Finite Group Databasebeta

← #512264

Subgroups — #512264

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33130C211yesmaximal, characteristic
2C4×#427C211yesmaximal, characteristic
3#13133C211yesmaximal, characteristic, complemented
4#13133C211yesmaximal, characteristic, complemented
5#13132C211yesmaximal, characteristic, complemented
6#13132C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic
12#12994C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic
15#12844C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2855—21no—
21C4×#117—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32#1969C2211yescharacteristic
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1933—21no—
37#1933—21no—
38#1892—21no—
39#1808C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#106—21no—
45C4×#106—21no—
46#1735—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C42D411yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#467—22no—
71#467—22no—
72#427—22no—
73#427—22no—
74#427—22no—
75#427—22no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×#106—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—21no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#391—21no—
96#389—21no—
97#389—21no—
98#387—41no—
99#387—41no—
100#385—41no—
101#381—41no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—41no—
108C4×C22⋊C4—41no—
109#376—41no—
110C43—41no—
111#367—22no—
112#367—22no—
113#366—22no—
114#366—22no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119C22⋊C16—22no—
120C22⋊C16—22no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C23×C4D811yescharacteristic
130C23×C4D811yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8#4112yes—
149C22×C8#4112yes—
150C22×C8C2411yescharacteristic
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153#117—24no—
154#117—24no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42D811yescharacteristic
180C2×C42D811yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
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—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107#4112yes—
206#107#4112yes—
207#107#4112yes—
208#107#4112yes—
209#106—24no—
210#106—24no—
211#103—42no—
212#103—42no—
213#103—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—41no—
221#95—41no—
222#95—41no—
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
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
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—41no—
255C22×C4—41no—
256C22×C4#12312yes—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4#13712yes—
263C22×C4C22×D411yescharacteristic
264Q16—41no—
265Q16—41no—
266Q16—41no—
267Q16—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—44no—
281C2×C8#12312yes—
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—44no—
316C42—44no—
317C42—82no—
318C42—82no—
319C42—82no—
320C42—82no—
321C16—41no—
322C16—41no—
323C16—41no—
324C16—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23#48012yes—
332C23C22×D811yescharacteristic
333Q8—81no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
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#48012yes—
380C2×C4#50812yes—
381C2×C4#51012yes—
382C2×C4C2×#12311yescharacteristic
383C2×C4#52511yescharacteristic
384C2×C4#52511yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57511yescharacteristic
387C2×C4#57511yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#52911yescharacteristic
392C8#52911yescharacteristic
393C8#52911yescharacteristic
394C8#52911yescharacteristic
395C22—22no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#48011yescharacteristic
403C22#310011yescharacteristic
404C22#310011yescharacteristic
405C22C2×#50811yescharacteristic
406C22C2×#51011yescharacteristic
407C22#341411yescharacteristic
408C22#341411yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#285311yescharacteristic
423C4#309311yescharacteristic
424C4#311711yescharacteristic
425C4#311711yescharacteristic
426C2—42no—
427C2#3309411yescharacteristic
428C2#3364511yescharacteristic
429C2#3364911yescharacteristic
430C2#3366211yescharacteristic
431C2#3366711yescharacteristic
432C2#3367111yescharacteristic
433C2#3367611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
3. #13133C2semidirect
4. #13133C2semidirect
5. #13132C2semidirect
6. #13132C2semidirect
7. #13112C2semidirect
8. #13111C2semidirect
9. #13110C2semidirect
10. #13109C2semidirect

Maximal subgroups

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

subgroupindex
#331302
C4×#4272
#131332
#131322
#131122
#131112
#131102
#131092
#129942
#129212
#128442