Finite Group Databasebeta

← #508612

Subgroups — #508612

Order 512.

493 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33319C211yesmaximal, characteristic, complemented
2#21282C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13538C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic
6#13526C211yesmaximal, characteristic
7#13453C211yesmaximal, characteristic, complemented
8#13451C211yesmaximal, characteristic, complemented
9#13365C211yesmaximal, characteristic
10#13363C211yesmaximal, characteristic
11#13178C211yesmaximal, characteristic, complemented
12#13176C211yesmaximal, characteristic, complemented
13#12880C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12811C211yesmaximal, characteristic
16C2×#501—21no—
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2660—21no—
21#2319—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32#1970C2211yescharacteristic
33#1970C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1936—21no—
37#1936—21no—
38#1847—21no—
39#1847—21no—
40#1846C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1762—21no—
46#1726C2211yescharacteristic
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×C4⋊Q8—21no—
61C2×#125—21no—
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#501—44no—
68#500—41no—
69#500—41no—
70#499—41no—
71#499—41no—
72C4×D8—22no—
73C4×D8Q814yescomplemented
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×#97D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#398—41no—
97#398—41no—
98#395—41no—
99#395—41no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#388—41no—
105#388—41no—
106#384—21no—
107#384—21no—
108#382—21no—
109#382—21no—
110#381—41no—
111#380—41no—
112C4×C4⋊C4—21no—
113#376D411yescharacteristic
114#367—41no—
115#366—41no—
116#366—41no—
117#366—41no—
118C4⋊C16—22no—
119C4⋊C16—22no—
120#357—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#340—21no—
125#340—21no—
126#340—21no—
127C22×Q8—41no—
128C22×D4—21no—
129C23×C4—41no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8C2×Q812yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142#125—42no—
143#125—42no—
144C4×D4—24no—
145C4×D4—42no—
146C2×C4⋊C4—21no—
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—41no—
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×C22⋊C4—41no—
163C2×C22⋊C4—41no—
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×C16—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—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—42no—
197#107—42no—
198#107—42no—
199#107C2×Q812yes—
200#106—42no—
201#106—42no—
202#106—42no—
203#102—24no—
204#102—42no—
205#102—42no—
206#97—22no—
207#97—22no—
208#97—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8C2×Q812yes—
214C4×C8#4112yes—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95D811yescharacteristic
223#95D811yescharacteristic
224#95D811yescharacteristic
225#95D811yescharacteristic
226C24—41no—
227C2×Q8—82no—
228C2×Q8—161no—
229C2×Q8—161no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C22×C4—41no—
237C22×C4—44no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—161no—
245C22×C4—161no—
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C22×D411yescharacteristic
253D8—22no—
254D8—22no—
255D8—41no—
256D8—41no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8C22⋊Q812yes—
264C2×C8AGL(1,ℤ/8)12yes—
265C2×C8#13712yes—
266C2×C8C22×Q811yescharacteristic
267C2×C8C2×#4111yescharacteristic
268C2×C8H(5,𝔽2)11yescharacteristic
269C2×C8H(5,𝔽2)11yescharacteristic
270C2×C82−1+411yescharacteristic
271C2×C82−1+411yescharacteristic
272C4⋊C4—22no—
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—41no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—161no—
290C4⋊C4—161no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C22⋊C4—44no—
296C22⋊C4—161no—
297C22⋊C4—161no—
298C22⋊C4—161no—
299C22⋊C4—161no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—161no—
305C42C22⋊Q812yes—
306C42#13712yes—
307C16—41no—
308C16—41no—
309C16—41no—
310C16—41no—
311C23—42no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23C22×D811yescharacteristic
319Q8—164no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
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—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
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—81no—
360C2×C4—81no—
361C2×C4—164no—
362C2×C4#47412yes—
363C2×C4#47412yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4#53511yescharacteristic
366C2×C4#53611yescharacteristic
367C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4C2×#13711yescharacteristic
370C8—22no—
371C8—22no—
372C8—22no—
373C8#54211yescharacteristic
374C8#54211yescharacteristic
375C8#54411yescharacteristic
376C8#54411yescharacteristic
377C22—82no—
378C22—82no—
379C22—82no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22C2×#47411yescharacteristic
385C22#320511yescharacteristic
386C22#322611yescharacteristic
387C22#341511yescharacteristic
388C22#341511yescharacteristic
389C22#341611yescharacteristic
390C22#341711yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4#262811yescharacteristic
405C4#307911yescharacteristic
406C4#321811yescharacteristic
407C4#323811yescharacteristic
408C2—162no—
409C2#2998011yescharacteristic
410C2#3363111yescharacteristic
411C2#3363311yescharacteristic
412C2#3378611yescharacteristic
413C2#3379111yescharacteristic
414C2#3380711yescharacteristic
415C2#3380911yescharacteristic
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. #33319C2semidirect
3. #13539C2semidirect
4. #13538C2semidirect
7. #13453C2semidirect
8. #13451C2semidirect
11. #13178C2semidirect
12. #13176C2semidirect
13. #12880C2semidirect
73. C4×D8Q8semidirect

Maximal subgroups

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

subgroupindex
#333192
#212822
#135392
#135382
#135282
#135262
#134532
#134512
#133652
#133632
#131782
#131762
#128802
#128132
#128112