Finite Group Databasebeta

← #505662

Subgroups — #505662

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C212yesmaximal
2C4×#466C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic, complemented
4#13507C212yesmaximal, complemented
5#13482C212yesmaximal, complemented
6#13177C211yesmaximal, characteristic, complemented
7#13176C211yesmaximal, characteristic, complemented
8#12868C212yesmaximal, complemented
9#12811C211yesmaximal, characteristic, complemented
10#12811C211yesmaximal, characteristic, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#466C2211yescharacteristic
13C2×C4×D8C2211yescharacteristic
14C2×#427—21no—
15C2×#427C2212yes—
16C22×C4×C8C2211yescharacteristic
17C4×#121—21no—
18#2285—21no—
19#2234—28no—
20C2×#367C2212yes—
21C2×#366C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2212yes—
27#1970C2211yescharacteristic
28#1942C2211yescharacteristic
29#1934—21no—
30#1893—21no—
31#1846—21no—
32#1846C2212yes—
33#1845C2211yescharacteristic
34#1845C2211yescharacteristic
35#1844—21no—
36#1808C2211yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#102—21no—
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1380C2212yes—
43#1374C2212yes—
44C22×D8C2311yescharacteristic
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#121—21no—
48C2×C4×D4—21no—
49C2×C4×D4—41no—
50C2×#117—22no—
51C22×C42D411yescharacteristic
52C22×C16C2312yes—
53C22×C16C2312yes—
54#466—22no—
55#466D414yes—
56C4×D8—22no—
57C4×D8—22no—
58#427—24no—
59#427—24no—
60#427—44no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#107C2312yes—
66C2×#107C2312yes—
67C2×#107C2311yescharacteristic
68C2×#106—22no—
69C2×#106—22no—
70C2×#102—21no—
71C2×#102—21no—
72C2×#102—21no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—21no—
80#389—21no—
81#388—41no—
82#386—41no—
83#384—21no—
84#384—21no—
85#382—22no—
86#380—21no—
87#380—41no—
88#380—41no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—41no—
91C4×C4⋊C4—41no—
92C4×C22⋊C4—41no—
93#367—24no—
94#366—24no—
95C4⋊C16—22no—
96C4⋊C16—22no—
97#357—24no—
98#357—24no—
99C22⋊C16—24no—
100#340—21no—
101#340—21no—
102#340—21no—
103C22×D4—21no—
104C22×D4—41no—
105C23×C4—41no—
106C23×C4D811yescharacteristic
107C23×C4D811yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×D8—21no—
110C2×D8—21no—
111C2×D8—21no—
112C2×D8—21no—
113C2×D8—21no—
114C2×D8C2×D412yes—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D412yes—
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8#4112yes—
127C22×C8C2411yescharacteristic
128#121—24no—
129#121—42no—
130C4×D4—42no—
131C4×D4—42no—
132C4×D4—84no—
133#117—44no—
134#117—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42—41no—
155C2×C42D811yescharacteristic
156C2×C42D811yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165#107—21no—
166#107—22no—
167#107—22no—
168#107—22no—
169#107—22no—
170#107—21no—
171#107—21no—
172#107—21no—
173#107—21no—
174#107—42no—
175#107—42no—
176#107C2×D412yes—
177#106—44no—
178#106—44no—
179#102—24no—
180#102—42no—
181#102—42no—
182C4×C8—22no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—22no—
186C4×C8#4112yes—
187C4×C8#4112yes—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—42no—
192#95—42no—
193C24—41no—
194C24C2×D811yescharacteristic
195C2×D4—22no—
196C2×D4—44no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—82no—
203C2×D4—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#12112yes—
223C22×C4C2×D811yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4AGL(1,ℤ/8)12yes—
229C22×C4C22×D411yescharacteristic
230D8—22no—
231D8—22no—
232D8—41no—
233D8—41no—
234C2×C8—28no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8#12112yes—
246C2×C8#12312yes—
247C2×C8#13712yes—
248C2×C8C22×D411yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C8C2×#4111yescharacteristic
251C2×C8C2×#4111yescharacteristic
252C2×C82−1+411yescharacteristic
253C2×C82−1+411yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—44no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—84no—
274C22⋊C4—82no—
275C42—21no—
276C42—21no—
277C42—22no—
278C42—22no—
279C42—84no—
280C42—82no—
281C42#12312yes—
282C42#13712yes—
283C16—42no—
284C16—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23#45912yes—
299C23C22×D811yescharacteristic
300D4—42no—
301D4—42no—
302D4—88no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—164no—
337C2×C4#45912yes—
338C2×C4#48012yes—
339C2×C4#48012yes—
340C2×C4#50812yes—
341C2×C4#50912yes—
342C2×C4C2×#12111yescharacteristic
343C2×C4C2×#12311yescharacteristic
344C2×C4#53611yescharacteristic
345C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C2×#13711yescharacteristic
347C2×C4C2×#13711yescharacteristic
348C8—22no—
349C8—22no—
350C8—22no—
351C8#54711yescharacteristic
352C8#54711yescharacteristic
353C8#55211yescharacteristic
354C8#55211yescharacteristic
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—161no—
369C22C2×#45911yescharacteristic
370C22C2×#48011yescharacteristic
371C22#322611yescharacteristic
372C22C2×#50811yescharacteristic
373C22C2×#50911yescharacteristic
374C22#341411yescharacteristic
375C22#341411yescharacteristic
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—82no—
381C4—161no—
382C4—162no—
383C4—162no—
384C4—161no—
385C4#223414yescomplemented
386C4#248611yescharacteristic
387C4#303911yescharacteristic
388C4#309211yescharacteristic
389C4#323811yescharacteristic
390C2—42no—
391C2—162no—
392C2#2764411yescharacteristic
393C2#3357911yescharacteristic
394C2#3358011yescharacteristic
395C2C2×#223411yescharacteristic
396C2#3364711yescharacteristic
397C2#3379911yescharacteristic
398C2#3380111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20797C2semidirect
4. #13507C2semidirect
5. #13482C2semidirect
6. #13177C2semidirect
7. #13176C2semidirect
8. #12868C2semidirect
9. #12811C2semidirect
10. #12811C2semidirect
11. #12783C2semidirect
385. C4#2234semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C4×#4662
#207972
#135072
#134822
#131772
#131762
#128682
#128112
#127832