Finite Group Databasebeta

← #508328

Subgroups — #508328

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic, complemented
2C2×#2234C211yesmaximal, characteristic
3#33173C211yesmaximal, characteristic
4#21151C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13507C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic, complemented
9#13475C211yesmaximal, characteristic, complemented
10#13474C211yesmaximal, characteristic, complemented
11#13408C211yesmaximal, characteristic, complemented
12#13408C211yesmaximal, characteristic, complemented
13#12994C211yesmaximal, characteristic, complemented
14#12992C211yesmaximal, characteristic
15#12783C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#465—21no—
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2850—21no—
24#2296—21no—
25#2237—24no—
26#2234—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C22⋊C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1968—21no—
46#1958—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1941—21no—
52#1941—21no—
53#1846C2211yescharacteristic
54#1735—21no—
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C42D411yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#495—41no—
77#495—41no—
78#467—22no—
79#467—22no—
80#466—22no—
81#466—22no—
82#465—44no—
83#427—22no—
84#427—41no—
85#427—41no—
86#427D414yes—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8—41no—
101C2×C4×C8—41no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—41no—
113#388—41no—
114#386—41no—
115#386—41no—
116#385—41no—
117#385—41no—
118#382—21no—
119C4×C4⋊C4—41no—
120C4×C4⋊C4—41no—
121#367—22no—
122#367—22no—
123#366—22no—
124#366—22no—
125C4⋊C16—41no—
126C4⋊C16—41no—
127#358—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#357—22no—
132#357—22no—
133#357—41no—
134#357—41no—
135C22⋊C16—22no—
136C22⋊C16—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×D4—21no—
143C22×D4—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—21no—
153C2×SD16—42no—
154C2×SD16—42no—
155C2×SD16—42no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—41no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D412yes—
172C22×C8C2×D411yescharacteristic
173C22×C8#4112yes—
174C22×C8C2411yescharacteristic
175#124—42no—
176#124—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179#121—42no—
180#121—42no—
181#117—24no—
182#117—42no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42D811yescharacteristic
209C2×C42D811yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
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×C16—21no—
223C2×C16—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—21no—
234#107C2×D412yes—
235#107C2×D412yes—
236#106—24no—
237#106—42no—
238#106—42no—
239#103—42no—
240#103—42no—
241#102—42no—
242#102—42no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—41no—
248C4×C8—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24—41no—
258C24C2×D811yescharacteristic
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
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—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#12112yes—
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4C22×D411yescharacteristic
299Q16—41no—
300Q16—41no—
301Q16—41no—
302Q16—41no—
303SD16—84no—
304D8—41no—
305D8—41no—
306D8—41no—
307D8—41no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—44no—
320C2×C8#12112yes—
321C2×C8#13712yes—
322C2×C8C22×D411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8H(5,𝔽2)11yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C82−1+411yescharacteristic
327C2×C82−1+411yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—22no—
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—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C22⋊C4—44no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—41no—
358C42—41no—
359C42—44no—
360C42—82no—
361C16—41no—
362C16—41no—
363C16—41no—
364C16—41no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23#45912yes—
379C23C22×D811yescharacteristic
380Q8—81no—
381Q8—81no—
382Q8—81no—
383Q8—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—164no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
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—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#45912yes—
430C2×C4#50812yes—
431C2×C4C2×#12111yescharacteristic
432C2×C4#53411yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57511yescharacteristic
437C8—22no—
438C8—22no—
439C8—41no—
440C8#53711yescharacteristic
441C8#53711yescharacteristic
442C8#53911yescharacteristic
443C8#53911yescharacteristic
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22C2×#45911yescharacteristic
459C22#318611yescharacteristic
460C22#322611yescharacteristic
461C22C2×#50811yescharacteristic
462C22#341411yescharacteristic
463C22#341411yescharacteristic
464C22#341411yescharacteristic
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—82no—
471C4—82no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#261811yescharacteristic
478C4#303711yescharacteristic
479C4#319311yescharacteristic
480C4#323411yescharacteristic
481C2—42no—
482C2—321no—
483C2#2975411yescharacteristic
484C2#3357911yescharacteristic
485C2#3358411yescharacteristic
486C2#3375711yescharacteristic
487C2#3376311yescharacteristic
488C2#3379911yescharacteristic
489C2#3380611yescharacteristic
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×#2237C2semidirect
4. #21151C2semidirect
5. #13507C2semidirect
6. #13507C2semidirect
7. #13477C2semidirect
8. #13476C2semidirect
9. #13475C2semidirect
10. #13474C2semidirect
11. #13408C2semidirect
12. #13408C2semidirect
13. #12994C2semidirect
15. #12783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#22342
#331732
#211512
#135072
#134772
#134762
#134752
#134742
#134082
#129942
#129922
#127832