Finite Group Databasebeta

← #390928

Subgroups — #390928

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic
2#14041C212yesmaximal
3#11316C212yesmaximal
4#11316C212yesmaximal
5#11232C211yesmaximal, characteristic
6#10907C212yesmaximal
7#10907C212yesmaximal
8#10880C212yesmaximal
9#10834C211yesmaximal, characteristic
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×C4×C4⋊C4C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#2087C418yescomplemented
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1445—22no—
29#1442—22no—
30#1436—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C23×C8C2312yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42—22no—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49C2×#107—24no—
50C2×#107—41no—
51C2×#107—41no—
52C2×#107—41no—
53C2×#107C2×C414yes—
54C2×C4⋊C8—41no—
55C2×C4⋊C8—41no—
56C2×C4⋊C8C2×C414yes—
57C2×C4×C8—22no—
58C2×C4×C8—41no—
59C2×C4×C8—41no—
60#389—24no—
61#389C2×C418yes—
62C4×C4⋊C4—41no—
63C4×C4⋊C4—41no—
64C4×C4⋊C4—42no—
65C4×C4⋊C4—42no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95D412yes—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—42no—
81#340—42no—
82#340—41no—
83#340C2×C414yes—
84#340C2×C418yes—
85#336—24no—
86#336—24no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C22×C412yes—
111C22×C8C22×C412yes—
112C22×C8#4114yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—82no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4Q1614yes—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4#4114yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—44no—
157C2×C42—82no—
158C2×C42—82no—
159C2×C42C22×C412yes—
160C2×C42#4112yes—
161#107—44no—
162#107—44no—
163C4⋊C8—44no—
164C4×C8—44no—
165#95—28no—
166#95—42no—
167#95—44no—
168#95—44no—
169#95—44no—
170#95—44no—
171#95—82no—
172#95—82no—
173#95—82no—
174#95—82no—
175#95—82no—
176C24C22×D411yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C4×D412yes—
227C22×C4C4×D412yes—
228C22×C4C2×Q1614yes—
229C22×C4C2×Q1612yes—
230C22×C4#13512yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C2×#4112yes—
233C22×C4C2×#4112yes—
234C22×C4C2×#4111yescharacteristic
235C22×C4H(5,𝔽2)11yescharacteristic
236C22×C4H(5,𝔽2)11yescharacteristic
237C22×C4H(5,𝔽2)11yescharacteristic
238C22×C42−1+412yes—
239C22×C42−1+412yes—
240C22×C42−1+411yescharacteristic
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—88no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—88no—
286C23C2×C4×D411yescharacteristic
287C23#53411yescharacteristic
288C23#53711yescharacteristic
289C23#54712yes—
290C23#54812yes—
291C23C22×Q1611yescharacteristic
292C23C2×#13511yescharacteristic
293C23#57511yescharacteristic
294C23#57511yescharacteristic
295C23#57612yes—
296C23#57712yes—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4C4×Q1614yes—
351C2×C4C4×#4112yes—
352C2×C4#51811yescharacteristic
353C2×C4#51912yes—
354C2×C4#52011yescharacteristic
355C2×C4#53811yescharacteristic
356C2×C4#53912yes—
357C2×C4#53912yes—
358C2×C4#53911yescharacteristic
359C2×C4#54112yes—
360C2×C4#54112yes—
361C2×C4#55111yescharacteristic
362C2×C4#55212yes—
363C2×C4#55312yes—
364C2×C4#55612yes—
365C2×C4#55911yescharacteristic
366C2×C4#56312yes—
367C2×C4#56311yescharacteristic
368C2×C4#56311yescharacteristic
369C8—42no—
370C8—42no—
371C8—44no—
372C22#232511yescharacteristic
373C22#236212yes—
374C22#272411yescharacteristic
375C22#272512yes—
376C22#272811yescharacteristic
377C22C2×C4×Q1611yescharacteristic
378C22#294311yescharacteristic
379C22#294612yes—
380C22#318911yescharacteristic
381C22#319111yescharacteristic
382C22#319911yescharacteristic
383C22#320411yescharacteristic
384C22#324911yescharacteristic
385C22#325011yescharacteristic
386C22#325712yes—
387C22#333812yes—
388C22#334012yes—
389C22#335112yes—
390C22#335512yes—
391C22#336112yes—
392C22#336312yes—
393C22#337012yes—
394C22#337112yes—
395C4—24no—
396C4—42no—
397C4—44no—
398C4—84no—
399C4—84no—
400C4—84no—
401C4—82no—
402C4—162no—
403C4#233612yes—
404C4#237411yescharacteristic
405C4#238111yescharacteristic
406C4#278712yes—
407C4#278911yescharacteristic
408C4#279411yescharacteristic
409C2#1459011yescharacteristic
410C2#2218211yescharacteristic
411C2#2220211yescharacteristic
412C2#2306312yes—
413C2#2307112yes—
414C2#3246011yescharacteristic
415C2#3246111yescharacteristic
416C2#3247312yes—
417C2#3247812yes—
418C2#3254511yescharacteristic
419C2#3254811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2087C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
#140412
#113162
#112322
#109072
#108802
#108342