Finite Group Databasebeta

← #497861

Subgroups — #497861

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2168C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic, complemented
3C22×#367C211yesmaximal, characteristic, complemented
4#22656C211yesmaximal, characteristic
5D4×#107C211yesmaximal, characteristic
6#13178C211yesmaximal, characteristic, complemented
7#13177C211yesmaximal, characteristic, complemented
8#12921C211yesmaximal, characteristic, complemented
9#12920C211yesmaximal, characteristic, complemented
10#12920C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic, complemented
12#12908C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23#2352—21no—
24D4×C4⋊C4—21no—
25#2168C418yescomplemented
26#2159—24no—
27C2×#367—24no—
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#367C2211yescharacteristic
33C2×#367C2211yescharacteristic
34C2×#366C2211yescharacteristic
35C2×#366C2211yescharacteristic
36C2×#366C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42#1949—21no—
43#1947—21no—
44#1854—21no—
45#1853—21no—
46#1852—21no—
47#1852—21no—
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C4×#107C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4×D4—41no—
65C2×C4×D4D411yescharacteristic
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C2×M32—22no—
71C2×M32—22no—
72C2×M32C2×C414yescomplemented
73C22×C16—22no—
74C22×C16—22no—
75C22×C16C2×C414yescomplemented
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C8×D4C2×C414yes—
81C8×D4Q814yes—
82#427—22no—
83#427—41no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—41no—
92C2×#107—41no—
93C2×#107—41no—
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#106—21no—
103C2×#106—21no—
104C2×C4⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#389—41no—
109#388—41no—
110#387—41no—
111#386—41no—
112#385—41no—
113#385—41no—
114#384—21no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C22⋊C4—41no—
123#367—22no—
124#367—22no—
125#367—22no—
126#367—22no—
127#367—41no—
128#367D414yescomplemented
129#366—22no—
130#366—22no—
131#366—22no—
132#366—41no—
133C4⋊C16C2×C414yescomplemented
134C22⋊C16C2×C414yescomplemented
135C22⋊C16C2×C414yescomplemented
136#346C2×C414yescomplemented
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143C22×D4D811yescharacteristic
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×Q812yes—
163C22×C8C2×Q812yes—
164C22×C8C2411yescharacteristic
165C4×D4—84no—
166C4×D4C4⋊C414yes—
167C4×D4Q1614yes—
168#117—42no—
169#117—81no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4D811yescharacteristic
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4D811yescharacteristic
196C2×C22⋊C4D811yescharacteristic
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201M32—28no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—28no—
209C2×C16C22×C412yes—
210C2×C16C22×C412yes—
211C2×C16C22×C412yes—
212C2×C16C22×C412yes—
213C2×C16C2×D412yes—
214C2×C16#4112yes—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
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—42no—
230#107—42no—
231#107—42no—
232#107—81no—
233#107C2×D412yes—
234#107C2×D412yes—
235#106—42no—
236#106—81no—
237C4⋊C8C4⋊C414yes—
238C22⋊C8C4⋊C414yes—
239C22⋊C8C4⋊C414yes—
240C4×C8C22×C412yes—
241C4×C8C2×Q812yes—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24C2×D811yescharacteristic
251C24C2×D811yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4#10714yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—161no—
278C22×C4—161no—
279C22×C4—161no—
280C22×C4C2×C4⋊C412yes—
281C22×C4C2×C4⋊C412yes—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×Q1612yes—
287C22×C4C2×Q1612yes—
288C22×C4C22×D411yescharacteristic
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8C2×C4⋊C412yes—
304C2×C8C2×C4⋊C412yes—
305C2×C8C2×C4⋊C412yes—
306C2×C8C23×C411yescharacteristic
307C2×C8C22×D411yescharacteristic
308C2×C8C22×Q811yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8H(5,𝔽2)11yescharacteristic
311C2×C82−1+411yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—161no—
333C4⋊C4#10714yes—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—161no—
339C22⋊C4#10714yes—
340C22⋊C4#10714yes—
341C42—82no—
342C42—82no—
343C42—161no—
344C42C2×C4⋊C412yes—
345C42C2×Q1612yes—
346C16—22no—
347C16—22no—
348C16—22no—
349C16C4×D414yescomplemented
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23C2×#10712yes—
361C23C2×#10712yes—
362C23#50912yes—
363C23#51012yes—
364C23C22×D811yescharacteristic
365D4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
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—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—164no—
407C2×C4—164no—
408C2×C4—164no—
409C2×C4C2×#10712yes—
410C2×C4C2×#10712yes—
411C2×C4C2×#10712yes—
412C2×C4C2×#10712yes—
413C2×C4C22×C4⋊C411yescharacteristic
414C2×C4D4211yescharacteristic
415C2×C4#54711yescharacteristic
416C2×C4C22×Q1611yescharacteristic
417C2×C4#57611yescharacteristic
418C2×C4#57811yescharacteristic
419C8—22no—
420C8—22no—
421C8—22no—
422C8C2×C4×D411yescharacteristic
423C8#52011yescharacteristic
424C8D4×Q811yescharacteristic
425C8#55411yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22#215914yescomplemented
440C22C22×#10711yescharacteristic
441C22D4×D811yescharacteristic
442C22#333411yescharacteristic
443C22C2×#50911yescharacteristic
444C22C2×#51011yescharacteristic
445C22#341611yescharacteristic
446C22#341611yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—82no—
452C4—82no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4D4×C4⋊C411yescharacteristic
460C4#291311yescharacteristic
461C4D4×Q1611yescharacteristic
462C4#333811yescharacteristic
463C2—24no—
464C2—42no—
465C2D4×#10711yescharacteristic
466C2C2×#215911yescharacteristic
467C2#3341111yescharacteristic
468C2#3383111yescharacteristic
469C2#3383211yescharacteristic
470C2#3391111yescharacteristic
471C2#3391211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C2×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
2. C2×#2159C2semidirect
3. C22×#367C2semidirect
6. #13178C2semidirect
7. #13177C2semidirect
8. #12921C2semidirect
9. #12920C2semidirect
10. #12920C2semidirect
11. #12919C2semidirect
12. #12908C2semidirect
13. #12836C2semidirect
14. #12834C2semidirect
15. #12674C2semidirect
25. #2168C4semidirect
72. C2×M32C2×C4semidirect
75. C22×C16C2×C4semidirect
128. #367D4semidirect
133. C4⋊C16C2×C4semidirect
134. C22⋊C16C2×C4semidirect
135. C22⋊C16C2×C4semidirect
136. #346C2×C4semidirect
349. C16C4×D4semidirect
439. C22#2159semidirect

Maximal subgroups

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

subgroupindex
C2×#21682
C2×#21592
C22×#3672
#226562
D4×#1072
#131782
#131772
#129212
#129202
#129192
#129082
#128362
#128342
#126742