Finite Group Databasebeta

← #96828

Subgroups — #96828

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21487C211yesmaximal, characteristic
2#10797C211yesmaximal, characteristic, complemented
3C4×#327C211yesmaximal, characteristic
4C8×C22⋊C8C211yesmaximal, characteristic, complemented
5#7565C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C2×#436C411yescharacteristic, complemented
9C2×#435—21no—
10C22×C4×C8C2211yescharacteristic
11C4×#121C2211yescharacteristic
12#1954C411yescharacteristic, complemented
13#1939C411yescharacteristic, complemented
14#1923—21no—
15#1869—21no—
16#1797C2211yescharacteristic
17#1770C411yescharacteristic, complemented
18#1753—21no—
19C4×C22⋊C8C2211yescharacteristic
20C4×C22⋊C8C2211yescharacteristic
21C2×#336—21no—
22C2×#336—21no—
23C2×#336C2211yescharacteristic
24C2×#327C412yes—
25C2×C82—21no—
26C4×#95C2211yescharacteristic
27#1294C412yes—
28#1273—21no—
29#1271C412yescomplemented
30#1270C412yescomplemented
31C22×M16—21no—
32C23×C8—21no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35C2×#121C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2×C411yescharacteristic
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#436—42no—
41#435—42no—
42#435—42no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8C2×C412yes—
52C2×#97—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—42no—
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8C2×C411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60#390C2×C411yescharacteristic
61#389C2×C411yescharacteristic
62#388—21no—
63#386—21no—
64#380C2×C411yescharacteristic
65C4×C4⋊C4C2×C411yescharacteristic
66C4×C22⋊C4—21no—
67C2×#95—21no—
68C2×#95C2×C412yes—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—42no—
75#336—42no—
76#336C2×C412yes—
77#336C2×C412yes—
78#327—24no—
79C82—42no—
80C22×D4—21no—
81C22×D4C4211yescharacteristic
82C23×C4—22no—
83C23×C4—21no—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C22×C8C22⋊C411yescharacteristic
110C22×C8C22⋊C411yescharacteristic
111C22×C8D812yes—
112C22×C8SD1612yes—
113#121—22no—
114#121—22no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C4211yescharacteristic
123C2×C4⋊C4C4211yescharacteristic
124C2×C4⋊C4C4211yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—21no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42C22×C411yescharacteristic
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—42no—
146C22⋊C8—24no—
147C22⋊C8—22no—
148C22⋊C8—42no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153C22⋊C8M1614yescomplemented
154#97—42no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163#95—24no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—22no—
168#95—22no—
169C24—21no—
170C24C4×D411yescharacteristic
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4C2×C4211yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C2×M1612yes—
211C22×C4#13112yes—
212C22×C4C2×D811yescharacteristic
213C22×C4C2×SD1611yescharacteristic
214C22×C4#13511yescharacteristic
215C22×C4#13511yescharacteristic
216M16—84no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—24no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8#9912yes—
250C2×C8#10212yes—
251C2×C8#10212yes—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—81no—
274C42—81no—
275C42—81no—
276C42—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C4×C22⋊C411yescharacteristic
289C23#43612yes—
290C23C4×D811yescharacteristic
291C23C4×SD1611yescharacteristic
292C23#44312yes—
293D4—81no—
294D4—81no—
295D4—82no—
296D4—82no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
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—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C4×M1611yescharacteristic
340C2×C4#40511yescharacteristic
341C2×C4C2×#9911yescharacteristic
342C2×C4#41011yescharacteristic
343C2×C4C2×#10211yescharacteristic
344C2×C4#41611yescharacteristic
345C2×C4#42111yescharacteristic
346C2×C4#42111yescharacteristic
347C2×C4#43512yes—
348C2×C4C4×D811yescharacteristic
349C2×C4C4×SD1611yescharacteristic
350C2×C4#44312yes—
351C8—22no—
352C8—22no—
353C8—22no—
354C8—44no—
355C8—44no—
356C8—41no—
357C8—82no—
358C8—81no—
359C8—81no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#158912yes—
376C22#159012yes—
377C22#175311yescharacteristic
378C22C4×#9911yescharacteristic
379C22#175711yescharacteristic
380C22#175911yescharacteristic
381C22C4×#10211yescharacteristic
382C22#176411yescharacteristic
383C22#176411yescharacteristic
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—84no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4#151611yescharacteristic
396C4#151711yescharacteristic
397C4#151911yescharacteristic
398C4#152111yescharacteristic
399C2—22no—
400C2—22no—
401C2—161no—
402C2#816111yescharacteristic
403C2#819711yescharacteristic
404C2#820611yescharacteristic
405C2#822011yescharacteristic
406C2#823811yescharacteristic
407C2#825011yescharacteristic
408C2#825111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC23C4×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10797C2semidirect
4. C8×C22⋊C8C2semidirect
6. #7029C2semidirect
7. #7028C2semidirect
8. C2×#436C4semidirect
12. #1954C4semidirect
13. #1939C4semidirect
17. #1770C4semidirect
29. #1271C4semidirect
30. #1270C4semidirect
153. C22⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#214872
#107972
C4×#3272
C8×C22⋊C82
#75652
#70292
#70282