Finite Group Databasebeta

← #386807

Subgroups — #386807

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1852C214yesmaximal, complemented
3#13872C212yesmaximal
4#11232C212yesmaximal, complemented
5#11096C214yesmaximal, complemented
6#10862C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C2212yes—
9C22×#107C2212yes—
10C22×#107C2212yes—
11C22×#107C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2214yes—
14C22×C4×C8C2211yescharacteristic
15C2×#386—22no—
16C2×#380C2214yes—
17C2×#380C2212yes—
18#1954C418yescomplemented
19#1852—216no—
20C2×#340C2214yes—
21C2×#340C2214yes—
22C2×#340C2212yes—
23#1441—22no—
24#1439—22no—
25C4⋊#95C2214yes—
26C22⋊#95—22no—
27C23×D4D411yescharacteristic
28C24×C4—22no—
29C24×C4C2312yes—
30C23×C8C2312yes—
31C23×C8C2312yes—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C414yes—
37C2×C4×D4Q814yes—
38C22×C4⋊C4—28no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4—44no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4C2314yes—
45C22×C4⋊C4C2314yes—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—24no—
48C22×C22⋊C4D412yes—
49C22×C42C2311yescharacteristic
50C2×#107—24no—
51C2×#107—28no—
52C2×#107—42no—
53C2×#107—42no—
54C2×#107—42no—
55C2×#107—42no—
56C2×#107—42no—
57C2×#107D418yescomplemented
58C2×C4⋊C8—41no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8C2×C414yescomplemented
61C2×C22⋊C8—28no—
62C2×C22⋊C8—28no—
63C2×C22⋊C8C2×C4116yescomplemented
64C2×C4×C8—41no—
65C2×C4×C8—41no—
66C2×C4×C8C2×C414yescomplemented
67#386—48no—
68#380—28no—
69#380—44no—
70#380—42no—
71#380—42no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95—22no—
75C2×#95—24no—
76#340—28no—
77#340—28no—
78#340—42no—
79C25C2×D412yes—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4C4⋊C414yes—
85C22×D4D814yes—
86C22×D4Q1614yes—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—48no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4—44no—
101C23×C4—84no—
102C23×C4C22×C414yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q814yes—
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8C22×C414yes—
117C22×C8C22×C414yes—
118C22×C8C2×D414yes—
119C22×C8#4114yes—
120C4×D4—44no—
121C4×D4—44no—
122C4×D4—44no—
123C4×D4—44no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—416no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4C4⋊C414yes—
143C2×C4⋊C4C2×D418yes—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—84no—
152C2×C22⋊C4C4⋊C418yes—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42C22×C412yes—
158C2×C42C2×Q812yes—
159#107—48no—
160#107—48no—
161#107—48no—
162C4⋊C8—44no—
163C22⋊C8—232no—
164C4×C8—44no—
165#95—48no—
166#95—44no—
167#95—48no—
168#95—84no—
169C24—22no—
170C24—22no—
171C24—24no—
172C24—24no—
173C24—22no—
174C24—22no—
175C24—24no—
176C24—24no—
177C24C2×C4⋊C414yes—
178C24C2×D814yes—
179C24C2×Q1614yes—
180C24C22×D411yescharacteristic
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4#107116yescomplemented
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—416no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—816no—
234C22×C4—816no—
235C22×C4—88no—
236C22×C4—88no—
237C22×C4C2×C4⋊C412yes—
238C22×C4C2×C4⋊C414yes—
239C22×C4C2×C4⋊C412yes—
240C22×C4C2×D812yes—
241C22×C4C2×Q1612yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4112yes—
246C22×C4H(5,𝔽2)11yescharacteristic
247C22×C4H(5,𝔽2)11yescharacteristic
248C22×C4H(5,𝔽2)11yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C4H(5,𝔽2)12yes—
251C22×C42−1+412yes—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—28no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8C4×D418yes—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—44no—
267C4⋊C4—416no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—816no—
276C22⋊C4—88no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C23—28no—
282C23—28no—
283C23—24no—
284C23—28no—
285C23—24no—
286C23—28no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—24no—
291C23—28no—
292C23—28no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—44no—
297C23—42no—
298C23—42no—
299C23—44no—
300C23—42no—
301C23—44no—
302C23—44no—
303C23C2×#10718yes—
304C23C22×C4⋊C411yescharacteristic
305C23#53411yescharacteristic
306C23#53411yescharacteristic
307C23D4212yes—
308C23#54712yes—
309C23C22×D811yescharacteristic
310C23C22×Q1611yescharacteristic
311C23#57511yescharacteristic
312C23#57511yescharacteristic
313C23#57612yes—
314C23#57812yes—
315D4—416no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—832no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4C2×#10714yes—
354C2×C4C2×C4×D412yes—
355C2×C4#51811yescharacteristic
356C2×C4#51811yescharacteristic
357C2×C4#52012yes—
358C2×C4#53412yes—
359C2×C4#53712yes—
360C2×C4#53812yes—
361C2×C4#53912yes—
362C2×C4#54311yescharacteristic
363C2×C4#54311yescharacteristic
364C2×C4D4211yescharacteristic
365C2×C4D4×Q812yes—
366C2×C4#55012yes—
367C2×C4#55412yes—
368C2×C4#55911yescharacteristic
369C2×C4H(3,𝔽4)11yescharacteristic
370C2×C4#56212yes—
371C2×C4#56311yescharacteristic
372C8—28no—
373C8—44no—
374C22—216no—
375C22—28no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—48no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—48no—
384C22—48no—
385C22#231511yescharacteristic
386C22D4×C4⋊C412yes—
387C22#267911yescharacteristic
388C22#268212yes—
389C22#272411yescharacteristic
390C22C22×#10711yescharacteristic
391C22#291111yescharacteristic
392C22#291312yes—
393C22#318611yescharacteristic
394C22#318611yescharacteristic
395C22#318911yescharacteristic
396C22#318911yescharacteristic
397C22#319911yescharacteristic
398C22#320211yescharacteristic
399C22#320312yes—
400C22#327312yes—
401C22#327712yes—
402C22D4×D812yes—
403C22D4×Q1612yes—
404C22#333412yes—
405C22#333812yes—
406C22#335212yes—
407C22#335512yes—
408C4—24no—
409C4—44no—
410C4—42no—
411C4—88no—
412C4—164no—
413C4#232512yes—
414C4#236611yescharacteristic
415C4#237911yescharacteristic
416C4#268311yescharacteristic
417C4#273212yes—
418C4#280411yescharacteristic
419C2—48no—
420C2#1444011yescharacteristic
421C2#2194011yescharacteristic
422C2#2194711yescharacteristic
423C2D4×#10712yes—
424C2#2266412yes—
425C2#3105611yescharacteristic
426C2#3106211yescharacteristic
427C2#3118812yes—
428C2#3119012yes—
429C2#3245811yescharacteristic
430C2#3246011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1852C2semidirect
4. #11232C2semidirect
5. #11096C2semidirect
6. #10862C2semidirect
18. #1954C4semidirect
57. C2×#107D4semidirect
60. C2×C4⋊C8C2×C4semidirect
63. C2×C22⋊C8C2×C4semidirect
66. C2×C4×C8C2×C4semidirect
187. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18522
#138722
#112322
#110962
#108622