Finite Group Databasebeta

← #505258

Subgroups — #505258

Order 512.

515 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27994C211yesmaximal, characteristic, complemented
2#22682C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13167C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13105C211yesmaximal, characteristic, complemented
10#12937C211yesmaximal, characteristic
11#12937C211yesmaximal, characteristic
12#12937C211yesmaximal, characteristic
13#12935C211yesmaximal, characteristic
14#12811C211yesmaximal, characteristic
15#12637C211yesmaximal, characteristic
16C2×C4×D8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2506—21no—
20#2349—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31#2099—21no—
32#2086—21no—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1918—21no—
40#1864—21no—
41#1864—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
44#1804—21no—
45C4×#107C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
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×D8C2311yescharacteristic
60C2×#125—41no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C4×D8—22no—
69C4×D8—22no—
70C4×D8—22no—
71C4×D8—22no—
72C4×D8—41no—
73C4×D8—41no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×C4⋊C8D411yescharacteristic
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#400—41no—
91#398—41no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#387—41no—
99#385—41no—
100#385—41no—
101#384—21no—
102#382—21no—
103#380—41no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4D411yescharacteristic
106C2×#95—41no—
107#367—22no—
108#367—22no—
109#367—41no—
110#366—41no—
111C4⋊C16—41no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357Q814yes—
116#346—41no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#336D411yescharacteristic
125#336D411yescharacteristic
126C22×D4—21no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—41no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8C2×Q812yes—
134C2×D8#4112yes—
135C2×D8#4112yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140#125—84no—
141C4×D4—24no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—81no—
146C4×D4—81no—
147C2×C4⋊C4—21no—
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—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4D811yescharacteristic
163C2×C4⋊C4D811yescharacteristic
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16C2×Q812yes—
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—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—42no—
196#107C2×Q812yes—
197#107#4112yes—
198#106—81no—
199C4⋊C8—41no—
200#102—24no—
201#102—42no—
202#102—42no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95D811yescharacteristic
219#95D811yescharacteristic
220C24—41no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—161no—
240C22×C4—161no—
241C22×C4—161no—
242C22×C4—161no—
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C22×D411yescharacteristic
250D8—22no—
251D8—22no—
252D8—22no—
253D8#12514yescomplemented
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8C22×Q811yescharacteristic
264C2×C8C2×#4111yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C2×C8C2×#4111yescharacteristic
267C2×C8H(5,𝔽2)11yescharacteristic
268C2×C82−1+411yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—82no—
303C16—22no—
304C16—41no—
305C16—41no—
306C16—41no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23C22×D811yescharacteristic
315D4—24no—
316D4—42no—
317D4—42no—
318D4—42no—
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—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
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—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—164no—
358C2×C4—164no—
359C2×C4—164no—
360C2×C4#50912yes—
361C2×C4#52511yescharacteristic
362C2×C4#54611yescharacteristic
363C2×C4D4×Q811yescharacteristic
364C2×C4#57511yescharacteristic
365C2×C4#57711yescharacteristic
366C2×C4#57811yescharacteristic
367C8—41no—
368C8—41no—
369C8—41no—
370C8C2×#12511yescharacteristic
371C8#53911yescharacteristic
372C8#55211yescharacteristic
373C8#55411yescharacteristic
374C22—44no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22#310011yescharacteristic
382C22#329411yescharacteristic
383C22Q8×D811yescharacteristic
384C22C2×#50911yescharacteristic
385C22#341411yescharacteristic
386C22#341611yescharacteristic
387C22#341611yescharacteristic
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—82no—
395C4—82no—
396C4—161no—
397C4—161no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4#247311yescharacteristic
402C4#312011yescharacteristic
403C4#331511yescharacteristic
404C4#338811yescharacteristic
405C2—84no—
406C2#2729411yescharacteristic
407C2#3367511yescharacteristic
408C2#3368311yescharacteristic
409C2#3387311yescharacteristic
410C2#3389011yescharacteristic
411C2#3395111yescharacteristic
412C2#3395511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27994C2semidirect
3. #13539C2semidirect
4. #13516C2semidirect
5. #13507C2semidirect
6. #13506C2semidirect
7. #13167C2semidirect
8. #13110C2semidirect
9. #13105C2semidirect
253. D8#125semidirect

Maximal subgroups

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

subgroupindex
#279942
#226822
#135392
#135162
#135072
#135062
#131672
#131102
#131052
#129372
#129352
#128112
#126372