Finite Group Databasebeta

← #499494

Subgroups — #499494

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2175C211yesmaximal, characteristic, complemented
2C2×#2163C211yesmaximal, characteristic, complemented
3C4×#467C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13449C211yesmaximal, characteristic, complemented
10#13382C211yesmaximal, characteristic, complemented
11#13380C211yesmaximal, characteristic
12#13305C211yesmaximal, characteristic
13#12674C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C4×C22⋊Q8—21no—
23C4×#121—21no—
24#2175—24no—
25#2163—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1934—21no—
46#1933—21no—
47#1893—21no—
48#1892—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#103—21no—
54C4×#102—21no—
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C22×C42D411yescharacteristic
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—22no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#467—22no—
79#467—22no—
80#466—22no—
81#466—22no—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×D8—22no—
85C4×D8—22no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#388—41no—
109#387—41no—
110#386—41no—
111#385—41no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119#367—22no—
120#366—22no—
121#366—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#346—22no—
133#346D414yescomplemented
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4—41no—
144C23×C4D811yescharacteristic
145C23×C4D811yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16#4112yes—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8#4112yes—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2411yescharacteristic
172C22⋊Q8—24no—
173C22⋊Q8—42no—
174#121—24no—
175#121—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—84no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42D811yescharacteristic
204C2×C42D811yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207SD32—28no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16C2×D412yes—
219C2×C16C2×D412yes—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107#4112yes—
231#107#4112yes—
232#103—24no—
233#103—42no—
234#103—42no—
235#102—24no—
236#102—42no—
237#102—42no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—22no—
242C4×C8C2×D412yes—
243C4×C8#4112yes—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—41no—
253C24C2×D811yescharacteristic
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—82no—
268C2×D4—82no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C22×D411yescharacteristic
297Q16—22no—
298Q16—22no—
299Q16—41no—
300Q16—41no—
301D8—22no—
302D8—22no—
303D8—41no—
304D8—41no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—44no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8#12112yes—
319C2×C8C22×D411yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—21no—
349C42—21no—
350C42—22no—
351C42—22no—
352C42—82no—
353C42—82no—
354C42—82no—
355C42#12112yes—
356C42#13512yes—
357C16—22no—
358C16—22no—
359C16—41no—
360C16—41no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23#50912yes—
375C23C22×D811yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—81no—
379Q8—81no—
380D4—42no—
381D4—42no—
382D4—88no—
383D4—81no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4#46612yes—
426C2×C4#46612yes—
427C2×C4#51012yes—
428C2×C4C2×#12111yescharacteristic
429C2×C4#52511yescharacteristic
430C2×C4#52511yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57511yescharacteristic
434C8—22no—
435C8—41no—
436C8—41no—
437C8#52511yescharacteristic
438C8#52911yescharacteristic
439C8#52911yescharacteristic
440C8#53211yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22C2×#46611yescharacteristic
456C22#310011yescharacteristic
457C22#310011yescharacteristic
458C22C2×#50911yescharacteristic
459C22C2×#51011yescharacteristic
460C22#341411yescharacteristic
461C22#341411yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4#239211yescharacteristic
475C4#305411yescharacteristic
476C4#310211yescharacteristic
477C4#310211yescharacteristic
478C2—42no—
479C2—162no—
480C2#2369611yescharacteristic
481C2#3360511yescharacteristic
482C2#3360611yescharacteristic
483C2#3367111yescharacteristic
484C2#3367211yescharacteristic
485C2#3367411yescharacteristic
486C2#3367611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2175C2semidirect
2. C2×#2163C2semidirect
3. C4×#467C2semidirect
5. #13515C2semidirect
6. #13492C2semidirect
7. #13476C2semidirect
8. #13475C2semidirect
9. #13449C2semidirect
10. #13382C2semidirect
13. #12674C2semidirect
14. #12638C2semidirect
15. #12637C2semidirect
133. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21752
C2×#21632
C4×#4672
C4×#4662
#135152
#134922
#134762
#134752
#134492
#133822
#133802
#133052
#126742
#126382
#126372