Finite Group Databasebeta

← #358515

Subgroups — #358515

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2C2×#1778C211yesmaximal, characteristic, complemented
3C2×C4×#108C211yesmaximal, characteristic, complemented
4#20377C211yesmaximal, characteristic
5C2×#1750C211yesmaximal, characteristic
6#11326C212yesmaximal, complemented
7#11269C212yesmaximal, complemented
8#11097C212yesmaximal
9#11097C212yesmaximal
10#10705C211yesmaximal, characteristic, complemented
11C4×#341C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13#2921—24no—
14#2921—24no—
15C2×#433—22no—
16C2×#433—22no—
17C22×#108C2212yes—
18C22×#108C2212yes—
19C2×#423C2212yes—
20C2×#407C2212yes—
21C2×#407C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#1778—24no—
31C4×#108—24no—
32#1750—22no—
33#1750—22no—
34#1750C418yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37C2×#341C2212yes—
38C2×#341C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117C2×C414yes—
58C2×#117D412yes—
59C2×#117D412yes—
60C22×C4⋊C4Q812yes—
61C22×C22⋊C4Q812yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#433—216no—
65#433—216no—
66C2×#108—24no—
67C2×#108—24no—
68C2×#108—24no—
69C2×#108—24no—
70C2×#108—24no—
71C2×#108—24no—
72#423—24no—
73#423—24no—
74C2×C4⋊C8—24no—
75#407—24no—
76#407—24no—
77#407—24no—
78#407—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8C2×C414yes—
84C2×C22⋊C8C2×C414yes—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16D414yes—
92C4×M16D414yes—
93C2×#97—22no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—22no—
97C2×#97—21no—
98C2×#97—21no—
99C2×#97C2×C414yes—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2×C414yes—
107#341—24no—
108#341—24no—
109#341—24no—
110#341—24no—
111#341—24no—
112#341—24no—
113#336—24no—
114#336C2×C414yes—
115#336C2×C414yes—
116C25C2×Q811yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×Q811yescharacteristic
128C23×C4C2×Q812yes—
129C23×C4C2×Q812yes—
130C23×C4C2411yescharacteristic
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16C2×D414yes—
152C2×M16C2×D414yes—
153C2×M16#4114yes—
154C2×M16#4114yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C412yes—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8C22×C412yes—
177C22×C8#4114yes—
178C22×C8#4114yes—
179#117—28no—
180#117—28no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4C4⋊C414yes—
188C2×C4⋊C4C4⋊C414yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4C4⋊C414yes—
192C2×C22⋊C4C4⋊C414yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206#108—216no—
207#108—216no—
208C4⋊C8—216no—
209C22⋊C8—216no—
210C22⋊C8—216no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213#97—28no—
214#97—24no—
215#97—24no—
216#97—42no—
217#97—42no—
218C4×C8—28no—
219C4×C8—24no—
220C4×C8—24no—
221C4×C8—42no—
222C4×C8—42no—
223C24—22no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—42no—
228C24—42no—
229C24C2×C4⋊C412yes—
230C24C22×Q811yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C2×C4⋊C414yes—
260C22×C4C2×C4⋊C412yes—
261C22×C4C2×C4⋊C412yes—
262C22×C4C2×C4⋊C412yes—
263C22×C4C22⋊Q814yes—
264C22×C4C22⋊Q814yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—24no—
280M16—24no—
281M16—24no—
282M16—24no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8#11714yes—
322C2×C8#11714yes—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42C4×D414yes—
332C42C4×D414yes—
333C42#12012yes—
334C42#12012yes—
335C42#12312yes—
336C42#12312yes—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—24no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—44no—
345C23—44no—
346C23—42no—
347C23—42no—
348C23—82no—
349C23—82no—
350C23—84no—
351C23C22×C4⋊C411yescharacteristic
352C23C2×C22⋊Q811yescharacteristic
353C23C2×C22⋊Q811yescharacteristic
354C23#53312yes—
355C23#53312yes—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#38514yes—
379C2×C4#38514yes—
380C2×C4C2×#11711yescharacteristic
381C2×C4C2×#11711yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×C4×D411yescharacteristic
384C2×C4C4×#4111yescharacteristic
385C2×C4C4×#4111yescharacteristic
386C2×C4C2×#12011yescharacteristic
387C2×C4C2×#12311yescharacteristic
388C2×C4#52512yes—
389C2×C4#52512yes—
390C2×C4#52512yes—
391C2×C4#52912yes—
392C2×C4#52912yes—
393C2×C4#52912yes—
394C2×C4#52912yes—
395C2×C4#53212yes—
396C2×C4#53212yes—
397C2×C4#53212yes—
398C8—24no—
399C8—24no—
400C8—44no—
401C8—44no—
402C8—44no—
403C8—44no—
404C8—42no—
405C8—42no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—24no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—44no—
415C22—44no—
416C22—84no—
417C22—84no—
418C22#228511yescharacteristic
419C22C2×#38511yescharacteristic
420C22#229611yescharacteristic
421C22#239612yes—
422C22#239612yes—
423C22#291612yes—
424C22#291612yes—
425C4—24no—
426C4—24no—
427C4—42no—
428C4—44no—
429C4#229011yescharacteristic
430C4#229011yescharacteristic
431C4#229311yescharacteristic
432C4#229311yescharacteristic
433C4#238812yes—
434C4#239512yes—
435C2—22no—
436C2—22no—
437C2—82no—
438C2#1384411yescharacteristic
439C2#2082811yescharacteristic
440C2#2082911yescharacteristic
441C2#2085212yes—
442C2#2137212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C4⋊C4
prime radicalC25C2×Q8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1778C2semidirect
3. C2×C4×#108C2semidirect
6. #11326C2semidirect
7. #11269C2semidirect
10. #10705C2semidirect
11. C4×#341C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
C2×#17782
C2×C4×#1082
#203772
C2×#17502
#113262
#112692
#110972
#107052
C4×#3412