Finite Group Databasebeta

← #509388

Subgroups — #509388

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×#2215C211yesmaximal, characteristic
3#27791C211yesmaximal, characteristic, complemented
4#25034C211yesmaximal, characteristic, complemented
5#13526C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13451C211yesmaximal, characteristic, complemented
8#13448C211yesmaximal, characteristic, complemented
9#13380C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic, complemented
11#13176C211yesmaximal, characteristic, complemented
12#13130C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic, complemented
14#12912C211yesmaximal, characteristic, complemented
15#12834C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2491—21no—
23#2422—21no—
24#2224—24no—
25#2215—24no—
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2071—21no—
41#2033—21no—
42#2033—21no—
43#2023—21no—
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic
48#1925—21no—
49#1862—21no—
50#1845C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×#124—41no—
66C2×#123—41no—
67C2×#121—21no—
68C2×#121—21no—
69C2×C4×D4—21no—
70C2×D16—22no—
71C2×D16—22no—
72C2×D16—41no—
73C22×C16—22no—
74C22×C16—22no—
75C22×C16—41no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#500—22no—
81#500—41no—
82#466—22no—
83#466—22no—
84#466—41no—
85#466—41no—
86C4×D8—22no—
87C4×D8—41no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#397—41no—
106#394—41no—
107#394—41no—
108#393—41no—
109#390—21no—
110#390—21no—
111#390—21no—
112#389—21no—
113#388—41no—
114#385—41no—
115#384—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C22⋊C4D411yescharacteristic
120#366—22no—
121#366—22no—
122#366—22no—
123#366—41no—
124#366—41no—
125C4⋊C16—22no—
126C4⋊C16—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130#357—41no—
131#357—41no—
132#357D414yes—
133C22⋊C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—41no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—41no—
144C22×D4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8C2×D412yes—
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
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171#124—84no—
172#123—84no—
173#121—42no—
174#121—42no—
175#121—81no—
176#121—81no—
177C4×D4—42no—
178C4×D4—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
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⋊C4D811yescharacteristic
201C2×C22⋊C4D811yescharacteristic
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205D16—44no—
206C2×C16—21no—
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—44no—
217C2×C16C2×D412yes—
218C2×C16#4112yes—
219#107—21no—
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—42no—
231#107—81no—
232#107C2×D412yes—
233#102—42no—
234#102—42no—
235#102—42no—
236#102—42no—
237#102—42no—
238#102—81no—
239C22⋊C8—41no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95D811yescharacteristic
249#95D811yescharacteristic
250C24—41no—
251C24—41no—
252C24C2×D811yescharacteristic
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—161no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—161no—
273C2×D4—161no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
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
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
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×C8C22×D411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8H(5,𝔽2)11yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C2×C82−1+411yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—41no—
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—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—161no—
338C22⋊C4—41no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—161no—
350C22⋊C4D1614yescomplemented
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—82no—
356C16—22no—
357C16—22no—
358C16—41no—
359C16—41no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—42no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23C2×D1612yes—
381C23C22×D811yescharacteristic
382Q8—164no—
383D4—42no—
384D4—42no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—164no—
392D4—164no—
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—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
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—164no—
429C2×C4C2×D1612yes—
430C2×C4C2×D1612yes—
431C2×C4#53711yescharacteristic
432C2×C4D4211yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#57511yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57811yescharacteristic
437C8—41no—
438C8—41no—
439C8—41no—
440C8#53611yescharacteristic
441C8#54211yescharacteristic
442C8#54611yescharacteristic
443C8#56211yescharacteristic
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22#324511yescharacteristic
466C22D4×D811yescharacteristic
467C22#333411yescharacteristic
468C22C22×D1611yescharacteristic
469C22#341411yescharacteristic
470C22#341511yescharacteristic
471C22#341711yescharacteristic
472C4—22no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#267211yescharacteristic
484C4#326211yescharacteristic
485C4#326911yescharacteristic
486C4#334711yescharacteristic
487C2—42no—
488C2—162no—
489C2—321no—
490C2#3066111yescharacteristic
491C2#3381111yescharacteristic
492C2#3382811yescharacteristic
493C2#3383511yescharacteristic
494C2D4×D1611yescharacteristic
495C2#3390111yescharacteristic
496C2#3391411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2215C2
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×#2224C2semidirect
3. #27791C2semidirect
4. #25034C2semidirect
5. #13526C2semidirect
6. #13474C2semidirect
7. #13451C2semidirect
8. #13448C2semidirect
9. #13380C2semidirect
10. #13380C2semidirect
11. #13176C2semidirect
12. #13130C2semidirect
13. #13012C2semidirect
14. #12912C2semidirect
15. #12834C2semidirect
350. C22⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×#22152
#277912
#250342
#135262
#134742
#134512
#134482
#133802
#131762
#131302
#130122
#129122
#128342