Finite Group Databasebeta

← #498014

Subgroups — #498014

Order 512.

541 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
1C4×#508C211yesmaximal, characteristic
2#22802C211yesmaximal, characteristic, complemented
3#22688C211yesmaximal, characteristic, complemented
4#13151C212yesmaximal, complemented
5#13113C212yesmaximal, complemented
6#13011C212yesmaximal
7#12891C211yesmaximal, characteristic, complemented
8#12815C211yesmaximal, characteristic, complemented
9#12815C211yesmaximal, characteristic, complemented
10C16×C4⋊C4C211yesmaximal, characteristic, complemented
11#12641C212yesmaximal, complemented
12C2×#508C2211yescharacteristic
13C2×#443C2212yes—
14C4×#135C2211yescharacteristic
15C4×#135C2211yescharacteristic
16#2925—21no—
17#2906—21no—
18C4×Q32C411yescharacteristic, complemented
19C4×Q32C411yescharacteristic, complemented
20C4×SD32C411yescharacteristic, complemented
21C4×SD32C411yescharacteristic, complemented
22C4×SD32C411yescharacteristic, complemented
23C4×SD32C411yescharacteristic, complemented
24C4×D16C411yescharacteristic, complemented
25C4×D16C411yescharacteristic, complemented
26#2158C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#359C2212yes—
29C2×#359C2212yes—
30#2138C2211yescharacteristic
31#2138C2211yescharacteristic
32C2×C4×C16C2212yes—
33C2×C4×C16C2211yescharacteristic
34#1974C2212yes—
35#1945C2212yes—
36#1929—22no—
37#1866—22no—
38#1848C2211yescharacteristic
39#1848C2211yescharacteristic
40#1806—21no—
41#1806—21no—
42#1772C2212yes—
43C8×C4⋊C4C2211yescharacteristic
44#1382C2212yes—
45#1382C2212yes—
46#1381C2212yes—
47#1381C2212yes—
48#1374C2212yes—
49C2×#135C2311yescharacteristic
50C2×#135C2311yescharacteristic
51C2×#131—22no—
52C4×#41—21no—
53C4×#41—21no—
54#508—22no—
55#508—22no—
56C2×Q32C2×C411yescharacteristic
57C2×SD32C2×C411yescharacteristic
58C2×SD32C2×C411yescharacteristic
59C2×D16C2×C411yescharacteristic
60C22×C16C2312yes—
61C22×C16C2311yescharacteristic
62C22×C16C2311yescharacteristic
63#443—24no—
64#443—42no—
65C4×Q16C2×C411yescharacteristic
66C4×Q16C2×C411yescharacteristic
67C4×Q16C2×C411yescharacteristic
68C4×Q16C2×C411yescharacteristic
69C4×SD16—21no—
70C4×SD16—21no—
71C4×SD16—21no—
72C4×SD16—21no—
73C4×D8C2×C411yescharacteristic
74C4×D8C2×C411yescharacteristic
75C4×D8C2×C411yescharacteristic
76C4×D8C2×C411yescharacteristic
77#433—42no—
78#432—41no—
79#431—41no—
80C2×#108—21no—
81C2×#108—21no—
82C2×#108C2312yes—
83C2×#108C2312yes—
84#427C2311yescharacteristic
85#427C2311yescharacteristic
86#424—22no—
87#424—42no—
88#423—21no—
89#423—21no—
90#423—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8D411yescharacteristic
93C2×#104—22no—
94C2×#104—22no—
95#416—21no—
96#416—21no—
97#408—22no—
98#405—22no—
99C2×C4×C8C2312yes—
100C2×C4×C8C2311yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102#367C2×C411yescharacteristic
103#367C2×C411yescharacteristic
104#366C2×C411yescharacteristic
105#366C2×C411yescharacteristic
106C4⋊C16—21no—
107C4⋊C16—21no—
108#359—24no—
109#359—42no—
110#358C2×C411yescharacteristic
111#358C2×C411yescharacteristic
112#358C2×C411yescharacteristic
113#358C2×C411yescharacteristic
114#357C2×C411yescharacteristic
115#357C2×C411yescharacteristic
116#357C2×C411yescharacteristic
117#357C2×C411yescharacteristic
118C4×C16—42no—
119C4×C16C2×C411yescharacteristic
120C4×C16C2×C411yescharacteristic
121C4×C16C2×C411yescharacteristic
122C4×C16C2×C411yescharacteristic
123#339—22no—
124#339—22no—
125#336D412yes—
126C2×#41—21no—
127C2×#41—21no—
128#135—21no—
129#135—21no—
130#135—21no—
131#135—21no—
132#135—21no—
133#135—21no—
134#135#4112yes—
135#135#4112yes—
136C2×Q16C22×C411yescharacteristic
137C2×Q16C22×C411yescharacteristic
138C2×SD16—21no—
139C2×SD16—21no—
140C2×D8C22×C411yescharacteristic
141C2×D8C22×C411yescharacteristic
142#131—44no—
143#131—82no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—42no—
147C22×C8—42no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D412yes—
150C22×C8C2411yescharacteristic
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—41no—
162C4×D4—41no—
163#117—21no—
164#117—21no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—21no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C2×D412yes—
171C2×C42C2×D411yescharacteristic
172Q32—22no—
173SD32—22no—
174SD32—22no—
175D16—22no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16C22×C411yescharacteristic
184C2×C16C22×C411yescharacteristic
185C2×C16C22×C411yescharacteristic
186C2×C16C22×C411yescharacteristic
187#108—22no—
188#108—22no—
189#108—22no—
190#108—22no—
191#108—81no—
192#108—81no—
193#107C22×C411yescharacteristic
194#107C22×C411yescharacteristic
195#107C22×C411yescharacteristic
196#107C22×C411yescharacteristic
197#106—21no—
198#106—21no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8D811yescharacteristic
206C4⋊C8D811yescharacteristic
207C4⋊C8D812yes—
208#104—44no—
209#104—82no—
210#103—21no—
211#103—21no—
212#103—21no—
213#103—21no—
214#102—21no—
215#102—21no—
216#102—21no—
217#102—21no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—42no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—42no—
224C4×C8C22×C411yescharacteristic
225C4×C8C22×C411yescharacteristic
226C4×C8#4111yescharacteristic
227C4×C8#4111yescharacteristic
228#95—21no—
229#41—22no—
230#41—22no—
231#41—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236#41—41no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4#13511yescharacteristic
251C22×C4#13511yescharacteristic
252C22×C4C22×D411yescharacteristic
253Q16—21no—
254Q16—21no—
255Q16—21no—
256Q16#11712yes—
257SD16—22no—
258SD16—22no—
259SD16—41no—
260SD16—41no—
261D8—21no—
262D8—21no—
263D8—21no—
264D8#11712yes—
265M16—42no—
266M16—42no—
267M16—42no—
268M16—42no—
269M16—84no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—84no—
277C2×C8C4×D411yescharacteristic
278C2×C8C4×D411yescharacteristic
279C2×C8C2×D812yes—
280C2×C8C2×D812yes—
281C2×C8C23×C411yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C2×C82−1+411yescharacteristic
287C4⋊C4—24no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C2×D811yescharacteristic
313C42C2×D811yescharacteristic
314C16—21no—
315C16—21no—
316C16—21no—
317C16—21no—
318C16—42no—
319C23—41no—
320C23—41no—
321C23C2×#13511yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326D4—22no—
327D4—42no—
328D4—42no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—161no—
352C2×C4—161no—
353C2×C4C4×D811yescharacteristic
354C2×C4C4×D811yescharacteristic
355C2×C4C4×D811yescharacteristic
356C2×C4C4×D811yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4#54611yescharacteristic
359C2×C4#54811yescharacteristic
360C2×C4C22×D811yescharacteristic
361C2×C4#57711yescharacteristic
362C2×C4#57711yescharacteristic
363C8—21no—
364C8—21no—
365C8—22no—
366C8—22no—
367C8—82no—
368C8—82no—
369C8C2×#11711yescharacteristic
370C8#52011yescharacteristic
371C8#55211yescharacteristic
372C8#55311yescharacteristic
373C22—21no—
374C22—21no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22C2×C4×D811yescharacteristic
382C22#330311yescharacteristic
383C22#336011yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—44no—
388C4—42no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4#217812yes—
395C4#235211yescharacteristic
396C4#295211yescharacteristic
397C4#329411yescharacteristic
398C4#335811yescharacteristic
399C2—21no—
400C2—21no—
401C2—82no—
402C2—161no—
403C2#2276611yescharacteristic
404C2C2×#217811yescharacteristic
405C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#508C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22802C2semidirect
3. #22688C2semidirect
4. #13151C2semidirect
5. #13113C2semidirect
7. #12891C2semidirect
8. #12815C2semidirect
9. #12815C2semidirect
10. C16×C4⋊C4C2semidirect
11. #12641C2semidirect
18. C4×Q32C4semidirect
19. C4×Q32C4semidirect
20. C4×SD32C4semidirect
21. C4×SD32C4semidirect
22. C4×SD32C4semidirect
23. C4×SD32C4semidirect
24. C4×D16C4semidirect
25. C4×D16C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5082
#228022
#226882
#131512
#131132
#130112
#128912
#128152
C16×C4⋊C42
#126412