Finite Group Databasebeta

← #499492

Subgroups — #499492

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2177C211yesmaximal, characteristic, complemented
2C2×#2163C211yesmaximal, characteristic, complemented
3C4×#466C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic, complemented
5#13516C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13448C211yesmaximal, characteristic, complemented
10#13381C211yesmaximal, characteristic, complemented
11#13380C211yesmaximal, characteristic, complemented
12#13304C211yesmaximal, characteristic
13#12674C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C4×#121—21no—
23C4×#121—21no—
24#2177—24no—
25#2163—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1934—21no—
46#1934—21no—
47#1893—21no—
48#1893—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#102—21no—
54C4×#102—21no—
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C22×C42D411yescharacteristic
72C2×D16—22no—
73C2×D16—22no—
74C2×D16—22no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#466—22no—
80#466—22no—
81#466—22no—
82#466—22no—
83C4×D8—22no—
84C4×D8—22no—
85C4×D8—22no—
86C4×D8—22no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#388—41no—
110#388—41no—
111#386—41no—
112#386—41no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119#367—22no—
120#367—22no—
121#366—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124#357—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#346—22no—
133#346D414yescomplemented
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—41no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4D811yescharacteristic
147C23×C4D811yescharacteristic
148C23×C4C2×D411yescharacteristic
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×D8—21no—
159C2×D8#4112yes—
160C2×D8#4112yes—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174#121—24no—
175#121—24no—
176#121—42no—
177#121—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—84no—
183C4×D4—84no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊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⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42D811yescharacteristic
207C2×C42D811yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210D16—28no—
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—
222C2×C16C2×D412yes—
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—21no—
233#107#4112yes—
234#107#4112yes—
235#102—24no—
236#102—24no—
237#102—42no—
238#102—42no—
239#102—42no—
240#102—42no—
241C4×C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—22no—
245C4×C8C2×D412yes—
246C4×C8#4112yes—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—41no—
254C24—41no—
255C24C2×D811yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—44no—
259C2×D4—44no—
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—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C22×D411yescharacteristic
304D8—22no—
305D8—22no—
306D8—22no—
307D8—22no—
308D8—41no—
309D8—41no—
310D8—41no—
311D8—41no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—44no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8#12112yes—
326C2×C8C22×D411yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8C2×#4111yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—21no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—82no—
361C42—82no—
362C42#12112yes—
363C42#13512yes—
364C16—22no—
365C16—22no—
366C16—41no—
367C16—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23#50912yes—
389C23C22×D811yescharacteristic
390D4—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—88no—
395D4—88no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
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—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
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—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—164no—
437C2×C4—164no—
438C2×C4#46612yes—
439C2×C4#46612yes—
440C2×C4#50912yes—
441C2×C4C2×#12111yescharacteristic
442C2×C4#52511yescharacteristic
443C2×C4#52511yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C8—22no—
448C8—41no—
449C8—41no—
450C8#52511yescharacteristic
451C8#52911yescharacteristic
452C8#52911yescharacteristic
453C8#53211yescharacteristic
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22C2×#46611yescharacteristic
476C22#310011yescharacteristic
477C22#310011yescharacteristic
478C22C2×#50911yescharacteristic
479C22C2×#50911yescharacteristic
480C22#341411yescharacteristic
481C22#341411yescharacteristic
482C4—22no—
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4#239211yescharacteristic
494C4#305411yescharacteristic
495C4#310211yescharacteristic
496C4#310211yescharacteristic
497C2—42no—
498C2—162no—
499C2—162no—
500C2#2369611yescharacteristic
501C2#3360511yescharacteristic
502C2#3360511yescharacteristic
503C2#3366911yescharacteristic
504C2#3367111yescharacteristic
505C2#3367411yescharacteristic
506C2#3367511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13304C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2177C2semidirect
2. C2×#2163C2semidirect
3. C4×#466C2semidirect
4. C4×#466C2semidirect
5. #13516C2semidirect
6. #13494C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13448C2semidirect
10. #13381C2semidirect
11. #13380C2semidirect
13. #12674C2semidirect
14. #12637C2semidirect
15. #12637C2semidirect
133. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21772
C2×#21632
C4×#4662
#135162
#134942
#134762
#134742
#134482
#133812
#133802
#133042
#126742
#126372