Finite Group Databasebeta

← #276992

Subgroups — #276992

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#28687C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4C4×#462C211yesmaximal, characteristic
5C2×#1672C212yesmaximal
6#10589C212yesmaximal
7#10544C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9216C212yesmaximal
11#8586C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#477C2212yes—
14C2×#462C2211yescharacteristic
15C2×C8×Q8C2212yes—
16C2×C4×Q16—21no—
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C42×Q8C2211yescharacteristic
20#2084—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1981C2211yescharacteristic
24#1948—21no—
25#1948—21no—
26#1947—21no—
27#1947—21no—
28#1947C2211yescharacteristic
29#1939—22no—
30#1939C2212yes—
31#1928—21no—
32#1878—21no—
33#1842—21no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1803—21no—
37#1803—21no—
38#1803C2211yescharacteristic
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43#1672—28no—
44C2×#334C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1277C2212yes—
50C22×Q16—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59#498—24no—
60#498—42no—
61#477—24no—
62#477—24no—
63#462—22no—
64#462D414yes—
65C8×Q8—24no—
66C8×Q8—42no—
67C4×Q16—44no—
68C2×#107—22no—
69C2×#107—21no—
70C2×#106—22no—
71C2×#106—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103—21no—
85C2×#103—22no—
86C2×#103—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×C4×C8—22no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#391—21no—
93#389—21no—
94#389—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384C2311yescharacteristic
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C43—21no—
109C43C2311yescharacteristic
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—21no—
115#340—21no—
116#336—22no—
117#336—21no—
118#336D412yes—
119#334—24no—
120#333—24no—
121#333—42no—
122#332—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8C2×D411yescharacteristic
126C2×Q16—24no—
127C2×Q16—42no—
128C2×Q16—42no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—82no—
143C4⋊Q8—82no—
144C4⋊Q8C2×D412yes—
145C4×Q8—22no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8SD1614yes—
153C4×Q8C2×D412yes—
154C4×Q8#4112yes—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#107—44no—
188#107—42no—
189#106—44no—
190#106—44no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—44no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C4⋊C8—81no—
204C4⋊C8—82no—
205C4⋊C8Q1614yescomplemented
206C4⋊C8C2×D412yes—
207#103—24no—
208#103—42no—
209#103—44no—
210#103—42no—
211#103—42no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—44no—
215C4×C8—42no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—41no—
220#95—41no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8#12112yes—
235C2×Q8C2×SD1612yes—
236C2×Q8#13712yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251Q16—48no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8C2×Q1614yes—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4#12112yes—
297C4⋊C4C2×SD1612yes—
298C42—22no—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—24no—
305C42—22no—
306C42—41no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—42no—
311C42—81no—
312C42—82no—
313C42C2×SD1612yes—
314C42C2×Q1612yes—
315C42#13512yes—
316C42#13712yes—
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C42H(5,𝔽2)11yescharacteristic
320C422−1+411yescharacteristic
321C23C2×#12111yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—22no—
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—162no—
330Q8—162no—
331Q8#46114yes—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4#46212yes—
363C2×C4#46212yes—
364C2×C4#46512yes—
365C2×C4#46512yes—
366C2×C4#47112yes—
367C2×C4C2×#12111yescharacteristic
368C2×C4#53711yescharacteristic
369C2×C4D4211yescharacteristic
370C2×C4#54711yescharacteristic
371C2×C4#54811yescharacteristic
372C2×C4#55311yescharacteristic
373C2×C4C22×SD1611yescharacteristic
374C2×C4C22×Q1611yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4#57511yescharacteristic
379C2×C4#57711yescharacteristic
380C2×C4#57811yescharacteristic
381C8—24no—
382C8—42no—
383C8—42no—
384C8—81no—
385C8—82no—
386C8—82no—
387C22#241611yescharacteristic
388C22C2×#46211yescharacteristic
389C22#304611yescharacteristic
390C22C2×#46511yescharacteristic
391C22#305611yescharacteristic
392C22C2×#47111yescharacteristic
393C22#306811yescharacteristic
394C4—24no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—81no—
403C4—82no—
404C4—162no—
405C4—161no—
406C4#167214yescomplemented
407C4C2×#46111yescharacteristic
408C4#303511yescharacteristic
409C4#324611yescharacteristic
410C4#324811yescharacteristic
411C4#327811yescharacteristic
412C4D4×Q1611yescharacteristic
413C4#336311yescharacteristic
414C4#337411yescharacteristic
415C2C2×#167211yescharacteristic
416C2#1917311yescharacteristic
417C2#1941411yescharacteristic
418C2#1950011yescharacteristic
419C2#2473211yescharacteristic
420C2#2475111yescharacteristic
421C2#2477311yescharacteristic
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
205. C4⋊C8Q16semidirect
406. C4#1672semidirect

Maximal subgroups

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

subgroupindex
#331982
#286872
Q8×C4⋊C82
C4×#4622
C2×#16722
#105892
#105442
#96912
#96022
#92162
#85862