Finite Group Databasebeta

← #279279

Subgroups — #279279

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4C4×#462C211yesmaximal, characteristic
5#10649C212yesmaximal
6#10594C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9601C211yesmaximal, characteristic
9#9218C212yesmaximal
10#8645C212yesmaximal
11#8586C211yesmaximal, characteristic
12C2×#477—21no—
13C2×#462C2211yescharacteristic
14C2×C8×Q8C2212yes—
15C2×C4×Q16—21no—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2057—22no—
20#1990C2212yes—
21#1987—21no—
22#1981C2211yescharacteristic
23#1947—21no—
24#1947—21no—
25#1947—21no—
26#1947C2212yes—
27#1947C2211yescharacteristic
28#1939—22no—
29#1939C2212yes—
30#1928—21no—
31#1878—21no—
32#1845C2212yes—
33#1842—21no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1803—21no—
37#1803C2212yes—
38#1803C2211yescharacteristic
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43C2×#335C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×Q16—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#477—44no—
59#462—22no—
60#462Q814yescomplemented
61C8×Q8—24no—
62C8×Q8—42no—
63C4×Q16—44no—
64C2×#107—22no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—21no—
81C2×#103—22no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—21no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#391—21no—
89#389—21no—
90#389—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384C2311yescharacteristic
95#384C2312yes—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D411yescharacteristic
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4C2311yescharacteristic
104C43—21no—
105C43C2311yescharacteristic
106#340—22no—
107#340—22no—
108#340—22no—
109#340—21no—
110#340—22no—
111#340—21no—
112#336—22no—
113#336—21no—
114#336D412yes—
115#335—42no—
116#332—42no—
117C22×Q8—21no—
118C22×Q8—41no—
119C22×Q8C2×D411yescharacteristic
120C2×Q16—24no—
121C2×Q16—42no—
122C2×Q16—42no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D412yes—
128C22×C8C2×D412yes—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—81no—
135C4⋊Q8—81no—
136C4⋊Q8—82no—
137C4⋊Q8C2×Q812yes—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8D814yes—
146C4×Q8C2×Q812yes—
147C4×Q8#4112yes—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—41no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42C2411yescharacteristic
180#107—42no—
181#107—44no—
182#107—82no—
183#107—82no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—44no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—81no—
197C4⋊C8—82no—
198C4⋊C8Q1614yescomplemented
199C4⋊C8C2×Q812yes—
200#103—24no—
201#103—42no—
202#103—42no—
203#103—44no—
204#103—42no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—44no—
208C4×C8—42no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—41no—
213#95—41no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8C22⋊Q812yes—
227C2×Q8C2×D812yes—
228C2×Q8#13712yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243Q16—48no—
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—42no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8C2×Q1614yes—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4C22⋊Q812yes—
289C4⋊C4C2×D812yes—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—24no—
297C42—22no—
298C42—41no—
299C42—42no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—81no—
304C42—82no—
305C42C2×D812yes—
306C42C2×Q1612yes—
307C42#13512yes—
308C42#13712yes—
309C42C22×Q811yescharacteristic
310C42C2×#4111yescharacteristic
311C42H(5,𝔽2)11yescharacteristic
312C422−1+411yescharacteristic
313C23C2×#12111yescharacteristic
314Q8—22no—
315Q8—22no—
316Q8—22no—
317Q8—42no—
318Q8—42no—
319Q8—81no—
320Q8—81no—
321Q8—164no—
322Q8#47414yes—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4#46212yes—
354C2×C4#46212yes—
355C2×C4#46612yes—
356C2×C4#46612yes—
357C2×C4#47312yes—
358C2×C4C2×C22⋊Q811yescharacteristic
359C2×C4#53711yescharacteristic
360C2×C4#54611yescharacteristic
361C2×C4D4×Q811yescharacteristic
362C2×C4#55211yescharacteristic
363C2×C4#55411yescharacteristic
364C2×C4C22×D811yescharacteristic
365C2×C4C22×Q1611yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4#57511yescharacteristic
370C2×C4#57611yescharacteristic
371C2×C4#57711yescharacteristic
372C8—24no—
373C8—42no—
374C8—42no—
375C8—82no—
376C8—81no—
377C8—82no—
378C22#241711yescharacteristic
379C22C2×#46211yescharacteristic
380C22#304511yescharacteristic
381C22C2×#46611yescharacteristic
382C22#305611yescharacteristic
383C22C2×#47311yescharacteristic
384C22#306811yescharacteristic
385C4—24no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—22no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4—161no—
396C4—162no—
397C4#167414yes—
398C4C2×#47411yescharacteristic
399C4#307611yescharacteristic
400C4#324511yescharacteristic
401C4#324811yescharacteristic
402C4#329911yescharacteristic
403C4#333311yescharacteristic
404C4Q8×Q1611yescharacteristic
405C4#338711yescharacteristic
406C2C2×#167411yescharacteristic
407C2#1917311yescharacteristic
408C2#1941011yescharacteristic
409C2#1949711yescharacteristic
410C2#2481711yescharacteristic
411C2#2483511yescharacteristic
412C2#2485711yescharacteristic
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
60. #462Q8semidirect
198. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#331982
#292082
Q8×C4⋊C82
C4×#4622
#106492
#105942
#98982
#96012
#92182
#86452
#85862