Finite Group Databasebeta

← #362075

Subgroups — #362075

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C8C211yesmaximal, characteristic
2C2×#1950C212yesmaximal, complemented
3C2×#1810C212yesmaximal
4#20626C211yesmaximal, characteristic
5C4×#407C211yesmaximal, characteristic
6#11233C214yesmaximal, complemented
7#10853C212yesmaximal, complemented
8#10723C212yesmaximal
9C24×C8C2212yes—
10C23×C42C2211yescharacteristic
11C22×#108C2214yes—
12C2×#423C2212yes—
13C22×C4⋊C8—28no—
14C22×C4⋊C8C414yes—
15C22×C4⋊C8C414yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C2×#407C2212yes—
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×C4×M16C2212yes—
23#1950—28no—
24#1810—28no—
25#1797—22no—
26#1797—22no—
27#1749—22no—
28C4×C22⋊C8—22no—
29C2×#341C2214yes—
30C2×#341C2214yes—
31C2×#341C2214yes—
32#1728C2212yes—
33#1728C2212yes—
34C24×C4Q811yescharacteristic
35C24×C4Q811yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16C2314yes—
38C22×M16C2312yes—
39C22×M16C2312yes—
40C23×C8—24no—
41C23×C8—24no—
42C23×C8—28no—
43C23×C8—44no—
44C23×C8C2×C418yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C42—24no—
48C22×C42—24no—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42Q811yescharacteristic
54C22×C42Q811yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#108—28no—
58C2×#108—28no—
59C2×#108—44no—
60#423—24no—
61#423—24no—
62C2×C4⋊C8—28no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—216no—
65C2×C4⋊C8—216no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8C2×C418yes—
69C2×C4⋊C8C2×C412yes—
70C2×C4⋊C8C2×C412yes—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8D418yes—
74#407—22no—
75#407—22no—
76#407—24no—
77#407—24no—
78#407D414yes—
79#407Q814yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C4×M16—24no—
84C4×M16—24no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#341—28no—
88#341—28no—
89#341—44no—
90#336—24no—
91#336—24no—
92C25C2×Q811yescharacteristic
93C23×C4—28no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—44no—
101C23×C4C4⋊C412yes—
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4C4⋊C412yes—
105C23×C4C22×C412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×Q811yescharacteristic
109C23×C4C2×Q811yescharacteristic
110C23×C4C2×Q811yescharacteristic
111C23×C4C2×Q812yes—
112C23×C4C2×Q811yescharacteristic
113C23×C4C2×Q811yescharacteristic
114C23×C4#4112yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16C2×D414yes—
134C2×M16C2×Q814yes—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—24no—
138C22×C8—216no—
139C22×C8—22no—
140C22×C8—24no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—24no—
145C22×C8—216no—
146C22×C8—24no—
147C22×C8—48no—
148C22×C8—416no—
149C22×C8—416no—
150C22×C8C22×C414yes—
151C22×C8C22×C414yes—
152C22×C8C2×D418yes—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—28no—
162C2×C42—28no—
163C2×C42C4⋊C412yes—
164C2×C42C4⋊C412yes—
165C2×C42C4⋊C412yes—
166C2×C42C4⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D412yes—
171C2×C42C2×D414yes—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4114yes—
175#108—416no—
176C4⋊C8—216no—
177C4⋊C8—24no—
178C4⋊C8—24no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181C4⋊C8C22⋊C418yes—
182C4⋊C8C22⋊C418yes—
183C22⋊C8—28no—
184C22⋊C8—28no—
185C22⋊C8—48no—
186#97—44no—
187C4×C8—44no—
188C24—22no—
189C24—22no—
190C24—24no—
191C24—24no—
192C24C2×C4⋊C412yes—
193C24#12512yes—
194C24C4⋊Q812yes—
195C24C22×Q811yescharacteristic
196C22×C4—232no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—416no—
231C22×C4#10818yescomplemented
232C22×C4C2×C4⋊C412yes—
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C412yes—
237C22×C4C2×C4⋊C412yes—
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4C4×D414yes—
240C22×C4C4×Q814yes—
241C22×C4#12114yes—
242C22×C4C22⋊Q814yes—
243C22×C4C22⋊Q814yes—
244C22×C4C22⋊Q814yes—
245C22×C4#12512yes—
246C22×C4C4⋊Q812yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4112yes—
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+412yes—
257M16—24no—
258M16—24no—
259M16—24no—
260M16—24no—
261M16—42no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—44no—
266M16—44no—
267M16—44no—
268M16—44no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—28no—
272C2×C8—28no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—432no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8C2×C22⋊C418yes—
289C42—22no—
290C42—22no—
291C42—24no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42C2×C22⋊C412yes—
297C42C2×C22⋊C412yes—
298C42#11712yes—
299C42#11712yes—
300C42#12114yes—
301C42#12114yes—
302C23—28no—
303C23—28no—
304C23—216no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—28no—
313C23—28no—
314C23—22no—
315C23#38914yes—
316C23C2×#10814yes—
317C23#43014yes—
318C23C22×C4⋊C411yescharacteristic
319C23C2×C22⋊Q812yes—
320C23C2×#12511yescharacteristic
321C23C2×C4⋊Q811yescharacteristic
322C23#54412yes—
323C2×C4—216no—
324C2×C4—28no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—216no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4#38012yes—
350C2×C4#38012yes—
351C2×C4#38012yes—
352C2×C4#38012yes—
353C2×C4#38914yes—
354C2×C4C2×#10814yes—
355C2×C4C2×#10814yes—
356C2×C4C22×C22⋊C411yescharacteristic
357C2×C4C2×#11711yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×C4×Q811yescharacteristic
360C2×C4#51811yescharacteristic
361C2×C4#51911yescharacteristic
362C2×C4C2×#12111yescharacteristic
363C2×C4C2×#12112yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4#53712yes—
366C2×C4D4212yes—
367C2×C4#54612yes—
368C2×C4#54712yes—
369C2×C4#54812yes—
370C2×C4D4×Q812yes—
371C2×C4#55212yes—
372C2×C4#55312yes—
373C2×C4#55412yes—
374C8—28no—
375C8—44no—
376C8—44no—
377C8—44no—
378C8—84no—
379C22—216no—
380C22—216no—
381C22—24no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—24no—
387C22—22no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22#195018yescomplemented
393C22C2×#38011yescharacteristic
394C22C2×#38911yescharacteristic
395C22#231611yescharacteristic
396C22#244012yes—
397C22#244212yes—
398C22C22×#10812yes—
399C22C2×#43012yes—
400C4—24no—
401C4—22no—
402C4—24no—
403C4—22no—
404C4—22no—
405C4—42no—
406C4#181014yescomplemented
407C4D4×C22⋊C411yescharacteristic
408C4Q8×C22⋊C411yescharacteristic
409C4#234411yescharacteristic
410C4#234511yescharacteristic
411C4#241612yes—
412C4#241712yes—
413C2—22no—
414C2—22no—
415C2—28no—
416C2#1390711yescharacteristic
417C2C2×#181011yescharacteristic
418C2#2066711yescharacteristic
419C2C2×#195012yes—
420C2#2196912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C4⋊C4
prime radicalC25C2×Q8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
6. #11233C2semidirect
7. #10853C2semidirect
231. C22×C4#108semidirect
392. C22#1950semidirect
406. C4#1810semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C82
C2×#19502
C2×#18102
#206262
C4×#4072
#112332
#108532
#107232