Finite Group Databasebeta

← #497014

Subgroups — #497014

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2#21525C211yesmaximal, characteristic, complemented
3#21524C211yesmaximal, characteristic
4#13133C211yesmaximal, characteristic, complemented
5#13130C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13062C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#13014C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15#12545C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×#405C2211yescharacteristic
24#2324—21no—
25#2323—21no—
26#2217C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1936—21no—
41#1935—21no—
42#1895—21no—
43#1894—21no—
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1763—21no—
49#1762—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×#121—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×#117D411yescharacteristic
69C2×SD32—41no—
70C2×SD32—41no—
71C2×SD32C2×C414yes—
72C2×M32—22no—
73C2×M32—41no—
74C2×M32—41no—
75C22×C16—41no—
76C22×C16—41no—
77C22×C16C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#467—22no—
83#467C2×C414yescomplemented
84#466—22no—
85#466C2×C414yes—
86C4×Q16—41no—
87C4×Q16—41no—
88C4×D8—41no—
89C4×D8—41no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102#405—41no—
103#405—41no—
104C2×#97D411yescharacteristic
105C2×#97D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390—21no—
110#390—41no—
111#389—21no—
112#389—21no—
113#389—41no—
114#388—41no—
115#388—41no—
116#388—41no—
117#387—41no—
118#386—41no—
119#385—41no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124#367C2×C414yescomplemented
125#366—41no—
126#366—41no—
127#358—41no—
128#358—41no—
129#358C2×C414yescomplemented
130#357—41no—
131#357—41no—
132#357C2×C414yes—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16C22×C412yes—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8C22×C412yes—
156C2×M16—22no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C22×C412yes—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8#4112yes—
168C22×C8C2411yescharacteristic
169C22⋊Q8—24no—
170C22⋊Q8—42no—
171#121—24no—
172#121—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—42no—
176C4×D4—42no—
177#117—41no—
178#117—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4D811yescharacteristic
190C2×C4⋊C4D811yescharacteristic
191C2×C22⋊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⋊C4D811yescharacteristic
199C2×C22⋊C4D811yescharacteristic
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205SD32—44no—
206M32—44no—
207C2×C16—21no—
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—44no—
218C2×C16C22×C412yes—
219C2×C16C22×C412yes—
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#107C22×C412yes—
231#107C22×C412yes—
232#103—24no—
233#103—42no—
234#103—42no—
235#102—24no—
236#102—42no—
237#102—42no—
238#97—41no—
239#97—41no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—41no—
253C24C2×D811yescharacteristic
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—161no—
268C2×D4—161no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—161no—
282C22×C4—161no—
283C22×C4C4×D412yes—
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4#13512yes—
290C22×C4C22×D411yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—41no—
294Q16—41no—
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299M16—44no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8C4×D412yes—
313C2×C8C23×C411yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C82−1+411yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—161no—
336C4⋊C4—161no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—161no—
343C22⋊C4—161no—
344C22⋊C4—161no—
345C22⋊C4—161no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—82no—
351C42—161no—
352C42—161no—
353C16—22no—
354C16—22no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23C4×D812yes—
371C23C22×D811yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—81no—
375Q8—81no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—164no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—164no—
421C2×C4C4×D812yes—
422C2×C4C2×C4×D411yescharacteristic
423C2×C4#53411yescharacteristic
424C2×C4#53711yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57511yescharacteristic
428C8—22no—
429C8—41no—
430C8—41no—
431C8#51811yescharacteristic
432C8#52011yescharacteristic
433C8#53811yescharacteristic
434C8#53911yescharacteristic
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22C2×C4×D811yescharacteristic
450C22#318611yescharacteristic
451C22#324511yescharacteristic
452C22#341511yescharacteristic
453C22#341611yescharacteristic
454C22#341611yescharacteristic
455C22#341711yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—82no—
462C4—82no—
463C4—82no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4#232511yescharacteristic
469C4#294311yescharacteristic
470C4#319111yescharacteristic
471C4#325011yescharacteristic
472C2—42no—
473C2—162no—
474C2#2218111yescharacteristic
475C2#3345711yescharacteristic
476C2#3345911yescharacteristic
477C2#3376411yescharacteristic
478C2#3376711yescharacteristic
479C2#3382311yescharacteristic
480C2#3382411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21525C2semidirect
4. #13133C2semidirect
5. #13130C2semidirect
6. #13111C2semidirect
7. #13110C2semidirect
8. #13062C2semidirect
10. #13014C2semidirect
11. #13013C2semidirect
12. #12936C2semidirect
13. #12935C2semidirect
14. #12834C2semidirect
15. #12545C2semidirect
26. #2217C4semidirect
77. C22×C16C2×C4semidirect
83. #467C2×C4semidirect
124. #367C2×C4semidirect
129. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
#215252
#215242
#131332
#131302
#131112
#131102
#130622
#130612
#130142
#130132
#129362
#129352
#128342
#125452