Finite Group Databasebeta

← #509363

Subgroups — #509363

Order 512.

564 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
1C2×#2242C211yesmaximal, characteristic, complemented
2C2×#2237C211yesmaximal, characteristic, complemented
3#26066C211yesmaximal, characteristic, complemented
4#25412C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13453C211yesmaximal, characteristic, complemented
8#13408C211yesmaximal, characteristic, complemented
9#13382C211yesmaximal, characteristic, complemented
10#13381C211yesmaximal, characteristic
11#13166C211yesmaximal, characteristic, complemented
12#13120C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic
14#12938C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2441—21no—
23#2426—21no—
24#2242—24no—
25#2237—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×C4×C16C2211yescharacteristic
42#2084—21no—
43#2059—21no—
44#2042—21no—
45#2000—21no—
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1880—21no—
51#1854—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×#125—21no—
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69C2×C4×Q8—21no—
70C2×#117—21no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77#495—22no—
78#495—41no—
79#467—22no—
80#467—41no—
81#466—22no—
82#466—41no—
83C4×Q16—22no—
84C4×Q16—41no—
85#427—22no—
86#427—41no—
87C2×#107—21no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#399—41no—
104#396—41no—
105#396—41no—
106#394—41no—
107#390—21no—
108#390—41no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—41no—
113#388—41no—
114#387—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—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4C2×D411yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16C2×D412yes—
150C2×SD16—42no—
151C2×SD16—42no—
152C2×SD16—81no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#125—42no—
170#125—81no—
171#124—42no—
172#124—81no—
173C22⋊Q8—42no—
174C22⋊Q8—81no—
175#121—42no—
176#121—81no—
177C4×Q8—42no—
178C4×Q8—81no—
179#117—42no—
180#117—81no—
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×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×C22⋊C4D811yescharacteristic
205C2×C22⋊C4D811yescharacteristic
206C2×C42—41no—
207C2×C42—41no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
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—21no—
220C2×C16—21no—
221C2×C16C2×D412yes—
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—81no—
233#107C2×D412yes—
234#107#4112yes—
235#106—42no—
236#106—42no—
237#103—42no—
238#103—42no—
239#102—42no—
240#102—81no—
241C22⋊C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95D811yescharacteristic
251#95D811yescharacteristic
252C24—41no—
253C24C2×D811yescharacteristic
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—161no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288Q16—22no—
289Q16—22no—
290Q16—41no—
291Q16—41no—
292SD16—84no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8#13712yes—
310C2×C8C22×D411yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C82−1+411yescharacteristic
315C2×C82−1+411yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—22no—
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—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—161no—
335C4⋊C4—161no—
336C4⋊C4—161no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—82no—
352C42—82no—
353C16—22no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23C22×D811yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—81no—
375Q8—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—164no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
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—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
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#50812yes—
421C2×C4#53611yescharacteristic
422C2×C4D4211yescharacteristic
423C2×C4#54611yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57711yescharacteristic
427C8—22no—
428C8—41no—
429C8—41no—
430C8#53711yescharacteristic
431C8#54211yescharacteristic
432C8#54711yescharacteristic
433C8#56211yescharacteristic
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22#322611yescharacteristic
449C22D4×D811yescharacteristic
450C22#329411yescharacteristic
451C22C2×#50811yescharacteristic
452C22#341411yescharacteristic
453C22#341611yescharacteristic
454C22#341611yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#267211yescharacteristic
468C4#324111yescharacteristic
469C4#326611yescharacteristic
470C4#333311yescharacteristic
471C2—81no—
472C2—321no—
473C2#3062911yescharacteristic
474C2#3379911yescharacteristic
475C2#3381011yescharacteristic
476C2#3383711yescharacteristic
477C2#3385011yescharacteristic
478C2#3386811yescharacteristic
479C2#3389011yescharacteristic
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×#2242C2semidirect
2. C2×#2237C2semidirect
3. #26066C2semidirect
5. #13527C2semidirect
6. #13474C2semidirect
7. #13453C2semidirect
8. #13408C2semidirect
9. #13382C2semidirect
11. #13166C2semidirect
12. #13120C2semidirect
14. #12938C2semidirect
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×#22422
C2×#22372
#260662
#254122
#135272
#134742
#134532
#134082
#133822
#133812
#131662
#131202
#129922
#129382
#128122