Finite Group Databasebeta

← #119215

Subgroups — #119215

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2#21332C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic
5#8014C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7854C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1996—22no—
16#1996—22no—
17#1987—21no—
18#1981—21no—
19#1872—22no—
20C4×#103C414yescomplemented
21#1740—21no—
22#1740—21no—
23C2×#339—21no—
24C2×#339—21no—
25#1550—22no—
26#1549—21no—
27#1548—21no—
28#1542—22no—
29#1294—22no—
30#1286—22no—
31#1274—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C4⋊C4—21no—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×#103—24no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8C2×C412yes—
76#384—21no—
77#382—21no—
78#380—21no—
79#380D412yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90#376—21no—
91C2×#95—21no—
92C43—21no—
93#342—24no—
94#342—24no—
95#340—24no—
96#339—42no—
97#339—42no—
98#333—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8C22⋊C412yes—
102C23×C4—41no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123#125—41no—
124#125—41no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—21no—
144#117—21no—
145#117—21no—
146#117—21no—
147#117—21no—
148#117—21no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157#117SD1612yes—
158#117Q1612yes—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209#103—48no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218#95—22no—
219#95—22no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224C24#12011yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—82no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4C2×SD1611yescharacteristic
262C22×C4C2×Q1611yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4#13511yescharacteristic
265C22×C4AGL(1,ℤ/8)11yescharacteristic
266C22×C4#13711yescharacteristic
267C22×C4#13711yescharacteristic
268C22×C4#13711yescharacteristic
269M16—42no—
270M16—42no—
271M16—81no—
272M16—81no—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—81no—
279C2×C8—81no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42—41no—
316C42—41no—
317C42—42no—
318C42—42no—
319C42—82no—
320C42—82no—
321C42—81no—
322C42—81no—
323C42—81no—
324C42—81no—
325C42#11711yescharacteristic
326C42#11711yescharacteristic
327C42C4×D411yescharacteristic
328C42C4×D411yescharacteristic
329C42#12011yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12411yescharacteristic
336C42#12411yescharacteristic
337C42#12611yescharacteristic
338C42#12611yescharacteristic
339C42#12611yescharacteristic
340C42#12611yescharacteristic
341C23—41no—
342C23—41no—
343C23#38611yescharacteristic
344C23#39411yescharacteristic
345C23#39411yescharacteristic
346C23#44811yescharacteristic
347C23#44911yescharacteristic
348C23#45111yescharacteristic
349C23#45211yescharacteristic
350Q8—44no—
351Q8—82no—
352Q8—82no—
353Q8—82no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4#38111yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39711yescharacteristic
393C2×C4#39711yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#39711yescharacteristic
396C2×C4C4×SD1611yescharacteristic
397C2×C4C4×Q1611yescharacteristic
398C2×C4#44011yescharacteristic
399C2×C4#44111yescharacteristic
400C2×C4#46011yescharacteristic
401C2×C4#46111yescharacteristic
402C2×C4#46211yescharacteristic
403C2×C4#46211yescharacteristic
404C2×C4#46311yescharacteristic
405C2×C4#46411yescharacteristic
406C2×C4#46411yescharacteristic
407C2×C4#46411yescharacteristic
408C2×C4#46511yescharacteristic
409C2×C4#46711yescharacteristic
410C2×C4#46811yescharacteristic
411C2×C4#47011yescharacteristic
412C2×C4#48311yescharacteristic
413C2×C4#48411yescharacteristic
414C2×C4#48511yescharacteristic
415C2×C4#48511yescharacteristic
416C2×C4#48711yescharacteristic
417C2×C4#48811yescharacteristic
418C2×C4#48911yescharacteristic
419C2×C4#49111yescharacteristic
420C8—82no—
421C8—82no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#144011yescharacteristic
434C22#187811yescharacteristic
435C22#188011yescharacteristic
436C22#203511yescharacteristic
437C22#203611yescharacteristic
438C22#204211yescharacteristic
439C22#204311yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—82no—
448C4—82no—
449C4—84no—
450C4—82no—
451C4—82no—
452C4—162no—
453C4#182712yes—
454C4#192711yescharacteristic
455C4#192811yescharacteristic
456C4#193311yescharacteristic
457C4#193511yescharacteristic
458C4#205511yescharacteristic
459C4#205611yescharacteristic
460C4#207211yescharacteristic
461C4#207411yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2#1114911yescharacteristic
469C2#1117811yescharacteristic
470C2#1119011yescharacteristic
subgroupG/H
centerC2×C4#388
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1440
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
#213852
#213322
#177082
#108522
#80142
#79572
#78542