Finite Group Databasebeta

← #415570

Subgroups — #415570

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21313C212yesmaximal
3#14155C211yesmaximal, 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—
20C2×#391—22no—
21C4×#117C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1988—22no—
25#1988—22no—
26#1987—22no—
27#1987—22no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30#1736C2214yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1445—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#391—48no—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#387—24no—
75#387—24no—
76#387—24no—
77#385—24no—
78#384—22no—
79#382—22no—
80#380—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92#376—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—21no—
96C2×#95D412yes—
97C43—22no—
98#343—24no—
99#343—42no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#339—28no—
105#339—44no—
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8—44no—
109C22×Q8—44no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C22×C8—24no—
127C4⋊Q8—28no—
128C4⋊Q8—28no—
129C4⋊Q8—28no—
130C4⋊Q8—28no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133#125—28no—
134#125—28no—
135C22⋊Q8—28no—
136C22⋊Q8—28no—
137C4×Q8—84no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117C2×Q814yes—
145#117#4114yes—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×Q814yes—
172C2×C4⋊C4C2×Q812yes—
173C2×C4⋊C4#4112yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—44no—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×Q812yes—
192C2×C42#4114yes—
193C2×C42#4112yes—
194C4⋊C8—44no—
195C4⋊C8—44no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—48no—
205#95—48no—
206#95—42no—
207C24C22×D411yescharacteristic
208C2×Q8—28no—
209C2×Q8—416no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—88no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4#12012yes—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4C22⋊Q812yes—
252C22×C4C22⋊Q814yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12412yes—
255C22×C4#12412yes—
256C22×C4C22×D412yes—
257C22×C4C22×Q811yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4112yes—
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C42−1+411yescharacteristic
265C22×C42−1+412yes—
266M16—44no—
267M16—44no—
268M16—44no—
269M16—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C4⋊C4—28no—
274C4⋊C4—216no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—28no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—48no—
289C4⋊C4—84no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4C22⋊Q814yes—
294C4⋊C4C22⋊Q814yes—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—24no—
300C42—24no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—44no—
306C42—44no—
307C42—44no—
308C42—82no—
309C42—82no—
310C42—82no—
311C42C22⋊Q814yes—
312C42C22⋊Q814yes—
313C42#12414yes—
314C42#12414yes—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C2×#12011yescharacteristic
320C23C2×C22⋊Q811yescharacteristic
321C23C2×C22⋊Q811yescharacteristic
322C23#52511yescharacteristic
323C23C2×#12411yescharacteristic
324C23#53611yescharacteristic
325C23#53711yescharacteristic
326Q8—88no—
327Q8—88no—
328Q8—88no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4#39314yes—
372C2×C4#39312yes—
373C2×C4#39312yes—
374C2×C4#45312yes—
375C2×C4#45412yes—
376C2×C4C2×C22⋊Q812yes—
377C2×C4C2×C22⋊Q812yes—
378C2×C4#52512yes—
379C2×C4C2×#12412yes—
380C2×C4#53612yes—
381C2×C4#53712yes—
382C2×C4#55612yes—
383C2×C4#55612yes—
384C2×C4#55612yes—
385C2×C4#55711yescharacteristic
386C2×C4#55712yes—
387C2×C4#55711yescharacteristic
388C2×C4#55712yes—
389C2×C4Q8211yescharacteristic
390C2×C4#55912yes—
391C2×C4#55911yescharacteristic
392C8—84no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22C2×#39311yescharacteristic
406C22#256811yescharacteristic
407C22#258111yescharacteristic
408C22#258211yescharacteristic
409C22#259211yescharacteristic
410C22#261611yescharacteristic
411C22#284811yescharacteristic
412C22C2×#45311yescharacteristic
413C22C2×#45411yescharacteristic
414C22#302211yescharacteristic
415C22#302211yescharacteristic
416C4—22no—
417C4—24no—
418C4—44no—
419C4—44no—
420C4—44no—
421C4—84no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—84no—
426C4#202714yes—
427C4#256812yes—
428C4#258112yes—
429C4#258212yes—
430C4#259212yes—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2#1520111yescharacteristic
436C2C2×#202711yescharacteristic
437C2#2346411yescharacteristic
438C2#2970311yescharacteristic
439C2#2970411yescharacteristic
440C2#3296911yescharacteristic
441C2#3297011yescharacteristic
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
#141552
#113922
#112962
#112762
#108522
#108472