Finite Group Databasebeta

← #287778

Subgroups — #287778

Order 512.

691 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28687C211yesmaximal, characteristic
2#28440C211yesmaximal, characteristic
3#28284C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5C2×#1672C212yesmaximal
6#10479C212yesmaximal
7#10273C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9218C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#494—21no—
14C2×#477C2212yes—
15C2×#462C2212yes—
16C2×#462C2212yes—
17C2×C8×Q8C2212yes—
18#2868C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#2084—22no—
22#2068—22no—
23#2016—22no—
24#1990C2212yes—
25#1984C2212yes—
26#1981—21no—
27#1956—21no—
28#1948—21no—
29#1948—21no—
30#1947—21no—
31#1939—21no—
32#1916—22no—
33#1842C2211yescharacteristic
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1672—28no—
40C2×#334C2212yes—
41C2×#333C2211yescharacteristic
42C2×#333C2211yescharacteristic
43C2×#332C2212yes—
44C2×#332C2212yes—
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1277C2212yes—
48C22×Q16—22no—
49C22×Q16—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8C2312yes—
57#498—24no—
58#498—42no—
59#494—44no—
60#477—24no—
61#477—42no—
62#462—24no—
63#462—24no—
64#462—42no—
65#462—42no—
66C8×Q8—24no—
67C8×Q8—42no—
68C2×#107—22no—
69C2×#106—22no—
70C2×#106—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—21no—
90#391D412yes—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384C2311yescharacteristic
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C43C2311yescharacteristic
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336D412yes—
114#334—24no—
115#333—22no—
116#333D414yes—
117#332—24no—
118#332—42no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D412yes—
122C2×Q16—44no—
123C2×Q16—44no—
124C2×Q16—44no—
125C2×Q16—44no—
126C2×Q16—82no—
127C2×Q16—82no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—21no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—81no—
146C4⋊Q8—81no—
147C4⋊Q8SD1614yes—
148C4⋊Q8C2×D412yes—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—82no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C2×D412yes—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181#107—44no—
182#106—44no—
183#106—44no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—44no—
193C4⋊C8—42no—
194C4⋊C8—81no—
195C4⋊C8Q1614yescomplemented
196C4⋊C8C2×D412yes—
197C4⋊C8C2×D412yes—
198C4⋊C8#4112yes—
199#103—44no—
200#103—44no—
201#103—44no—
202#103—82no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—44no—
206C4×C8—42no—
207C4×C8—42no—
208#95—22no—
209#95—22no—
210#95—41no—
211#95—41no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—81no—
226C2×Q8—81no—
227C2×Q8C2×SD1614yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4111yescharacteristic
240Q16—88no—
241Q16—88no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8C2×Q1614yes—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—41no—
295C42—42no—
296C42—42no—
297C42—82no—
298C42C2×SD1612yes—
299C42C2×Q1612yes—
300C42#13512yes—
301C42#13712yes—
302C42C22×D411yescharacteristic
303C42C2×#4111yescharacteristic
304C42H(5,𝔽2)11yescharacteristic
305C42H(5,𝔽2)11yescharacteristic
306C23C2×#12111yescharacteristic
307Q8—24no—
308Q8—24no—
309Q8—42no—
310Q8—42no—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314Q8—82no—
315Q8—82no—
316Q8—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4#46212yes—
342C2×C4#46212yes—
343C2×C4#46512yes—
344C2×C4#46512yes—
345C2×C4#47112yes—
346C2×C4#53211yescharacteristic
347C2×C4#54011yescharacteristic
348C2×C4D4211yescharacteristic
349C2×C4D4211yescharacteristic
350C2×C4#54611yescharacteristic
351C2×C4#55011yescharacteristic
352C2×C4C22×SD1611yescharacteristic
353C2×C4C22×Q1611yescharacteristic
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57611yescharacteristic
357C2×C4#57711yescharacteristic
358C2×C4#57811yescharacteristic
359C2×C4#57811yescharacteristic
360C8—24no—
361C8—42no—
362C8—42no—
363C8—42no—
364C8—42no—
365C8—81no—
366C8—82no—
367C22#242811yescharacteristic
368C22C2×#46211yescharacteristic
369C22#304411yescharacteristic
370C22C2×#46511yescharacteristic
371C22#306211yescharacteristic
372C22C2×#47111yescharacteristic
373C22#306911yescharacteristic
374C4—22no—
375C4—21no—
376C4—21no—
377C4—22no—
378C4—44no—
379C4—41no—
380C4—82no—
381C4—82no—
382C4—162no—
383C4—161no—
384C4#167214yescomplemented
385C4#327611yescharacteristic
386C4D4×SD1611yescharacteristic
387C4D4×Q1611yescharacteristic
388C4#328711yescharacteristic
389C4#329511yescharacteristic
390C4#330411yescharacteristic
391C4#339511yescharacteristic
392C4#339811yescharacteristic
393C2C2×#167211yescharacteristic
394C2#1924011yescharacteristic
395C2#1940211yescharacteristic
396C2#1957011yescharacteristic
397C2#2537011yescharacteristic
398C2#2538711yescharacteristic
399C2#2540511yescharacteristic
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
195. C4⋊C8Q16semidirect
384. C4#1672semidirect

Maximal subgroups

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

subgroupindex
#286872
#284402
#282842
#231492
C2×#16722
#104792
#102732
#96922
#96022
#92182
C4×#3332