Finite Group Databasebeta

← #509376

Subgroups — #509376

Order 512.

590 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×#2243C211yesmaximal, characteristic, complemented
2C2×#2235C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic, complemented
4#25412C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13452C211yesmaximal, characteristic, complemented
8#13408C211yesmaximal, characteristic, complemented
9#13382C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic
11#13168C211yesmaximal, characteristic, complemented
12#13121C211yesmaximal, characteristic, complemented
13#12991C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2426—21no—
24#2243—24no—
25#2235—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42#2083—21no—
43#2057—21no—
44#2042—21no—
45#2000—21no—
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1880—21no—
51#1852—21no—
52#1845C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—41no—
69C2×#121—21no—
70C2×C4×Q8—21no—
71C22×C4⋊C4—21no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#495—22no—
79#495—41no—
80#467—22no—
81#467—41no—
82#466—22no—
83#466—41no—
84C4×Q16—22no—
85C4×Q16—41no—
86C2×#107—22no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—41no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#398—41no—
105#396—41no—
106#395—41no—
107#394—41no—
108#390—21no—
109#390—41no—
110#389—21no—
111#389—21no—
112#389—21no—
113#388—41no—
114#385—41no—
115#384—21no—
116#380—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—41no—
121#366—22no—
122C4⋊C16—41no—
123#358—22no—
124#358—22no—
125#358—41no—
126#358D414yescomplemented
127#357—22no—
128#357—22no—
129#357—41no—
130#357—41no—
131C22⋊C16—22no—
132C22⋊C16—41no—
133#346—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—41no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16C2×D412yes—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×SD16—81no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2411yescharacteristic
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173#124—42no—
174#124—81no—
175C22⋊Q8—42no—
176C22⋊Q8—84no—
177C22⋊Q8—81no—
178#121—42no—
179#121—81no—
180C4×Q8—42no—
181C4×Q8—81no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—81no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4D811yescharacteristic
207C2×C22⋊C4D811yescharacteristic
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16C2×D412yes—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—42no—
234#107—42no—
235#107—81no—
236#107C2×D412yes—
237#107#4112yes—
238#103—42no—
239#103—42no—
240#102—42no—
241#102—81no—
242C22⋊C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95D811yescharacteristic
252#95D811yescharacteristic
253C24—41no—
254C24C2×D811yescharacteristic
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—161no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—161no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C22×D411yescharacteristic
295Q16—22no—
296Q16—22no—
297Q16—41no—
298Q16—41no—
299SD16—84no—
300D8—41no—
301D8—41no—
302D8—41no—
303D8—41no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8#13712yes—
317C2×C8C22×D411yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C2×C82−1+411yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—161no—
343C4⋊C4—161no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—161no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—82no—
358C16—22no—
359C16—41no—
360C16—41no—
361C16—41no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23#50912yes—
376C23C22×D811yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381Q8—164no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—164no—
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—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
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—81no—
425C2×C4—81no—
426C2×C4—164no—
427C2×C4#51012yes—
428C2×C4#53611yescharacteristic
429C2×C4D4211yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4C2×#13711yescharacteristic
432C2×C4#57611yescharacteristic
433C2×C4#57711yescharacteristic
434C8—22no—
435C8—41no—
436C8—41no—
437C8#53711yescharacteristic
438C8#54211yescharacteristic
439C8#54711yescharacteristic
440C8#56211yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22#322611yescharacteristic
456C22D4×D811yescharacteristic
457C22#329411yescharacteristic
458C22C2×#50911yescharacteristic
459C22C2×#51011yescharacteristic
460C22#341611yescharacteristic
461C22#341611yescharacteristic
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4#267211yescharacteristic
475C4#324111yescharacteristic
476C4#326611yescharacteristic
477C4#333311yescharacteristic
478C2—42no—
479C2—321no—
480C2#3062911yescharacteristic
481C2#3380411yescharacteristic
482C2#3381011yescharacteristic
483C2#3383111yescharacteristic
484C2#3384411yescharacteristic
485C2#3387311yescharacteristic
486C2#3388311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2243C2semidirect
2. C2×#2235C2semidirect
3. #26065C2semidirect
5. #13527C2semidirect
6. #13476C2semidirect
7. #13452C2semidirect
8. #13408C2semidirect
9. #13382C2semidirect
11. #13168C2semidirect
12. #13121C2semidirect
14. #12936C2semidirect
15. #12812C2semidirect
126. #358D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C2×#22352
#260652
#254122
#135272
#134762
#134522
#134082
#133822
#133802
#131682
#131212
#129912
#129362
#128122