Finite Group Databasebeta

← #501874

Subgroups — #501874

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27661C211yesmaximal, characteristic
2D4×#107C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13505C211yesmaximal, characteristic, complemented
5#13492C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13109C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#12976C211yesmaximal, characteristic
10#12919C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic, complemented
12#12866C211yesmaximal, characteristic, complemented
13#12866C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15C2×#1380C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2486—21no—
22D4×C4⋊C4—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#2086—21no—
31#2075—21no—
32#1947—21no—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1852—21no—
36#1852—21no—
37#1845C2211yescharacteristic
38#1823—21no—
39#1808C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#123—41no—
62C2×C4×D4—41no—
63C2×C4×D4D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C16—22no—
68C22×C16—22no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C8×D4—22no—
75C8×D4—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—41no—
85C2×#107—41no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#398—41no—
99#397—41no—
100#389—21no—
101#389—21no—
102#389—41no—
103#385—41no—
104#385—41no—
105#384—21no—
106#381—41no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113#366—22no—
114#366—22no—
115#366—41no—
116#366—41no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119C22⋊C16—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4D811yescharacteristic
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8#4112yes—
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150#123—84no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—84no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D811yescharacteristic
182C2×C22⋊C4D811yescharacteristic
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—44no—
197C2×C16#4112yes—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—42no—
214#107—42no—
215#107—42no—
216#107—81no—
217#107#4112yes—
218C4⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24C2×D811yescharacteristic
232C24C2×D811yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4D1614yes—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—161no—
262C22×C4—161no—
263C22×C4#12312yes—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8#12312yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C82−1+411yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—161no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—82no—
327C16—22no—
328C16—41no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23#48012yes—
342C23C2×D1612yes—
343C23C2×D1612yes—
344C23#51012yes—
345C23C22×D811yescharacteristic
346D4—28no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—164no—
390C2×C4—164no—
391C2×C4—164no—
392C2×C4#48012yes—
393C2×C4C2×D1612yes—
394C2×C4C2×#12311yescharacteristic
395C2×C4#54611yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57711yescharacteristic
399C2×C4#57811yescharacteristic
400C8—22no—
401C8—41no—
402C8—41no—
403C8#52911yescharacteristic
404C8#53911yescharacteristic
405C8#55111yescharacteristic
406C8#55211yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22#223214yes—
421C22C2×#48011yescharacteristic
422C22#329411yescharacteristic
423C22#333411yescharacteristic
424C22C22×D1611yescharacteristic
425C22C2×#51011yescharacteristic
426C22#341611yescharacteristic
427C22#341711yescharacteristic
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4#241911yescharacteristic
441C4#309711yescharacteristic
442C4#331211yescharacteristic
443C4#334411yescharacteristic
444C2—24no—
445C2—42no—
446C2#2493311yescharacteristic
447C2C2×#223211yescharacteristic
448C2#3365111yescharacteristic
449C2#3386111yescharacteristic
450C2#3387911yescharacteristic
451C2#3390111yescharacteristic
452C2#3391111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13537C2semidirect
4. #13505C2semidirect
5. #13492C2semidirect
6. #13492C2semidirect
7. #13109C2semidirect
8. #13109C2semidirect
10. #12919C2semidirect
11. #12919C2semidirect
12. #12866C2semidirect
13. #12866C2semidirect
14. #12834C2semidirect
15. C2×#1380C2semidirect

Maximal subgroups

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

subgroupindex
#276612
D4×#1072
#135372
#135052
#134922
#131092
#129762
#129192
#128662
#128342
C2×#13802