Finite Group Databasebeta

← #358689

Subgroups — #358689

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2955C211yesmaximal, characteristic
2#20436C211yesmaximal, characteristic, complemented
3C2×#1765C212yesmaximal, complemented
4C2×#1757C212yesmaximal, complemented
5C2×#1747C211yesmaximal, characteristic, complemented
6#11368C211yesmaximal, characteristic, complemented
7#11367C211yesmaximal, characteristic, complemented
8#11086C212yesmaximal, complemented
9#11085C212yesmaximal, complemented
10#11039C211yesmaximal, characteristic
11#11038C211yesmaximal, characteristic
12C2×#575C2211yescharacteristic
13C22×#131C2212yes—
14#2955—24no—
15#2955C418yescomplemented
16C2×#444—22no—
17C2×#443—22no—
18C2×#430C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#421C2212yes—
21C22×#104C2212yes—
22C2×#416C2212yes—
23C2×#413C2212yes—
24C2×#410C2212yes—
25C2×#405C2211yescharacteristic
26C2×#405C2212yes—
27C2×#405C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2296C2211yescharacteristic
30#1771—22no—
31#1770—22no—
32#1765—28no—
33#1757—28no—
34#1747—24no—
35C2×#344C2212yes—
36C2×#344C2212yes—
37C2×#341C2212yes—
38C2×#339C2212yes—
39C2×#338C2211yescharacteristic
40C2×H(3,ℤ/4)C2211yescharacteristic
41C2×H(3,ℤ/4)C2212yes—
42#1729C2211yescharacteristic
43#1728C2211yescharacteristic
44C22×#41C2312yes—
45#575—24no—
46#575—22no—
47#575C2×C414yes—
48C2×#137—22no—
49C2×AGL(1,ℤ/8)—22no—
50C2×#135—22no—
51C2×#131—24no—
52C2×#131—22no—
53C2×#131—22no—
54C2×#131—24no—
55C2×#131—24no—
56C2×#131—22no—
57C2×#131—22no—
58C2×#131C2×C418yes—
59C22×M16—22no—
60C22×M16C2312yes—
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8—22no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#117C2312yes—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#444—216no—
72#443—216no—
73#430—22no—
74#430C2×C414yes—
75C2×#108—22no—
76#424—22no—
77#424—22no—
78C2×C4⋊C8—22no—
79#421—24no—
80#421C2×C418yes—
81C2×#104—24no—
82C2×#104—24no—
83C2×#104—24no—
84C2×#104—24no—
85#416—24no—
86#416—24no—
87C2×#103—22no—
88C2×#102—22no—
89#413—24no—
90#413—24no—
91C2×#101—22no—
92C2×#100—22no—
93#410—24no—
94#410—24no—
95C2×#99—24no—
96#405—24no—
97#405—24no—
98#405C2×C414yes—
99#405D418yes—
100#405D414yes—
101C2×#97—22no—
102C2×#97—24no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97D412yes—
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109C2×C4×C8D412yes—
110#389—22no—
111#385—22no—
112C4×C4⋊C4—22no—
113#344—24no—
114#344—24no—
115#341—24no—
116#339—24no—
117#338—22no—
118H(3,ℤ/4)—22no—
119H(3,ℤ/4)—24no—
120#336—22no—
121#336—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41C22×C414yes—
133C2×#41#4114yes—
134C22×Q8#4112yes—
135C22×D4—22no—
136C22×D4#4112yes—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C2411yescharacteristic
140#137—28no—
141#137—28no—
142AGL(1,ℤ/8)—28no—
143AGL(1,ℤ/8)—28no—
144#135—28no—
145#135—28no—
146C2×Q16—22no—
147C2×SD16—24no—
148C2×D8—22no—
149#131—28no—
150#131—28no—
151#131—28no—
152#131—28no—
153#131—44no—
154#131—44no—
155#131—44no—
156#131—44no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—21no—
162C2×M16—28no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—44no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16C22×C414yes—
177C2×M16C22×C412yes—
178C2×M16C2×D414yes—
179C2×M16C2×D412yes—
180C2×M16#4112yes—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—28no—
189C22×C8—44no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8C22×C412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D414yes—
198C22×C8#4112yes—
199C22×C8#4112yes—
200C22×C8#4112yes—
201#117—21no—
202#117—21no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117C22×C412yes—
209#117C2×D414yes—
210#117C2×D412yes—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4#4112yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225#108—28no—
226C4⋊C8—28no—
227#104—216no—
228#104—216no—
229#103—28no—
230#102—28no—
231#101—28no—
232#100—28no—
233#99—216no—
234#97—28no—
235#97—42no—
236#97—42no—
237#97—42no—
238#97C22⋊C418yes—
239C4×C8—28no—
240C4×C8—42no—
241C4×C8C22⋊C418yes—
242#95—24no—
243C24—22no—
244C24C2×#4111yescharacteristic
245#41—24no—
246#41—24no—
247#41—24no—
248#41—24no—
249#41—24no—
250#41—24no—
251#41—24no—
252#41—24no—
253#41—42no—
254#41—42no—
255#41—42no—
256#41—42no—
257#41—42no—
258#41—42no—
259#41—42no—
260#41—42no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8#11714yes—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4#11714yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C2×#4112yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302Q16—28no—
303SD16—216no—
304D8—28no—
305M16—24no—
306M16—24no—
307M16—24no—
308M16—24no—
309M16—24no—
310M16—24no—
311M16—24no—
312M16—42no—
313M16—42no—
314M16—42no—
315M16—42no—
316M16—42no—
317M16—48no—
318M16—48no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—48no—
336C2×C8—48no—
337C2×C8C2×C22⋊C414yes—
338C2×C8C2×C22⋊C412yes—
339C2×C8C2×C22⋊C412yes—
340C2×C8C4×D414yes—
341C2×C8C4×D414yes—
342C2×C8#12114yes—
343C2×C8#12114yes—
344C2×C8#12411yescharacteristic
345C2×C8#12412yes—
346C2×C8#12411yescharacteristic
347C2×C8#12712yes—
348C2×C8#12712yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—84no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—42no—
360C42—22no—
361C42—24no—
362C42—24no—
363C42—41no—
364C42—41no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42C2×C22⋊C414yes—
369C23—24no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—84no—
374C23C4×#4111yescharacteristic
375C23#52511yescharacteristic
376C23#52512yes—
377C23#52912yes—
378C23#53211yescharacteristic
379Q8—24no—
380Q8—24no—
381Q8—42no—
382Q8—42no—
383D4—24no—
384D4—24no—
385D4—44no—
386D4—44no—
387D4—44no—
388D4—44no—
389D4—42no—
390D4—42no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4C22×C22⋊C411yescharacteristic
409C2×C4C2×#11712yes—
410C2×C4C2×C4×D412yes—
411C2×C4C4×#4111yescharacteristic
412C2×C4C2×#12112yes—
413C2×C4#52512yes—
414C2×C4#52512yes—
415C2×C4#52512yes—
416C2×C4#52511yescharacteristic
417C2×C4#52511yescharacteristic
418C2×C4#52512yes—
419C2×C4#52511yescharacteristic
420C2×C4C2×#12411yescharacteristic
421C2×C4#52912yes—
422C2×C4#52912yes—
423C2×C4#52912yes—
424C2×C4C2×#12711yescharacteristic
425C2×C4#53211yescharacteristic
426C8—24no—
427C8—44no—
428C8—42no—
429C8—42no—
430C8—44no—
431C8#39012yes—
432C8#39014yes—
433C8#39012yes—
434C22—22no—
435C22—24no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—44no—
440C22—84no—
441C22—84no—
442C22—84no—
443C22#229012yes—
444C22#229711yescharacteristic
445C22#239212yes—
446C22#239711yescharacteristic
447C22#239811yescharacteristic
448C4—44no—
449C4—44no—
450C4—82no—
451C4—82no—
452C4—84no—
453C4#228011yescharacteristic
454C4#229012yes—
455C4C2×#39011yescharacteristic
456C4#239212yes—
457C4#239711yescharacteristic
458C4#239811yescharacteristic
459C2—21no—
460C2—21no—
461C2—22no—
462C2—22no—
463C2—84no—
464C2#1384711yescharacteristic
465C2#2383412yes—
subgroupG/H
centerC2×C4C2×#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2398
prime radicalC2×#575C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20436C2semidirect
3. C2×#1765C2semidirect
4. C2×#1757C2semidirect
5. C2×#1747C2semidirect
6. #11368C2semidirect
7. #11367C2semidirect
8. #11086C2semidirect
9. #11085C2semidirect
15. #2955C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29552
#204362
C2×#17652
C2×#17572
C2×#17472
#113682
#113672
#110862
#110852
#110392
#110382