Finite Group Databasebeta

← #512347

Subgroups — #512347

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic
2#20455C211yesmaximal, characteristic
3#13277C211yesmaximal, characteristic, complemented
4#13277C211yesmaximal, characteristic, complemented
5#13177C211yesmaximal, characteristic, complemented
6#13176C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic
12#12921C211yesmaximal, characteristic
13#12867C211yesmaximal, characteristic
14#12866C211yesmaximal, characteristic
15#12784C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2629—21no—
21#2274—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#1992—21no—
31#1943—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1845C2211yescharacteristic
35#1809—21no—
36#1808C2211yescharacteristic
37#1808C2211yescharacteristic
38#1776—21no—
39#1776—21no—
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44#1736—21no—
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
59C22×M16D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#125—21no—
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#499—22no—
70#499—22no—
71#427—22no—
72#427—22no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—21no—
76C2×#107Q814yes—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87#405—22no—
88#405—22no—
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#398—41no—
94#398—41no—
95#396—41no—
96#396—41no—
97#389—21no—
98#389—21no—
99#384—21no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—41no—
107C4×C4⋊C4—41no—
108C4×C22⋊C4—41no—
109C4×C22⋊C4—41no—
110#367—22no—
111#367—22no—
112#366—22no—
113#366—22no—
114C4⋊C16—22no—
115C4⋊C16—22no—
116C22⋊C16—22no—
117C22⋊C16—22no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—41no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×Q812yes—
139C22×C8#4112yes—
140C22×C8C2411yescharacteristic
141#125—42no—
142#125—42no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—42no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4D811yescharacteristic
165C2×C4⋊C4D811yescharacteristic
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4D811yescharacteristic
170C2×C22⋊C4D811yescharacteristic
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190#107—24no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—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—42no—
204#107C2×Q812yes—
205#107C2×Q812yes—
206#107#4112yes—
207#107#4112yes—
208#106—24no—
209#106—42no—
210#97—22no—
211#97—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8#4112yes—
215C4×C8#4112yes—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224C24C2×D811yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—161no—
239C22×C4—161no—
240C22×C4C22⋊Q812yes—
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4#13712yes—
247C22×C4C22×D411yescharacteristic
248M16—44no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8C22⋊Q812yes—
261C2×C8#12312yes—
262C2×C8C22×Q811yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C2×C8C2×#4111yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C2×C8C2×#4111yescharacteristic
267C2×C8C2×#4111yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—161no—
290C4⋊C4—161no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—44no—
296C22⋊C4—161no—
297C22⋊C4—161no—
298C42—21no—
299C42—21no—
300C42—44no—
301C42—161no—
302C42—161no—
303C42—161no—
304C42—161no—
305C42#12312yes—
306C42#13712yes—
307C16—41no—
308C16—41no—
309C16—41no—
310C16—41no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23#47412yes—
318C23C22×D811yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—164no—
360C2×C4#47412yes—
361C2×C4#48012yes—
362C2×C4#48012yes—
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4C2×#12311yescharacteristic
365C2×C4#52511yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57511yescharacteristic
369C8—22no—
370C8—22no—
371C8—41no—
372C8#52911yescharacteristic
373C8#52911yescharacteristic
374C8#53311yescharacteristic
375C8#53311yescharacteristic
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C2×#47411yescharacteristic
384C22C2×#48011yescharacteristic
385C22#310011yescharacteristic
386C22#341511yescharacteristic
387C22#341611yescharacteristic
388C22#341611yescharacteristic
389C22#341711yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4#285511yescharacteristic
404C4#307811yescharacteristic
405C4#309511yescharacteristic
406C4#311311yescharacteristic
407C2—42no—
408C2#3314611yescharacteristic
409C2#3363111yescharacteristic
410C2#3363211yescharacteristic
411C2#3365211yescharacteristic
412C2#3365311yescharacteristic
413C2#3367911yescharacteristic
414C2#3368011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×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
3. #13277C2semidirect
4. #13277C2semidirect
5. #13177C2semidirect
6. #13176C2semidirect
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
#332162
#204552
#132772
#131772
#131762
#131122
#131112
#131102
#131092
#129212
#128672
#128662
#127842