Finite Group Databasebeta

← #495926

Subgroups — #495926

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C211yesmaximal, characteristic
2C4×#466C211yesmaximal, characteristic
3C4×#427C211yesmaximal, characteristic, complemented
4#13131C211yesmaximal, characteristic, complemented
5#13130C211yesmaximal, characteristic, complemented
6#13110C211yesmaximal, characteristic, complemented
7#13109C211yesmaximal, characteristic, complemented
8#12992C211yesmaximal, characteristic
9#12921C211yesmaximal, characteristic, complemented
10#12868C211yesmaximal, characteristic, complemented
11#12674C211yesmaximal, characteristic, complemented
12#12674C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12599C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C4×#121—21no—
22C4×#117—21no—
23#2234C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1934—21no—
39#1893—21no—
40#1808C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#106—21no—
47C4×#106—21no—
48C4×#102—21no—
49#1735—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8D411yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×C4×D4—41no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C42D411yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#466—22no—
74#466C2×C414yes—
75C4×D8—22no—
76C4×D8—22no—
77#427—22no—
78#427—22no—
79#427—22no—
80#427C2×C414yescomplemented
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#390—21no—
100#389—21no—
101#388—41no—
102#386—41no—
103#380—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C22⋊C4—41no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4—41no—
112#376—41no—
113C43—41no—
114#367—22no—
115#367C2×C414yescomplemented
116#366—22no—
117#366C2×C414yescomplemented
118#357—22no—
119#357—22no—
120#357C2×C414yescomplemented
121#357C2×C414yescomplemented
122C22⋊C16—22no—
123C22⋊C16C2×C414yescomplemented
124#346—22no—
125#346—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4D811yescharacteristic
137C23×C4D811yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—41no—
152C22×C8C22×C412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8#4112yes—
157C22×C8C2411yescharacteristic
158#121—24no—
159#121—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—84no—
163#117—24no—
164#117—24no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42D811yescharacteristic
192C2×C42D811yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16C22×C412yes—
206C2×C16C22×C412yes—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107C22×C412yes—
217#107C22×C412yes—
218#107C22×C412yes—
219#107#4112yes—
220#107#4112yes—
221#106—24no—
222#106—24no—
223#102—24no—
224#102—42no—
225#102—42no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—22no—
230C4×C8#4112yes—
231C4×C8#4112yes—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239C24—41no—
240C24C2×D811yescharacteristic
241C2×D4—22no—
242C2×D4—44no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C4×D412yes—
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4#13512yes—
277C22×C4C22×D411yescharacteristic
278D8—22no—
279D8—22no—
280D8—41no—
281D8—41no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—44no—
294C2×C8C4×D412yes—
295C2×C8#12312yes—
296C2×C8C23×C411yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—82no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—44no—
329C42—44no—
330C42—82no—
331C42—82no—
332C42—82no—
333C42—82no—
334C42—82no—
335C42#12312yes—
336C42#13712yes—
337C16—22no—
338C16—22no—
339C16—41no—
340C16—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23C4×D812yes—
355C23C22×D811yescharacteristic
356D4—42no—
357D4—42no—
358D4—88no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—164no—
400C2×C4C4×D812yes—
401C2×C4#48012yes—
402C2×C4#48012yes—
403C2×C4#50912yes—
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×#12311yescharacteristic
406C2×C4#52511yescharacteristic
407C2×C4C2×#13511yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4#57511yescharacteristic
410C8—22no—
411C8—22no—
412C8—41no—
413C8C4×#4111yescharacteristic
414C8C4×#4111yescharacteristic
415C8#52911yescharacteristic
416C8#52911yescharacteristic
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22C2×C4×D811yescharacteristic
432C22C2×#48011yescharacteristic
433C22#310011yescharacteristic
434C22C2×#50911yescharacteristic
435C22#341411yescharacteristic
436C22#341411yescharacteristic
437C22#341411yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—161no—
449C4—161no—
450C4C4×#12311yescharacteristic
451C4C4×#13711yescharacteristic
452C4#309111yescharacteristic
453C4#310811yescharacteristic
454C2—42no—
455C2—162no—
456C2C4×#48011yescharacteristic
457C2#3345111yescharacteristic
458C2C4×#50911yescharacteristic
459C2#3364711yescharacteristic
460C2#3364811yescharacteristic
461C2#3367111yescharacteristic
462C2#3367511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×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
3. C4×#427C2semidirect
4. #13131C2semidirect
5. #13130C2semidirect
6. #13110C2semidirect
7. #13109C2semidirect
9. #12921C2semidirect
10. #12868C2semidirect
11. #12674C2semidirect
12. #12674C2semidirect
13. #12637C2semidirect
14. #12637C2semidirect
15. #12599C2semidirect
23. #2234C4semidirect
80. #427C2×C4semidirect
115. #367C2×C4semidirect
117. #366C2×C4semidirect
120. #357C2×C4semidirect
121. #357C2×C4semidirect
123. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C4×#4662
C4×#4272
#131312
#131302
#131102
#131092
#129922
#129212
#128682
#126742
#126372
#125992