Finite Group Databasebeta

← #505672

Subgroups — #505672

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2236C211yesmaximal, characteristic
2C2×#2237C211yesmaximal, characteristic
3#21525C211yesmaximal, characteristic
4#20803C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13483C211yesmaximal, characteristic, complemented
8#13481C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13176C211yesmaximal, characteristic, complemented
11#12868C211yesmaximal, characteristic
12#12866C211yesmaximal, characteristic
13#12814C211yesmaximal, characteristic, complemented
14#12812C211yesmaximal, characteristic, complemented
15#12784C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×#428—21no—
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2323—21no—
23#2317—21no—
24#2237—24no—
25#2236—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1935—21no—
41#1894—21no—
42#1847—21no—
43#1847—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1763—21no—
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#467—22no—
74#467D414yes—
75C4×Q16—22no—
76C4×Q16—22no—
77#428—44no—
78#427—22no—
79#427—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98#405—22no—
99#405—22no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#389—21no—
106#389—41no—
107#389—41no—
108#389—41no—
109#388—41no—
110#388—41no—
111#387—41no—
112#385—41no—
113#385—41no—
114#384—21no—
115#384—21no—
116#382—21no—
117#382—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120#367—22no—
121#366—22no—
122#366—22no—
123#366—22no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135C22×Q8—21no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16C2×D412yes—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8#4112yes—
156C22×C8C2411yescharacteristic
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—22no—
162#117—22no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4D811yescharacteristic
188C2×C22⋊C4D811yescharacteristic
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—42no—
220#107—42no—
221#107—42no—
222#107C2×D412yes—
223#106—42no—
224#106—42no—
225#106—42no—
226#103—24no—
227#103—42no—
228#103—42no—
229#97—22no—
230#97—22no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8#4112yes—
234C4×C8#4112yes—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244C24C2×D811yescharacteristic
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—82no—
264C22×C4—161no—
265C22×C4—161no—
266C22×C4#12112yes—
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4C22×D411yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—41no—
277Q16—41no—
278M16—28no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8#12112yes—
290C2×C8#12312yes—
291C2×C8#13712yes—
292C2×C8C22×D411yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C22⋊C4—161no—
327C42—21no—
328C42—21no—
329C42—82no—
330C42—82no—
331C42—161no—
332C42#12312yes—
333C42#13712yes—
334C16—41no—
335C16—41no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#45912yes—
345C23C22×D811yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
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—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—164no—
391C2×C4#45912yes—
392C2×C4#48012yes—
393C2×C4#48012yes—
394C2×C4C2×#12111yescharacteristic
395C2×C4C2×#12311yescharacteristic
396C2×C4#53611yescharacteristic
397C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C8—22no—
401C8—22no—
402C8—22no—
403C8#54711yescharacteristic
404C8#54711yescharacteristic
405C8#55211yescharacteristic
406C8#55211yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22C2×#45911yescharacteristic
415C22C2×#48011yescharacteristic
416C22#322611yescharacteristic
417C22#341511yescharacteristic
418C22#341611yescharacteristic
419C22#341711yescharacteristic
420C22#341711yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4—82no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#248611yescharacteristic
435C4#303911yescharacteristic
436C4#309211yescharacteristic
437C4#323811yescharacteristic
438C2—42no—
439C2#2764411yescharacteristic
440C2#3358811yescharacteristic
441C2#3358911yescharacteristic
442C2#3365111yescharacteristic
443C2#3365211yescharacteristic
444C2#3380811yescharacteristic
445C2#3380911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13507C2semidirect
6. #13505C2semidirect
7. #13483C2semidirect
8. #13481C2semidirect
9. #13178C2semidirect
10. #13176C2semidirect
13. #12814C2semidirect
14. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22362
C2×#22372
#215252
#208032
#135072
#135052
#134832
#134812
#131782
#131762
#128682
#128662
#128142
#128122
#127842