Finite Group Databasebeta

← #287775

Subgroups — #287775

Order 512.

717 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28689C211yesmaximal, characteristic
2#28440C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5C2×#1674C212yesmaximal
6#10478C212yesmaximal
7#10265C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9218C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#494—21no—
14C2×#462—21no—
15C2×#462C2212yes—
16C2×#462C2212yes—
17C2×#462C2212yes—
18C2×C8×Q8C2212yes—
19#2868C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2083—22no—
23#2056—22no—
24#2016—22no—
25#1990C2212yes—
26#1984C2212yes—
27#1956—21no—
28#1947—21no—
29#1947—21no—
30#1939—21no—
31#1916—22no—
32#1842C2211yescharacteristic
33#1803—21no—
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1674—28no—
40C2×#335C2212yes—
41C2×#332C2212yes—
42C2×#332C2211yescharacteristic
43C2×#332C2212yes—
44C2×#332C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1277C2212yes—
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8D411yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8C2312yes—
58#500—24no—
59#500—42no—
60#494—44no—
61#462—24no—
62#462—24no—
63#462—24no—
64#462—42no—
65#462—42no—
66#462—42no—
67#462—44no—
68C8×Q8—24no—
69C8×Q8—42no—
70C2×#107—22no—
71C2×#107—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#391—21no—
91#391D412yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384C2311yescharacteristic
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C43C2311yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#336—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#336D412yes—
115#335—24no—
116#332—24no—
117#332—22no—
118#332—42no—
119#332D414yes—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8C2×D412yes—
123C2×Q16—44no—
124C2×Q16—44no—
125C2×Q16—44no—
126C2×Q16—44no—
127C2×Q16—44no—
128C2×Q16—44no—
129C2×Q16—82no—
130C2×Q16—82no—
131C2×Q16—82no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—21no—
142C4⋊Q8—22no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150C4⋊Q8—81no—
151C4⋊Q8D814yes—
152C4⋊Q8C2×D412yes—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—42no—
158C4×Q8—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4C2×D412yes—
177C2×C42—22no—
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#107—44no—
187#107—44no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—44no—
197C4⋊C8—42no—
198C4⋊C8—81no—
199C4⋊C8Q1614yescomplemented
200C4⋊C8C2×D412yes—
201C4⋊C8C2×D412yes—
202C4⋊C8#4112yes—
203#103—44no—
204#103—44no—
205#103—44no—
206#103—44no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—44no—
210C4×C8—42no—
211C4×C8—42no—
212#95—22no—
213#95—22no—
214#95—41no—
215#95—41no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8C2×D814yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4111yescharacteristic
246Q16—88no—
247Q16—88no—
248Q16—88no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8C2×Q1614yes—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—41no—
302C42—42no—
303C42—42no—
304C42—82no—
305C42C2×D812yes—
306C42C2×Q1612yes—
307C42#13512yes—
308C42#13712yes—
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C42H(5,𝔽2)11yescharacteristic
312C42H(5,𝔽2)11yescharacteristic
313C23C2×#12111yescharacteristic
314Q8—24no—
315Q8—24no—
316Q8—42no—
317Q8—42no—
318Q8—42no—
319Q8—42no—
320Q8—81no—
321Q8—81no—
322Q8—82no—
323Q8—82no—
324Q8—82no—
325Q8—82no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4#46212yes—
351C2×C4#46212yes—
352C2×C4#46612yes—
353C2×C4#46612yes—
354C2×C4#47312yes—
355C2×C4#53211yescharacteristic
356C2×C4#54011yescharacteristic
357C2×C4D4211yescharacteristic
358C2×C4D4211yescharacteristic
359C2×C4#54611yescharacteristic
360C2×C4#55011yescharacteristic
361C2×C4C22×D811yescharacteristic
362C2×C4C22×Q1611yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4#57611yescharacteristic
366C2×C4#57711yescharacteristic
367C2×C4#57811yescharacteristic
368C2×C4#57811yescharacteristic
369C8—24no—
370C8—42no—
371C8—42no—
372C8—42no—
373C8—42no—
374C8—81no—
375C8—82no—
376C22#242811yescharacteristic
377C22C2×#46211yescharacteristic
378C22#304411yescharacteristic
379C22C2×#46611yescharacteristic
380C22#306211yescharacteristic
381C22C2×#47311yescharacteristic
382C22#306911yescharacteristic
383C4—22no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—44no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—162no—
392C4—161no—
393C4#167414yescomplemented
394C4#327611yescharacteristic
395C4D4×D811yescharacteristic
396C4#328411yescharacteristic
397C4D4×Q1611yescharacteristic
398C4#329411yescharacteristic
399C4#330511yescharacteristic
400C4#339511yescharacteristic
401C4#339811yescharacteristic
402C2C2×#167411yescharacteristic
403C2#1924011yescharacteristic
404C2#1940411yescharacteristic
405C2#1957211yescharacteristic
406C2#2537011yescharacteristic
407C2#2538411yescharacteristic
408C2#2540611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
199. C4⋊C8Q16semidirect
393. C4#1674semidirect

Maximal subgroups

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

subgroupindex
#286892
#284402
#282552
#231492
C2×#16742
#104782
#102652
#96922
#96012
#92182
C4×#3322