Finite Group Databasebeta

← #508863

Subgroups — #508863

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic, complemented
2#33182C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13408C211yesmaximal, characteristic, complemented
8#13408C211yesmaximal, characteristic, complemented
9#13366C211yesmaximal, characteristic, complemented
10#13365C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic, complemented
12#13061C211yesmaximal, characteristic
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic
15#12545C211yesmaximal, 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—
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34#1992—21no—
35#1973—21no—
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1959—21no—
39#1943—21no—
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1809—21no—
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×#124—21no—
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×#117D411yescharacteristic
69C2×M32—22no—
70C2×M32—41no—
71C2×M32—41no—
72C22×C16—22no—
73C22×C16—41no—
74C22×C16—41no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#499—41no—
80#499—41no—
81#495—41no—
82#495—41no—
83#468—44no—
84#467—22no—
85#467—22no—
86#466—22no—
87#466—22no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#106—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101#405—41no—
102#405—41no—
103C2×#97D411yescharacteristic
104C2×#97D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#398—41no—
108#398—41no—
109#396—41no—
110#396—41no—
111#396—41no—
112#396—41no—
113#394—41no—
114#394—41no—
115#391—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#384—21no—
120#384—21no—
121#380—21no—
122#380—21no—
123#358—41no—
124#358—41no—
125#358—41no—
126#358—41no—
127#357—41no—
128#357—41no—
129#357—41no—
130#357—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135C22×Q8—21no—
136C22×D4—21no—
137C22×D4—41no—
138C23×C4C2×D411yescharacteristic
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×SD16—42no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16SD1614yes—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8#4112yes—
165C22×C8#4112yes—
166C22×C8C2411yescharacteristic
167#125—42no—
168#125—42no—
169#124—42no—
170#124—42no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173#121—42no—
174#121—42no—
175#117—41no—
176#117—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4D811yescharacteristic
193C2×C4⋊C4D811yescharacteristic
194C2×C22⋊C4—41no—
195C2×C22⋊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⋊C4D811yescharacteristic
203C2×C22⋊C4D811yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206M32—44no—
207C2×C16—21no—
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×C16—21no—
219C2×C16—44no—
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—21no—
231#107—21no—
232#107—42no—
233#107—42no—
234#106—42no—
235#103—42no—
236#103—42no—
237#102—42no—
238#102—42no—
239#97—41no—
240#97—41no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C24C2×D811yescharacteristic
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—161no—
267C2×D4—161no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4#12412yes—
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×SD1612yes—
286C22×C4C22×D411yescharacteristic
287Q16—41no—
288Q16—41no—
289Q16—41no—
290Q16—41no—
291SD16—84no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296M16—44no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8#12412yes—
308C2×C8C2×SD1612yes—
309C2×C8C2×SD1612yes—
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C82−1+411yescharacteristic
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—161no—
334C4⋊C4—161no—
335C4⋊C4—161no—
336C4⋊C4—161no—
337C4⋊C4—161no—
338C4⋊C4—161no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C22⋊C4—161no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C16—41no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23#48612yes—
371C23C22×D811yescharacteristic
372Q8—81no—
373Q8—81no—
374Q8—81no—
375Q8—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—164no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
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—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
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#48612yes—
421C2×C4C2×#12411yescharacteristic
422C2×C4#53411yescharacteristic
423C2×C4#53711yescharacteristic
424C2×C4C22×SD1611yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C8—22no—
428C8—22no—
429C8—22no—
430C8#53811yescharacteristic
431C8#53811yescharacteristic
432C8#53911yescharacteristic
433C8#53911yescharacteristic
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22C2×#48611yescharacteristic
449C22#318611yescharacteristic
450C22#324511yescharacteristic
451C22#341511yescharacteristic
452C22#341611yescharacteristic
453C22#341611yescharacteristic
454C22#341711yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#263911yescharacteristic
468C4#313511yescharacteristic
469C4#318711yescharacteristic
470C4#324611yescharacteristic
471C2—42no—
472C2—321no—
473C2#3018511yescharacteristic
474C2#3370211yescharacteristic
475C2#3370311yescharacteristic
476C2#3376911yescharacteristic
477C2#3377011yescharacteristic
478C2#3382311yescharacteristic
479C2#3382411yescharacteristic
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. #33216C2semidirect
3. #13538C2semidirect
4. #13537C2semidirect
5. #13481C2semidirect
6. #13480C2semidirect
7. #13408C2semidirect
8. #13408C2semidirect
9. #13366C2semidirect
10. #13365C2semidirect
11. #13062C2semidirect
13. #13060C2semidirect
15. #12545C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#331822
#135382
#135372
#134812
#134802
#134082
#133662
#133652
#130622
#130612
#130602
#130592
#125452