Finite Group Databasebeta

← #508523

Subgroups — #508523

Order 512.

616 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
1#33171C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic, complemented
3#13528C212yesmaximal, complemented
4#13482C212yesmaximal, complemented
5#13381C212yesmaximal, complemented
6#13380C212yesmaximal, complemented
7#13252C211yesmaximal, characteristic, complemented
8#13252C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13176C211yesmaximal, characteristic, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2856—21no—
17#2283—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×C4⋊C16C2211yescharacteristic
21C2×C4⋊C16C2211yescharacteristic
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2212yes—
27#1970—21no—
28#1970C2212yes—
29#1957—21no—
30#1846C2211yescharacteristic
31#1846C2211yescharacteristic
32#1845C2211yescharacteristic
33#1835—21no—
34#1808C2211yescharacteristic
35#1808C2211yescharacteristic
36#1807—21no—
37C4×#107C2211yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#106—21no—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×D8—21no—
44C22×D8C2312yes—
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#127—21no—
48C2×#121—22no—
49C2×#121—41no—
50C2×#121—41no—
51C22×C42D411yescharacteristic
52C22×C16C2312yes—
53C22×C16C2312yes—
54#493—22no—
55#493—22no—
56#466—42no—
57#466—42no—
58#466—44no—
59C2×#107—21no—
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2312yes—
65C2×#106—21no—
66C2×#106—21no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#390—22no—
76#390—41no—
77#390—41no—
78#384—21no—
79#383—41no—
80#382—21no—
81#382—21no—
82#381—41no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C22⋊C4—41no—
88#376—41no—
89#367—42no—
90#366—42no—
91C4⋊C16—22no—
92C4⋊C16—22no—
93#357—42no—
94#357—42no—
95#357—42no—
96#357—42no—
97C22⋊C16—42no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102C22×D4—22no—
103C22×D4—42no—
104C22×D4—41no—
105C22×D4—41no—
106C23×C4D811yescharacteristic
107C23×C4D811yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8—42no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—41no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2411yescharacteristic
131#127—42no—
132#127—42no—
133#121—44no—
134#121—44no—
135#121—84no—
136#121—84no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42D811yescharacteristic
154C2×C42D811yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163#107—21no—
164#107—21no—
165#107—21no—
166#107—21no—
167#107—21no—
168#107—22no—
169#107—21no—
170#107—22no—
171#107—21no—
172#107—21no—
173#107—42no—
174#106—42no—
175#106—42no—
176#102—44no—
177#102—44no—
178C4×C8—22no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—22no—
182C4×C8#4112yes—
183C4×C8#4112yes—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192C24—42no—
193C24C2×D811yescharacteristic
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—84no—
203C2×D4—84no—
204C2×D4—84no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C2×D4—82no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C22×D411yescharacteristic
232D8—42no—
233D8—42no—
234D8—42no—
235D8—42no—
236D8—84no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—44no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8#12312yes—
250C2×C8AGL(1,ℤ/8)12yes—
251C2×C8C2×#4111yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8H(5,𝔽2)11yescharacteristic
254C2×C8H(5,𝔽2)11yescharacteristic
255C2×C8H(5,𝔽2)11yescharacteristic
256C2×C82−1+411yescharacteristic
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—82no—
279C42—82no—
280C42#12312yes—
281C42#13712yes—
282C16—42no—
283C16—42no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23#50912yes—
298C23C22×D811yescharacteristic
299D4—82no—
300D4—82no—
301D4—82no—
302D4—82no—
303D4—164no—
304D4—168no—
305D4—164no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4#48012yes—
339C2×C4#48012yes—
340C2×C4#50812yes—
341C2×C4#50912yes—
342C2×C4C2×#12311yescharacteristic
343C2×C4#53511yescharacteristic
344C2×C4#53711yescharacteristic
345C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C2×#13711yescharacteristic
347C2×C4#57511yescharacteristic
348C8—22no—
349C8—22no—
350C8—41no—
351C8#54011yescharacteristic
352C8#54011yescharacteristic
353C8#54211yescharacteristic
354C8#54211yescharacteristic
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—162no—
363C22—162no—
364C22—162no—
365C22—162no—
366C22—162no—
367C22—162no—
368C22—162no—
369C22C2×#48011yescharacteristic
370C22#320511yescharacteristic
371C22#324511yescharacteristic
372C22C2×#50811yescharacteristic
373C22C2×#50911yescharacteristic
374C22C2×#50911yescharacteristic
375C22#341411yescharacteristic
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—161no—
381C4—161no—
382C4—161no—
383C4—161no—
384C4—162no—
385C4#223414yes—
386C4#262411yescharacteristic
387C4#309411yescharacteristic
388C4#321611yescharacteristic
389C4#325411yescharacteristic
390C2—42no—
391C2—322no—
392C2#2992011yescharacteristic
393C2C2×#223411yescharacteristic
394C2#3364811yescharacteristic
395C2#3377811yescharacteristic
396C2#3378311yescharacteristic
397C2#3381311yescharacteristic
398C2#3381711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33171C2
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
2. #20818C2semidirect
3. #13528C2semidirect
4. #13482C2semidirect
5. #13381C2semidirect
6. #13380C2semidirect
7. #13252C2semidirect
8. #13252C2semidirect
9. #13177C2semidirect
10. #13176C2semidirect
11. #12783C2semidirect

Maximal subgroups

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

subgroupindex
#331712
#208182
#135282
#134822
#133812
#133802
#132522
#131772
#131762
#127832