Finite Group Databasebeta

← #512199

Subgroups — #512199

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2242C211yesmaximal, characteristic, complemented
2C2×#2108C211yesmaximal, characteristic, complemented
3#33216C211yesmaximal, characteristic, complemented
4#33182C211yesmaximal, characteristic
5#13284C211yesmaximal, characteristic, complemented
6#13205C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic, complemented
12#13061C211yesmaximal, characteristic
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic
15#12888C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#468—21no—
21C2×#467C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×#405C2211yescharacteristic
24#2629—21no—
25#2620—21no—
26#2242—24no—
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35#2108—24no—
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#1992—21no—
41#1973—21no—
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1959—21no—
45#1943—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1809—21no—
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×M16D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#125—21no—
67C2×#124—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#117D411yescharacteristic
71C2×M32—22no—
72C2×M32—22no—
73C2×M32—22no—
74C22×C16—22no—
75C22×C16—22no—
76C22×C16—22no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#499—22no—
82#499—22no—
83#495—22no—
84#495—22no—
85#468—44no—
86#467—22no—
87#467—22no—
88#466—22no—
89#466—22no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#106—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103#405—22no—
104#405—22no—
105C2×#97D411yescharacteristic
106C2×#97D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#398—41no—
110#398—41no—
111#396—41no—
112#396—41no—
113#396—41no—
114#396—41no—
115#394—41no—
116#394—41no—
117#391—21no—
118#390—21no—
119#389—21no—
120#389—21no—
121#384—21no—
122#384—21no—
123#380—21no—
124#380—21no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—22no—
133#346—22no—
134#346—22no—
135C4×C16—22no—
136C4×C16D414yescomplemented
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4C2×D411yescharacteristic
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×SD16—42no—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16SD1614yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8#4112yes—
171C22×C8#4112yes—
172C22×C8C2411yescharacteristic
173#125—42no—
174#125—42no—
175#124—42no—
176#124—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179#121—42no—
180#121—42no—
181#117—22no—
182#117—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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⋊C4D811yescharacteristic
199C2×C4⋊C4D811yescharacteristic
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⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4D811yescharacteristic
209C2×C22⋊C4D811yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212M32—28no—
213C2×C16—28no—
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—
223C2×C16C2×D412yes—
224C2×C16#4112yes—
225C2×C16#4112yes—
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#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—42no—
239#107—42no—
240#106—42no—
241#103—42no—
242#103—42no—
243#102—42no—
244#102—42no—
245#97—22no—
246#97—22no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8C2×D412yes—
250C4×C8#4112yes—
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—161no—
273C2×D4—161no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4#12412yes—
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×SD1612yes—
292C22×C4C22×D411yescharacteristic
293Q16—41no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297SD16—84no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302M16—28no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8#12112yes—
312C2×C8#12412yes—
313C2×C8C2×SD1612yes—
314C2×C8C2×SD1612yes—
315C2×C8#13512yes—
316C2×C8C22×D411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
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—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—161no—
340C4⋊C4—161no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C4⋊C4—161no—
344C4⋊C4—161no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C22⋊C4—161no—
354C22⋊C4—161no—
355C42—21no—
356C42—21no—
357C42#12112yes—
358C42#13512yes—
359C16—22no—
360C16—22no—
361C16—22no—
362C16—22no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23#48612yes—
377C23C22×D811yescharacteristic
378Q8—81no—
379Q8—81no—
380Q8—81no—
381Q8—81no—
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—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—81no—
404C2×C4—81no—
405C2×C4—81no—
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#46612yes—
425C2×C4#46612yes—
426C2×C4#48612yes—
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12411yescharacteristic
429C2×C4#53211yescharacteristic
430C2×C4C22×SD1611yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C8—22no—
434C8—22no—
435C8#46514yes—
436C8#52511yescharacteristic
437C8#52511yescharacteristic
438C8#52911yescharacteristic
439C8#52911yescharacteristic
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22C2×#46611yescharacteristic
455C22C2×#48611yescharacteristic
456C22#316911yescharacteristic
457C22#341511yescharacteristic
458C22#341611yescharacteristic
459C22#341611yescharacteristic
460C22#341711yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4#285011yescharacteristic
474C4C2×#46511yescharacteristic
475C4#313111yescharacteristic
476C4#316711yescharacteristic
477C2—42no—
478C2—321no—
479C2#3304211yescharacteristic
480C2#3361011yescharacteristic
481C2#3361111yescharacteristic
482C2#3370211yescharacteristic
483C2#3370311yescharacteristic
484C2#3374411yescharacteristic
485C2#3374511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
1. C2×#2242C2semidirect
2. C2×#2108C2semidirect
3. #33216C2semidirect
5. #13284C2semidirect
6. #13205C2semidirect
7. #13150C2semidirect
8. #13149C2semidirect
9. #13148C2semidirect
10. #13147C2semidirect
11. #13062C2semidirect
13. #13060C2semidirect
15. #12888C2semidirect
136. C4×C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22422
C2×#21082
#332162
#331822
#132842
#132052
#131502
#131492
#131482
#131472
#130622
#130612
#130602
#130592
#128882