Finite Group Databasebeta

← #116831

Subgroups — #116831

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#21177C211yesmaximal, characteristic
3#17486C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7#7848C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#430—21no—
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2285C2211yescharacteristic
16#1986—22no—
17#1986—22no—
18#1950—22no—
19#1950—22no—
20#1738—21no—
21#1733—21no—
22#1733—21no—
23#1493—22no—
24#1493—22no—
25#1493C414yescomplemented
26#1487—22no—
27#1481—22no—
28C2×Q8⋊C8—22no—
29#1286—22no—
30#1279—22no—
31#1273—22no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16D411yescharacteristic
35C22×M16D411yescharacteristic
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533D412yes—
45C2×C4⋊Q8Q811yescharacteristic
46C2×C4⋊Q8Q811yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#430—42no—
60#430—42no—
61C2×#108—24no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68#423D412yes—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C2×#97—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2×C412yes—
88C2×C4×C8D412yes—
89#384—21no—
90#382—21no—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95#341—24no—
96#339—24no—
97#339—24no—
98Q8⋊C8—28no—
99Q8⋊C8—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16SD1612yes—
128C2×M16Q1612yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—44no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8C4⋊C412yes—
145C4⋊Q8C4⋊C412yes—
146#125—22no—
147#125—22no—
148#125—41no—
149#125—41no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166#117—22no—
167#117—22no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—41no—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199#108—48no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8C22⋊C414yes—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#97—41no—
213#97—41no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222#95—22no—
223#95—41no—
224C24#12411yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—82no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4#11711yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4C2×SD1611yescharacteristic
255C22×C4C2×Q1611yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259C22×C4#13711yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262M16—22no—
263M16—22no—
264M16—22no—
265M16—22no—
266M16—42no—
267M16—42no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—41no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—81no—
277M16—81no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—42no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—81no—
296C2×C8—81no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42C2×C22⋊C411yescharacteristic
322C42C2×C4⋊C411yescharacteristic
323C42C4×D411yescharacteristic
324C42C4×D411yescharacteristic
325C42#12011yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12611yescharacteristic
336C42#12711yescharacteristic
337C23—41no—
338C23—41no—
339C23#38811yescharacteristic
340C23#39311yescharacteristic
341C23#39411yescharacteristic
342C23#48711yescharacteristic
343C23#48811yescharacteristic
344C23#48911yescharacteristic
345C23#49111yescharacteristic
346Q8—44no—
347Q8—44no—
348Q8—82no—
349Q8—82no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#38011yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38611yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#39011yescharacteristic
375C2×C4#39211yescharacteristic
376C2×C4#39411yescharacteristic
377C2×C4#39611yescharacteristic
378C2×C4#39611yescharacteristic
379C2×C4#39711yescharacteristic
380C2×C4#39911yescharacteristic
381C2×C4C4×SD1611yescharacteristic
382C2×C4C4×Q1611yescharacteristic
383C2×C4#44011yescharacteristic
384C2×C4#44111yescharacteristic
385C2×C4#44811yescharacteristic
386C2×C4#44911yescharacteristic
387C2×C4#45011yescharacteristic
388C2×C4#45111yescharacteristic
389C2×C4#45111yescharacteristic
390C2×C4#45211yescharacteristic
391C2×C4#45211yescharacteristic
392C2×C4#45211yescharacteristic
393C2×C4#46111yescharacteristic
394C2×C4#46211yescharacteristic
395C2×C4#46311yescharacteristic
396C2×C4#46411yescharacteristic
397C2×C4#46511yescharacteristic
398C2×C4#46711yescharacteristic
399C2×C4#46811yescharacteristic
400C2×C4#47011yescharacteristic
401C2×C4#47511yescharacteristic
402C2×C4#47711yescharacteristic
403C2×C4#47911yescharacteristic
404C2×C4#47911yescharacteristic
405C8—44no—
406C8—44no—
407C8—42no—
408C8—42no—
409C8—82no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22#143911yescharacteristic
422C22#193311yescharacteristic
423C22#193511yescharacteristic
424C22#202411yescharacteristic
425C22#202611yescharacteristic
426C22#204211yescharacteristic
427C22#204311yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—82no—
435C4—82no—
436C4—162no—
437C4#186112yes—
438C4#186311yescharacteristic
439C4#186511yescharacteristic
440C4#187711yescharacteristic
441C4#187811yescharacteristic
442C4#200011yescharacteristic
443C4#200111yescharacteristic
444C4#206611yescharacteristic
445C4#206811yescharacteristic
446C4#209812yes—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2#1106711yescharacteristic
454C2#1111211yescharacteristic
455C2#1111411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#124
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
25. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#211772
#174862
C2×#14932
#79572
#78972
#78482