Finite Group Databasebeta

← #505695

Subgroups — #505695

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2168C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic, complemented
3#27661C211yesmaximal, characteristic
4#27660C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13494C211yesmaximal, characteristic, complemented
8#13492C211yesmaximal, characteristic, complemented
9#13168C211yesmaximal, characteristic, complemented
10#13167C211yesmaximal, characteristic, complemented
11#12868C211yesmaximal, characteristic, complemented
12#12867C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15#12834C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2486—21no—
22#2485—21no—
23#2168—24no—
24#2159—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2088—21no—
36#2086—21no—
37#2077—21no—
38#2075—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1825—21no—
44#1823—21no—
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#123—41no—
64C2×C4×D4D411yescharacteristic
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C2×M32—22no—
68C2×M32—22no—
69C2×M32D414yes—
70C22×C16—22no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C8×D4—22no—
78C8×D4—22no—
79#427—22no—
80#427—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#398—41no—
98#397—41no—
99#397—41no—
100#394—41no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#387—41no—
107#381—41no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#367—22no—
115#367—22no—
116#367—22no—
117#366—22no—
118#366—22no—
119#366—22no—
120C4⋊C16—22no—
121C22⋊C16—22no—
122C22⋊C16—22no—
123#346—22no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×D4D811yescharacteristic
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D412yes—
152C22×C8#4112yes—
153C22×C8C2411yescharacteristic
154#123—84no—
155C4×D4—22no—
156C4×D4—22no—
157#117—42no—
158#117—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4D811yescharacteristic
183C2×C22⋊C4D811yescharacteristic
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186M32—28no—
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×C16—44no—
196C2×C16C2×D412yes—
197C2×C16C2×D412yes—
198C2×C16#4112yes—
199C2×C16#4112yes—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—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—42no—
217#106—42no—
218C4⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223#95—41no—
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—
233C24C2×D811yescharacteristic
234C24C2×D811yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4#12112yes—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13512yes—
265C22×C4C22×D411yescharacteristic
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8#12112yes—
283C2×C8C22×D411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C82−1+411yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—161no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—161no—
319C22⋊C4—161no—
320C42—21no—
321C42—21no—
322C42—82no—
323C16—22no—
324C16—41no—
325C16—41no—
326C16#12314yescomplemented
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#46612yes—
340C23#50912yes—
341C23C22×D811yescharacteristic
342D4—44no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
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—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
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—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4#46612yes—
388C2×C4C2×#12111yescharacteristic
389C2×C4#54711yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4#57811yescharacteristic
393C2×C4#57811yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8C2×#12311yescharacteristic
398C8#53611yescharacteristic
399C8#55211yescharacteristic
400C8#55211yescharacteristic
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22C2×#46611yescharacteristic
416C22#333411yescharacteristic
417C22#333411yescharacteristic
418C22C2×#50911yescharacteristic
419C22#341411yescharacteristic
420C22#341611yescharacteristic
421C22#341711yescharacteristic
422C4—22no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#248611yescharacteristic
435C4#305911yescharacteristic
436C4#333611yescharacteristic
437C4#333611yescharacteristic
438C2—42no—
439C2—42no—
440C2#2766111yescharacteristic
441C2#3360511yescharacteristic
442C2#3361311yescharacteristic
443C2#3391011yescharacteristic
444C2#3391211yescharacteristic
445C2#3391511yescharacteristic
446C2#3391611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2159C2semidirect
5. #13516C2semidirect
6. #13515C2semidirect
7. #13494C2semidirect
8. #13492C2semidirect
9. #13168C2semidirect
10. #13167C2semidirect
11. #12868C2semidirect
12. #12867C2semidirect
13. #12836C2semidirect
14. #12835C2semidirect
15. #12834C2semidirect
326. C16#123semidirect

Maximal subgroups

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

subgroupindex
C2×#21682
C2×#21592
#276612
#276602
#135162
#135152
#134942
#134922
#131682
#131672
#128682
#128672
#128362
#128352
#128342