Finite Group Databasebeta

← #498007

Subgroups — #498007

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2175C211yesmaximal, characteristic
2#22808C211yesmaximal, characteristic, complemented
3#22682C211yesmaximal, characteristic
4#13150C211yesmaximal, characteristic, complemented
5#13148C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13109C211yesmaximal, characteristic, complemented
8#13014C211yesmaximal, characteristic
9#13014C211yesmaximal, characteristic
10#12890C211yesmaximal, characteristic, complemented
11#12814C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic
13#12663C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2355—21no—
22#2349—21no—
23#2175C418yescomplemented
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1969C2211yescharacteristic
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1928—21no—
40#1928—21no—
41#1864—21no—
42#1864—21no—
43#1846C2211yescharacteristic
44#1845C2211yescharacteristic
45#1804—21no—
46#1803—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4×Q8—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×SD32—22no—
67C2×SD32—22no—
68C2×SD32C2×C414yes—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C4×Q16—22no—
74C4×Q16C2×C414yescomplemented
75C4×D8—22no—
76C4×D8—22no—
77C4×D8—22no—
78C4×D8—41no—
79C4×D8—41no—
80C4×D8C2×C414yes—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#391—21no—
101#389—21no—
102#389—21no—
103#387—41no—
104#387—41no—
105#385—41no—
106#385—41no—
107#384—21no—
108#382—21no—
109#382—41no—
110#380—41no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—41no—
114C4×C4⋊C4D411yescharacteristic
115C2×#95—41no—
116#366C2×C414yescomplemented
117C4⋊C16—22no—
118#358—41no—
119#358—41no—
120#358C2×C414yescomplemented
121#357—22no—
122#357—22no—
123#357C2×C414yes—
124#346—41no—
125#346C2×C414yescomplemented
126C4×C16—41no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×Q8—21no—
134C22×D4—21no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16C22×C412yes—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C22×C412yes—
148C2×D8#4112yes—
149C2×D8#4112yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2411yescharacteristic
154C4×Q8—24no—
155C4×Q8—42no—
156C4×D4—24no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—81no—
161C4×D4—81no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4D811yescharacteristic
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184SD32—28no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C22×C412yes—
196C2×C16C22×C412yes—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—81no—
208#107C22×C412yes—
209#107C22×C412yes—
210#106—81no—
211C4⋊C8—22no—
212#103—24no—
213#103—81no—
214#103—81no—
215#102—24no—
216#102—42no—
217#102—42no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8C22×C412yes—
223C4×C8#4112yes—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24—41no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—161no—
262C22×C4—161no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C22×D411yescharacteristic
272Q16—22no—
273Q16—22no—
274Q16—41no—
275Q16—41no—
276D8—22no—
277D8—22no—
278D8—22no—
279D8#11714yes—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C4×D412yes—
289C2×C8C23×C411yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C82−1+411yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—44no—
322C42—161no—
323C42—161no—
324C42C4×D412yes—
325C42#13512yes—
326C16—22no—
327C16—22no—
328C16—41no—
329C16—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C22×D811yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342D4—24no—
343D4—42no—
344D4—42no—
345D4—42no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—164no—
383C2×C4—164no—
384C2×C4—164no—
385C2×C4C4×D812yes—
386C2×C4C4×D812yes—
387C2×C4#51012yes—
388C2×C4C2×C4×D411yescharacteristic
389C2×C4#54611yescharacteristic
390C2×C4#54811yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4#57711yescharacteristic
393C2×C4#57711yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8C2×#11711yescharacteristic
398C8#52011yescharacteristic
399C8#55211yescharacteristic
400C8#55311yescharacteristic
401C22—44no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22C2×C4×D811yescharacteristic
409C22#329411yescharacteristic
410C22#335811yescharacteristic
411C22C2×#51011yescharacteristic
412C22#341411yescharacteristic
413C22#341511yescharacteristic
414C22#341611yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#235211yescharacteristic
429C4#295211yescharacteristic
430C4#330311yescharacteristic
431C4#336011yescharacteristic
432C2—84no—
433C2#2276611yescharacteristic
434C2C4×#51011yescharacteristic
435C2#3345411yescharacteristic
436C2#3387811yescharacteristic
437C2#3388111yescharacteristic
438C2#3393711yescharacteristic
439C2#3393911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22808C2semidirect
4. #13150C2semidirect
5. #13148C2semidirect
6. #13112C2semidirect
7. #13109C2semidirect
10. #12890C2semidirect
11. #12814C2semidirect
13. #12663C2semidirect
23. #2175C4semidirect
74. C4×Q16C2×C4semidirect
116. #366C2×C4semidirect
120. #358C2×C4semidirect
125. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21752
#228082
#226822
#131502
#131482
#131122
#131092
#130142
#128902
#128142
#128112
#126632
#126372
C4×#3572