Finite Group Databasebeta

← #504344

Subgroups — #504344

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2232C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic, complemented
3#25034C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic, complemented
5#13526C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic, complemented
7#13380C211yesmaximal, characteristic, complemented
8#13304C211yesmaximal, characteristic
9#13176C211yesmaximal, characteristic, complemented
10#13165C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13120C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic, complemented
15#12919C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2422—21no—
23#2361—21no—
24#2232—24no—
25C4×D16—24no—
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2071—21no—
42#2033—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1925—21no—
48#1874—21no—
49#1862—21no—
50#1862—21no—
51#1845C2211yescharacteristic
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×#123—41no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C22×C4⋊C4—21no—
71C2×D16—22no—
72C2×D16—22no—
73C2×D16—22no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#466—22no—
79#466—22no—
80#466—22no—
81#466—41no—
82C4×D8—22no—
83C4×D8—22no—
84C4×D8—41no—
85C4×D8—41no—
86C2×#107—22no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—41no—
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#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#386—41no—
113#385—41no—
114#385—41no—
115#384—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4D411yescharacteristic
119#366—22no—
120#366—22no—
121#366—22no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130#357D414yescomplemented
131C22⋊C16—22no—
132C22⋊C16—22no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—41no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
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—
159C2×D8#4112yes—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170#123—84no—
171#121—24no—
172#121—42no—
173#121—42no—
174#121—81no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—81no—
178C4×D4—81no—
179C4×D4—84no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—81no—
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⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4D811yescharacteristic
203C2×C22⋊C4D811yescharacteristic
204C2×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207D16—28no—
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×C16C2×D412yes—
219C2×C16#4112yes—
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#107#4112yes—
234#102—24no—
235#102—42no—
236#102—42no—
237#102—42no—
238#102—42no—
239#102—81no—
240C22⋊C8—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#95D811yescharacteristic
250#95D811yescharacteristic
251C24—41no—
252C24—41no—
253C24C2×D811yescharacteristic
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—44no—
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—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—161no—
272C22×C4—21no—
273C22×C4—21no—
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—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—161no—
291C22×C4—161no—
292C22×C4—161no—
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C22×D411yescharacteristic
299D8—22no—
300D8—22no—
301D8—22no—
302D8—22no—
303D8—41no—
304D8—41no—
305D8—41no—
306D8—41no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8C22×D411yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8H(5,𝔽2)11yescharacteristic
325C2×C82−1+411yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—22no—
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—41no—
337C4⋊C4—41no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—161no—
343C22⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—161no—
353C22⋊C4D1614yescomplemented
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—82no—
359C16—22no—
360C16—22no—
361C16—41no—
362C16—41no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23C2×D1612yes—
384C23C22×D811yescharacteristic
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—88no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—164no—
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—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
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—81no—
430C2×C4—164no—
431C2×C4—164no—
432C2×C4—164no—
433C2×C4C2×D1612yes—
434C2×C4C2×D1612yes—
435C2×C4#52511yescharacteristic
436C2×C4D4211yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57811yescharacteristic
441C8—41no—
442C8—41no—
443C8—41no—
444C8#52511yescharacteristic
445C8#53911yescharacteristic
446C8#54711yescharacteristic
447C8#55911yescharacteristic
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22#310011yescharacteristic
470C22D4×D811yescharacteristic
471C22#333411yescharacteristic
472C22C22×D1611yescharacteristic
473C22#341411yescharacteristic
474C22#341511yescharacteristic
475C22#341711yescharacteristic
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#245511yescharacteristic
488C4#312411yescharacteristic
489C4#326911yescharacteristic
490C4#335511yescharacteristic
491C2—42no—
492C2—162no—
493C2—162no—
494C2#2654611yescharacteristic
495C2#3366011yescharacteristic
496C2#3368411yescharacteristic
497C2#3383511yescharacteristic
498C2D4×D1611yescharacteristic
499C2#3390111yescharacteristic
500C2#3392211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13304C2
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×#2232C2semidirect
2. C2×C4×D16C2semidirect
3. #25034C2semidirect
4. #23025C2semidirect
5. #13526C2semidirect
6. #13448C2semidirect
7. #13380C2semidirect
9. #13176C2semidirect
10. #13165C2semidirect
11. #13130C2semidirect
12. #13120C2semidirect
13. #12935C2semidirect
14. #12935C2semidirect
15. #12919C2semidirect
130. #357D4semidirect
353. C22⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22322
C2×C4×D162
#250342
#230252
#135262
#134482
#133802
#133042
#131762
#131652
#131302
#131202
#129352
#129192