Finite Group Databasebeta

← #415571

Subgroups — #415571

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21313C212yesmaximal
3#14161C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11296C212yesmaximal
6#11276C214yesmaximal
7#10852C212yesmaximal
8#10847C211yesmaximal, characteristic
9C2×#544C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#423C2212yes—
12#2331—22no—
13#2330—22no—
14#2329—22no—
15#2317C2212yes—
16#2316C2211yescharacteristic
17C4×C4⋊Q8—22no—
18C4×#125—22no—
19C4×C22⋊Q8—22no—
20C4×#117C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1988—22no—
24#1988—22no—
25#1987—22no—
26#1987—22no—
27C2×#343C2212yes—
28C2×#343C2212yes—
29#1736C2214yes—
30#1736C2214yes—
31C2×#339C2214yes—
32C2×#339C2214yes—
33#1446—22no—
34C4⋊#95—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×Q8—21no—
38C22×M16C2314yes—
39#544—24no—
40#544—22no—
41#544Q814yescomplemented
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8Q814yescomplemented
48C2×#125—22no—
49C2×C22⋊Q8—22no—
50C2×C4×Q8—22no—
51C2×#117C2312yes—
52C2×#117C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C42D412yes—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61#423—24no—
62#423—24no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65#391—24no—
66#391—22no—
67#389—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#387—24no—
74#387—24no—
75#387—24no—
76#385—24no—
77#384—22no—
78#382—22no—
79#380—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91#376—22no—
92C2×#95—21no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95D412yes—
96C43—22no—
97#343—24no—
98#343—42no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#339—28no—
104#339—44no—
105C22×Q8—22no—
106C22×Q8—24no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—82no—
110C22×Q8—82no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C22×C8—24no—
125C4⋊Q8—28no—
126C4⋊Q8—28no—
127C4⋊Q8—28no—
128C4⋊Q8—28no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#125—28no—
132#125—28no—
133C22⋊Q8—28no—
134C22⋊Q8—28no—
135C4×Q8—84no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117C2×Q814yes—
143#117#4114yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×Q814yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4#4112yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—44no—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×Q812yes—
191C2×C42#4114yes—
192C2×C42#4112yes—
193C4⋊C8—44no—
194C4⋊C8—44no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—42no—
204#95—84no—
205#95—84no—
206C24C22×D411yescharacteristic
207C2×Q8—28no—
208C2×Q8—48no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—88no—
214C2×Q8—88no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#12011yescharacteristic
245C22×C4#12012yes—
246C22×C4#12011yescharacteristic
247C22×C4C22⋊Q812yes—
248C22×C4C22⋊Q814yes—
249C22×C4C22⋊Q812yes—
250C22×C4#12412yes—
251C22×C4#12412yes—
252C22×C4C22×D412yes—
253C22×C4C22×Q811yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4H(5,𝔽2)11yescharacteristic
260C22×C42−1+411yescharacteristic
261C22×C42−1+412yes—
262M16—44no—
263M16—44no—
264M16—44no—
265M16—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—28no—
270C4⋊C4—216no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—28no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—84no—
290C4⋊C4C22⋊Q814yes—
291C4⋊C4C22⋊Q814yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—24no—
297C42—24no—
298C42—42no—
299C42—42no—
300C42—44no—
301C42—44no—
302C42—44no—
303C42—44no—
304C42—44no—
305C42—82no—
306C42—82no—
307C42—82no—
308C42C22⋊Q814yes—
309C42C22⋊Q814yes—
310C42#12414yes—
311C42#12414yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23C2×#12011yescharacteristic
317C23C2×C22⋊Q811yescharacteristic
318C23C2×C22⋊Q811yescharacteristic
319C23#52511yescharacteristic
320C23C2×#12411yescharacteristic
321C23#53611yescharacteristic
322C23#53711yescharacteristic
323Q8—816no—
324Q8—88no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#39312yes—
368C2×C4#39314yes—
369C2×C4#39312yes—
370C2×C4#45512yes—
371C2×C4#45612yes—
372C2×C4C2×C22⋊Q812yes—
373C2×C4C2×C22⋊Q812yes—
374C2×C4#52512yes—
375C2×C4C2×#12412yes—
376C2×C4#53612yes—
377C2×C4#53712yes—
378C2×C4#55612yes—
379C2×C4#55612yes—
380C2×C4#55612yes—
381C2×C4#55711yescharacteristic
382C2×C4#55712yes—
383C2×C4#55711yescharacteristic
384C2×C4#55712yes—
385C2×C4Q8211yescharacteristic
386C2×C4#55912yes—
387C2×C4#55911yescharacteristic
388C8—84no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22C2×#39311yescharacteristic
402C22#256811yescharacteristic
403C22#258111yescharacteristic
404C22#258211yescharacteristic
405C22#259211yescharacteristic
406C22#261611yescharacteristic
407C22#284811yescharacteristic
408C22C2×#45511yescharacteristic
409C22C2×#45611yescharacteristic
410C22#302111yescharacteristic
411C22#302111yescharacteristic
412C4—22no—
413C4—24no—
414C4—44no—
415C4—44no—
416C4—44no—
417C4—84no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—84no—
422C4#202814yes—
423C4#256812yes—
424C4#258112yes—
425C4#258212yes—
426C4#259212yes—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2#1520111yescharacteristic
432C2C2×#202811yescharacteristic
433C2#2346311yescharacteristic
434C2#2970511yescharacteristic
435C2#2970611yescharacteristic
436C2#3297111yescharacteristic
437C2#3297211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
41. #544Q8semidirect
47. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
#213132
#141612
#113922
#112962
#112762
#108522
#108472