Finite Group Databasebeta

← #415733

Subgroups — #415733

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14282C211yesmaximal, characteristic
2#14277C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11358C212yesmaximal
5#11268C212yesmaximal
6#11267C212yesmaximal
7#10881C212yesmaximal
8#10880C212yesmaximal
9#10694C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#97C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13C2×#389C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#376C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1446—21no—
29#1446—21no—
30#1445—21no—
31#1445—21no—
32#1444—21no—
33#1444—21no—
34#1442—21no—
35#1442—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#97—22no—
58C2×#97—41no—
59C2×#97—41no—
60C2×C4×C8—22no—
61C2×C4×C8—41no—
62C2×C4×C8—41no—
63#389—24no—
64#389—24no—
65#384—41no—
66#384—41no—
67#382—24no—
68#382—24no—
69#376—41no—
70#376—41no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D411yescharacteristic
83C2×#95D411yescharacteristic
84C2×#95D411yescharacteristic
85C2×#95D411yescharacteristic
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#336—24no—
95#336Q818yescomplemented
96C22×Q8—24no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—82no—
100C22×Q8—82no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8C2×Q814yes—
125C22×C8C2×Q814yes—
126C4⋊Q8—48no—
127C4⋊Q8—48no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4#4114yes—
161C2×C4⋊C4#4114yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C2×Q814yes—
173C2×C42#4114yes—
174C2×C42#4112yes—
175C2×C42#4112yes—
176#97—44no—
177C4×C8—44no—
178#95—24no—
179#95—24no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—42no—
185#95—42no—
186#95—44no—
187#95—82no—
188#95—82no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24C22×D411yescharacteristic
194C2×Q8—48no—
195C2×Q8—88no—
196C2×Q8—88no—
197C2×Q8—84no—
198C2×Q8—84no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4#12412yes—
252C22×C4#12412yes—
253C22×C4#12412yes—
254C22×C4#12412yes—
255C22×C4C4⋊Q812yes—
256C22×C4C4⋊Q812yes—
257C22×C4C22×Q812yes—
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4112yes—
262C22×C4C2×#4112yes—
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)11yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C42−1+411yescharacteristic
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—48no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—88no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C23#52511yescharacteristic
310C23#52511yescharacteristic
311C23C2×#12411yescharacteristic
312C23C2×#12411yescharacteristic
313C23C2×C4⋊Q811yescharacteristic
314C23#53411yescharacteristic
315C23#53711yescharacteristic
316C23#57612yes—
317C23#57712yes—
318C23#57712yes—
319C23#57812yes—
320Q8—816no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4#52912yes—
380C2×C4#52912yes—
381C2×C4#53312yes—
382C2×C4#53312yes—
383C2×C4#53811yescharacteristic
384C2×C4#53811yescharacteristic
385C2×C4#53911yescharacteristic
386C2×C4#53911yescharacteristic
387C2×C4#55112yes—
388C2×C4#55112yes—
389C2×C4#55412yes—
390C2×C4#55412yes—
391C2×C4#55412yes—
392C2×C4#55612yes—
393C2×C4#55712yes—
394C2×C4#55912yes—
395C8—44no—
396C8—44no—
397C22#239811yescharacteristic
398C22#256111yescharacteristic
399C22#258311yescharacteristic
400C22#258411yescharacteristic
401C22#259211yescharacteristic
402C22#263911yescharacteristic
403C22#286211yescharacteristic
404C22#311812yes—
405C22#312112yes—
406C22#312412yes—
407C22#312512yes—
408C22#313612yes—
409C22#313612yes—
410C22#313712yes—
411C22#313712yes—
412C22#316312yes—
413C22#316412yes—
414C22#319911yescharacteristic
415C22#320011yescharacteristic
416C22#320011yescharacteristic
417C22#320211yescharacteristic
418C22#325712yes—
419C22#325812yes—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—84no—
428C4—162no—
429C4—162no—
430C4#258812yes—
431C4#258812yes—
432C4#258912yes—
433C4#258912yes—
434C2#1520911yescharacteristic
435C2#2382712yes—
436C2#2911611yescharacteristic
437C2#2911811yescharacteristic
438C2#2931011yescharacteristic
439C2#2931111yescharacteristic
440C2#2932512yes—
441C2#2936312yes—
442C2#3019211yescharacteristic
443C2#3019311yescharacteristic
444C2#3329212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
95. #336Q8semidirect

Maximal subgroups

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

subgroupindex
#142822
#142772
#113842
#113582
#112682
#112672
#108812
#108802
#106942