Finite Group Databasebeta

← #499622

Subgroups — #499622

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21723C211yesmaximal, characteristic, complemented
2#21282C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13474C211yesmaximal, characteristic
9#13453C211yesmaximal, characteristic, complemented
10#13365C211yesmaximal, characteristic
11#12674C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2330—21no—
19#2319—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#1992—21no—
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1944—21no—
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1936—21no—
41#1936—21no—
42#1845C2211yescharacteristic
43#1776—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47#1762—21no—
48#1726C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×#125—21no—
61C2×C4×D4—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#499—22no—
67#499—22no—
68C4×D8—22no—
69C4×D8Q814yes—
70C2×#107—21no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#390—21no—
89#390—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—41no—
95#389—41no—
96#388—41no—
97#388—41no—
98#387—41no—
99#387—41no—
100#384—21no—
101#381—41no—
102#380—41no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106#376D411yescharacteristic
107#367—22no—
108#366—22no—
109#366—22no—
110#366—22no—
111C4⋊C16—22no—
112C4⋊C16—22no—
113#357—22no—
114#357—22no—
115#357—22no—
116#357—22no—
117#346—22no—
118C4×C16—22no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×D4—21no—
127C23×C4—41no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8C2×Q812yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2411yescharacteristic
138#125—24no—
139#125—42no—
140C4×D4—24no—
141C4×D4—42no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16#4112yes—
179C2×C16#4112yes—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—42no—
192#107C2×Q812yes—
193#107#4112yes—
194#107#4112yes—
195#106—24no—
196#102—24no—
197#102—42no—
198#102—42no—
199#97—22no—
200#97—22no—
201#97—41no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8C2×Q812yes—
207C4×C8#4112yes—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95D811yescharacteristic
216#95D811yescharacteristic
217#95D811yescharacteristic
218#95D811yescharacteristic
219C24—41no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—161no—
235C22×C4—161no—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C22×D411yescharacteristic
243D8—22no—
244D8—22no—
245D8—41no—
246D8—41no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C22⋊Q812yes—
255C2×C8#13512yes—
256C2×C8C22×Q811yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—161no—
279C4⋊C4—161no—
280C4⋊C4—161no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C22⋊C4—44no—
285C22⋊C4—161no—
286C22⋊C4—161no—
287C22⋊C4—161no—
288C22⋊C4—161no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—161no—
294C42—161no—
295C42—161no—
296C42C22⋊Q812yes—
297C42#13712yes—
298C16—22no—
299C16—22no—
300C16—41no—
301C16—41no—
302C23—42no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23C22×D811yescharacteristic
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
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—164no—
352C2×C4#47412yes—
353C2×C4#47412yes—
354C2×C4C2×C22⋊Q811yescharacteristic
355C2×C4#52511yescharacteristic
356C2×C4#53211yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C2×C4C2×#13711yescharacteristic
359C2×C4#57511yescharacteristic
360C8—22no—
361C8—22no—
362C8—41no—
363C8#52911yescharacteristic
364C8#52911yescharacteristic
365C8#53311yescharacteristic
366C8#53311yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22C2×#47411yescharacteristic
375C22#310011yescharacteristic
376C22#316911yescharacteristic
377C22#341511yescharacteristic
378C22#341511yescharacteristic
379C22#341611yescharacteristic
380C22#341711yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4#239611yescharacteristic
395C4#307611yescharacteristic
396C4#310811yescharacteristic
397C4#317211yescharacteristic
398C2—162no—
399C2#2377411yescharacteristic
400C2#3363111yescharacteristic
401C2#3363311yescharacteristic
402C2#3367911yescharacteristic
403C2#3368411yescharacteristic
404C2#3374011yescharacteristic
405C2#3374511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21723C2semidirect
3. #13538C2semidirect
5. #13515C2semidirect
6. #13514C2semidirect
9. #13453C2semidirect
11. #12674C2semidirect
12. C4×#366C2semidirect
13. #12654C2semidirect

Maximal subgroups

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

subgroupindex
#217232
#212822
#135382
#135262
#135152
#135142
#134762
#134742
#134532
#133652
#126742
C4×#3662
#126542
#126372
C4×#3572