Finite Group Databasebeta

← #500433

Subgroups — #500433

Order 512.

938 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
1C2×#3334C211yesmaximal, characteristic
2C2×#2225C211yesmaximal, characteristic, complemented
3C2×C4×SD32C211yesmaximal, characteristic, complemented
4#28228C211yesmaximal, characteristic, complemented
5#13381C212yesmaximal, complemented
6#13305C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13189C211yesmaximal, characteristic, complemented
9#12993C212yesmaximal, complemented
10#12920C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C22×SD32C2211yescharacteristic
14#3334—24no—
15#3334—24no—
16C2×#500C2211yescharacteristic
17C2×#480—22no—
18C2×#474—21no—
19C2×#467C2212yes—
20C2×#466C2212yes—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×#107C2212yes—
25#2538—21no—
26#2225—24no—
27C4×SD32—24no—
28C2×#367C2212yes—
29C2×#367C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2212yes—
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2212yes—
35C2×C22⋊C16C2212yes—
36C2×C22⋊C16C2212yes—
37C2×C4×C16C2211yescharacteristic
38#2036—22no—
39#1845C2211yescharacteristic
40#1808C2212yes—
41#1803—21no—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45C22×#41—41no—
46C22×Q16C2311yescharacteristic
47C22×D8C2311yescharacteristic
48C23×C8C2312yes—
49#547—28no—
50#547—44no—
51C2×C4⋊Q8—21no—
52C2×#123—42no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—41no—
55C2×#121—22no—
56C2×C4×Q8—21no—
57C2×C4×D4—21no—
58C2×C4×D4D411yescharacteristic
59C22×C4⋊C4—22no—
60C2×SD32—22no—
61C2×SD32—22no—
62C2×SD32—22no—
63C22×C16C2311yescharacteristic
64C22×C16C2312yes—
65C22×C16C2311yescharacteristic
66#500—22no—
67#500D414yes—
68#480—216no—
69#474—28no—
70#467—24no—
71#467—42no—
72#466—24no—
73#466—24no—
74C4×Q16—22no—
75C4×Q16—22no—
76C4×D8—22no—
77C4×D8D414yes—
78C8×D4—22no—
79C8×D4—22no—
80C2×#107—21no—
81C2×#107—22no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2312yes—
87C2×#107C2312yes—
88C2×#107C2311yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×#103—22no—
91C2×#103—21no—
92C2×#102—21no—
93C2×#102—22no—
94C2×C22⋊C8D412yes—
95C2×C4×C8C2311yescharacteristic
96#396—42no—
97#390—41no—
98#389—41no—
99#384—21no—
100#380—22no—
101#367—22no—
102#367—24no—
103C4⋊C16—22no—
104#358—22no—
105#358—24no—
106#357—22no—
107#357—42no—
108C22⋊C16—24no—
109C22⋊C16—42no—
110C4×C16—22no—
111#340—22no—
112C2×#41—48no—
113C2×#41—84no—
114C2×#41—84no—
115C22×Q8—21no—
116C22×Q8—41no—
117C22×D4—21no—
118C22×D4—42no—
119C22×D4D811yescharacteristic
120C23×C4—42no—
121C23×C4—41no—
122C23×C4C2×D412yes—
123C2×Q16—21no—
124C2×Q16—21no—
125C2×Q16—21no—
126C2×Q16—21no—
127C2×Q16—21no—
128C2×Q16#4112yescomplemented
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8C2×D412yes—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D411yescharacteristic
143C22×C8C2411yescharacteristic
144C4⋊Q8—24no—
145C4⋊Q8—42no—
146#123—416no—
147C22⋊Q8—48no—
148C22⋊Q8—44no—
149C22⋊Q8—82no—
150#121—28no—
151#121—44no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×D4—22no—
155C4×D4—24no—
156C4×D4—22no—
157C4×D4—42no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—41no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4D812yes—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178SD32—28no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—21no—
185C2×C16—22no—
186C2×C16—21no—
187C2×C16#4112yescomplemented
188#107—28no—
189#107—24no—
190#107—21no—
191#107—22no—
192#107—21no—
193#107—21no—
194#107—22no—
195#107—22no—
196#107—81no—
197#107C2×D412yes—
198#107C2×D414yes—
199C4⋊C8—22no—
200#103—44no—
201#103—42no—
202#102—28no—
203#102—24no—
204C22⋊C8—24no—
205C4×C8C2×D412yes—
206C4×C8C2×D412yes—
207#95—42no—
208C24—41no—
209C24C2×D812yes—
210#41—832no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—48no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—84no—
230C2×D4—84no—
231C2×D4SD3214yescomplemented
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—48no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4C2×D812yes—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C22×D411yescharacteristic
253Q16—22no—
254Q16—22no—
255Q16—41no—
256Q16—41no—
257D8—22no—
258D8—22no—
259D8—41no—
260D8—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8#12012yes—
271C2×C8C22×D411yescharacteristic
272C2×C8C22×D411yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8H(5,𝔽2)11yescharacteristic
276C2×C82−1+411yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—48no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—84no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—161no—
292C22⋊C4—24no—
293C22⋊C4—48no—
294C22⋊C4—44no—
295C22⋊C4—84no—
296C22⋊C4—162no—
297C42—82no—
298C42#12012yes—
299C42AGL(1,ℤ/8)12yes—
300C16—22no—
301C16—41no—
302C16—42no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23C2×SD3214yes—
317C23C22×D811yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—81no—
321Q8—88no—
322Q8—81no—
323D4—28no—
324D4—42no—
325D4—42no—
326D4—816no—
327D4—81no—
328D4—81no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—816no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—88no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4#44712yes—
356C2×C4#44712yes—
357C2×C4C2×SD3212yes—
358C2×C4#50912yes—
359C2×C4C2×#12011yescharacteristic
360C2×C4#54611yescharacteristic
361C2×C4#54711yescharacteristic
362C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
363C2×C4#57711yescharacteristic
364C2×C4#57811yescharacteristic
365C8—22no—
366C8—42no—
367C8#52511yescharacteristic
368C8#53711yescharacteristic
369C8#54811yescharacteristic
370C8#55011yescharacteristic
371C22—24no—
372C22—24no—
373C22—24no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22C2×#44711yescharacteristic
386C22#329411yescharacteristic
387C22#333411yescharacteristic
388C22C22×SD3211yescharacteristic
389C22C2×#50911yescharacteristic
390C22#341611yescharacteristic
391C22#341711yescharacteristic
392C4—22no—
393C4—42no—
394C4—82no—
395C4—82no—
396C4—84no—
397C4—162no—
398C4—161no—
399C4#218714yes—
400C4#240711yescharacteristic
401C4#301311yescharacteristic
402C4#333011yescharacteristic
403C4#335411yescharacteristic
404C2—44no—
405C2—162no—
406C2#2411611yescharacteristic
407C2C2×#218711yescharacteristic
408C2#3352711yescharacteristic
409C2#3386511yescharacteristic
410C2#3389911yescharacteristic
411C2#3390211yescharacteristic
412C2#3392011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3334C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2225C2semidirect
3. C2×C4×SD32C2semidirect
4. #28228C2semidirect
5. #13381C2semidirect
6. #13305C2semidirect
7. #13282C2semidirect
8. #13189C2semidirect
9. #12993C2semidirect
10. #12920C2semidirect
11. #12811C2semidirect
128. C2×Q16#41semidirect
187. C2×C16#41semidirect
231. C2×D4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#33342
C2×#22252
C2×C4×SD322
#282282
#133812
#133052
#132822
#131892
#129932
#129202
#128112