C8—24no—
278#103—28no—
279#103—28no—
280#103—28no—
281#103—28no—
282#103—28no—
283#103—216no—
284#102—28no—
285#101—416no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—28no—
290#97—24no—
291#97—22no—
292#97—22no—
293#97—22no—
294#97C2×D411yescharacteristic
295#97C2×D411yescharacteristic
296#97C2×D412yes—
297#97C2×D412yes—
298#97C2×D412yes—
299C4×C8—24no—
300C4×C8—22no—
301C4×C8—22no—
302C4×C8—22no—
303C4×C8C2×D414yes—
304C4×C8C2×D414yes—
305C24C2×#4111yescharacteristic
306#41—28no—
307#41—84no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—216no—
312C2×Q8—416no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—44no—
316C2×Q8—84no—
317C2×Q8—88no—
318C2×Q8—84no—
319C2×Q8C22×D412yes—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4C22×D412yes—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—84no—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2511yescharacteristic
342Q16—88no—
343SD16—88no—
344M16—28no—
345M16—28no—
346M16—28no—
347M16—44no—
348M16—44no—
349M16—48no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—48no—
356C2×C8C22×D412yes—
357C2×C8C22×D412yes—
358C2×C8C22×D414yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4C22×D414yes—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—44no—
381C42—24no—
382C42—22no—
383C42—44no—
384C42#12112yes—
385C42#12112yes—
386C42#12112yes—
387C42#12112yes—
388C42C22×D411yescharacteristic
389C42C22×D411yescharacteristic
390C42C22×D412yes—
391C42C22×D411yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×D412yes—
394C23—22no—
395C23—22no—
396C23—42no—
397C23C2×#12112yes—
398C23C22×#4111yescharacteristic
399Q8—24no—
400Q8—44no—
401Q8—48no—
402Q8—48no—
403Q8—88no—
404Q8—84no—
405Q8—84no—
406D4—24no—
407D4—44no—
408D4—44no—
409D4—84no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—84no—
426C2×C4—88no—
427C2×C4—84no—
428C2×C4C2×#12112yes—
429C2×C4C2×#12112yes—
430C2×C4C2×#12112yes—
431C2×C4C2×#12112yes—
432C2×C4C2×#12112yes—
433C2×C4C23×D411yescharacteristic
434C2×C4C23×D411yescharacteristic
435C2×C4#58514yes—
436C8—22no—
437C8—24no—
438C8—22no—
439C8—44no—
440C8—44no—
441C8—44no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—44no—
447C22—44no—
448C22—42no—
449C22C22×#12111yescharacteristic
450C22#351012yes—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—84no—
458C4—84no—
459C4#306914yes—
460C4#344912yes—
461C4#350012yes—
462C2—22no—
463C2—42no—
464C2—82no—
465C2#3532711yescharacteristic
466C2C2×#306912yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#3139C2semidirect
5. #33065C2semidirect
6. #29862C2semidirect
7. #28669C2semidirect
8. #28631C2semidirect
9. #28138C2semidirect
10. #21831C2semidirect
11. #21799C2semidirect
12. #20827C2semidirect

Maximal subgroups

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

subgroupindex
#608622
#608002
C2×#31392
#602082
#330652
#298622
#286692
#286312
#281382
#218312
#217992
#208272