Finite Group Databasebeta

← #96802

Subgroups — #96802

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#121C211yesmaximal, characteristic
2C8×#95C211yesmaximal, characteristic, complemented
3C4×#327C211yesmaximal, characteristic
4#8145C211yesmaximal, characteristic, complemented
5#7565C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C2×C8×D4—21no—
9C2×C8×D4C411yescharacteristic, complemented
10C22×C4×C8—21no—
11C22×C4×C8C2211yescharacteristic
12C4×#121C2211yescharacteristic
13#1954C411yescharacteristic, complemented
14#1939C411yescharacteristic, complemented
15#1923—21no—
16#1869—21no—
17#1797C2211yescharacteristic
18C8×C4⋊C4C411yescharacteristic, complemented
19C8×C22⋊C4—21no—
20C4×C22⋊C8C2211yescharacteristic
21C4×C22⋊C8C2211yescharacteristic
22C2×#336—21no—
23C2×#336C2211yescharacteristic
24C2×#327C412yes—
25C2×#322—21no—
26C4×#95C2211yescharacteristic
27#1294C412yes—
28#1273—21no—
29#1271C412yescomplemented
30#1270C412yescomplemented
31C23×C8—21no—
32C23×C8—21no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35C2×#121C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2×C411yescharacteristic
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40C8×D4—42no—
41C8×D4—42no—
42C8×D4—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×C4×C8—24no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—42no—
57C2×C4×C8—42no—
58C2×C4×C8—42no—
59C2×C4×C8C2×C411yescharacteristic
60C2×C4×C8C2×C411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63#390C2×C411yescharacteristic
64#389C2×C411yescharacteristic
65#388—21no—
66#386—21no—
67#380C2×C411yescharacteristic
68C4×C4⋊C4C2×C411yescharacteristic
69C4×C22⋊C4—21no—
70C2×#95—21no—
71C2×#95C2×C412yes—
72#336—24no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336C2×C412yes—
79#336C2×C412yes—
80#327C818yescomplemented
81#322—42no—
82C22×D4—21no—
83C22×D4C4211yescharacteristic
84C23×C4—22no—
85C23×C4—21no—
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—42no—
111C22×C8C22⋊C411yescharacteristic
112C22×C8C22⋊C411yescharacteristic
113C22×C8C22⋊C411yescharacteristic
114C22×C8C22⋊C411yescharacteristic
115#121—22no—
116#121C2×C814yes—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4C4211yescharacteristic
125C2×C4⋊C4C4211yescharacteristic
126C2×C4⋊C4C4211yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—21no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C22×C411yescharacteristic
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—42no—
148C22⋊C8—22no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8C2×C814yescomplemented
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—48no—
165C4×C8—42no—
166#95—24no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—22no—
171#95C2×C814yescomplemented
172C24—21no—
173C24C4×D411yescharacteristic
174C2×D4—24no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—42no—
181C2×D4C4×C812yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C2×C4211yescharacteristic
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4#11711yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C22×C812yes—
214C22×C4#13112yes—
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4AGL(1,ℤ/8)11yescharacteristic
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4#13711yescharacteristic
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—24no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—84no—
252C2×C8#9912yes—
253C2×C8#10412yes—
254C2×C8#10412yes—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4C4×C812yescomplemented
261C22⋊C4—24no—
262C22⋊C4—42no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—41no—
279C42—41no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C4×C22⋊C411yescharacteristic
292C23C8×D412yes—
293C23#44011yescharacteristic
294C23#44211yescharacteristic
295C23#44412yes—
296D4—42no—
297D4—42no—
298D4—44no—
299D4—44no—
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—28no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4C2×C4×C811yescharacteristic
343C2×C4#40511yescharacteristic
344C2×C4C2×#9911yescharacteristic
345C2×C4#41011yescharacteristic
346C2×C4#41711yescharacteristic
347C2×C4#41811yescharacteristic
348C2×C4C2×#10411yescharacteristic
349C2×C4C2×#10411yescharacteristic
350C2×C4C8×D412yes—
351C2×C4#44011yescharacteristic
352C2×C4#44211yescharacteristic
353C2×C4#44312yes—
354C8—22no—
355C8—22no—
356C8—22no—
357C8—44no—
358C8—44no—
359C8—41no—
360C8—82no—
361C8—81no—
362C8—81no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22SD16⋊C812yes—
379C22#158012yes—
380C22C8×C22⋊C411yescharacteristic
381C22C4×#9911yescharacteristic
382C22#175711yescharacteristic
383C22C4×#10411yescharacteristic
384C22#176211yescharacteristic
385C22#176411yescharacteristic
386C22#176511yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—48no—
394C4—42no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4#150411yescharacteristic
399C4#150511yescharacteristic
400C4#150711yescharacteristic
401C4#150911yescharacteristic
402C2—22no—
403C2—22no—
404C2—82no—
405C2C8×#9911yescharacteristic
406C2C8×#10411yescharacteristic
407C2#820011yescharacteristic
408C2#821811yescharacteristic
409C2#824011yescharacteristic
410C2#824911yescharacteristic
411C2#825711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×C42C23
socleC23C4×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C2×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. C8×#95C2semidirect
4. #8145C2semidirect
6. #7029C2semidirect
7. #7028C2semidirect
9. C2×C8×D4C4semidirect
13. #1954C4semidirect
14. #1939C4semidirect
18. C8×C4⋊C4C4semidirect
29. #1271C4semidirect
30. #1270C4semidirect
80. #327C8semidirect
155. C22⋊C8C2×C8semidirect
171. #95C2×C8semidirect
181. C2×D4C4×C8semidirect
260. C4⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
C8×#1212
C8×#952
C4×#3272
#81452
#75652
#70292
#70282