Finite Group Databasebeta

← #370458

Subgroups — #370458

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic
4#11293C212yesmaximal
5#11292C212yesmaximal
6#10843C211yesmaximal, characteristic
7#10842C211yesmaximal, characteristic
8#10841C211yesmaximal, characteristic
9#10807C212yesmaximal
10#10806C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×#97C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×#376C2211yescharacteristic
22#1836C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1441—22no—
34#1441—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#97—22no—
57C2×#97—22no—
58C2×#97C2×C414yes—
59C2×C4×C8—22no—
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62#389—42no—
63#389—42no—
64#384—22no—
65#384—22no—
66#384C2×C414yes—
67#384Q814yes—
68#382—22no—
69#382—22no—
70#382—22no—
71#382C2×C414yes—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74#376—22no—
75#376—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95D412yes—
87#340—24no—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340C2×C418yes—
94#340C2×C418yes—
95#336—24no—
96#336—24no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C22×C414yes—
121C22×C8#4114yes—
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—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2×Q814yes—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—44no—
168C2×C42—82no—
169C2×C42—82no—
170C2×C42C22×C412yes—
171C2×C42C2×Q812yes—
172C2×C42#4114yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C2×C42#4112yes—
176C2×C42#4112yes—
177#97—28no—
178C4×C8—44no—
179#95—28no—
180#95—24no—
181#95—24no—
182#95—42no—
183#95—42no—
184#95—44no—
185#95—44no—
186#95—82no—
187#95—82no—
188#95—82no—
189#95—82no—
190C24C22×D411yescharacteristic
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C4×D414yes—
246C22×C4C22⋊Q814yes—
247C22×C4#12314yes—
248C22×C4#12414yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4112yes—
256C22×C4C2×#4111yescharacteristic
257C22×C4H(5,𝔽2)12yes—
258C22×C42−1+412yes—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—88no—
290C4⋊C4—88no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C42—24no—
294C42—24no—
295C42—42no—
296C42—42no—
297C42—88no—
298C42#11714yes—
299C42C4×Q814yes—
300C42#12514yes—
301C42#12614yes—
302C23C2×C4×D411yescharacteristic
303C23C2×C22⋊Q811yescharacteristic
304C23C2×#12311yescharacteristic
305C23C2×#12411yescharacteristic
306C23#53211yescharacteristic
307C23#53511yescharacteristic
308C23#53611yescharacteristic
309C23#57612yes—
310C23#57712yes—
311C23#57712yes—
312C23#57812yes—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4#38718yes—
371C2×C4C2×#11711yescharacteristic
372C2×C4C2×C4×Q811yescharacteristic
373C2×C4C4×#4112yes—
374C2×C4#52012yes—
375C2×C4C2×#12511yescharacteristic
376C2×C4C2×#12611yescharacteristic
377C2×C4#52912yes—
378C2×C4#54212yes—
379C2×C4#55212yes—
380C2×C4#55212yes—
381C2×C4#55212yes—
382C2×C4#55312yes—
383C2×C4#55412yes—
384C2×C4#55512yes—
385C2×C4#55512yes—
386C2×C4#55712yes—
387C8—44no—
388C8—44no—
389C22C2×#38711yescharacteristic
390C22C4×#12411yescharacteristic
391C22#232611yescharacteristic
392C22#250011yescharacteristic
393C22#250311yescharacteristic
394C22#256311yescharacteristic
395C22#256511yescharacteristic
396C22#294612yes—
397C22#294712yes—
398C22#308112yes—
399C22#308212yes—
400C22#309812yes—
401C22#309912yes—
402C22#313612yes—
403C22#313712yes—
404C22#317312yes—
405C22#317512yes—
406C22#322011yescharacteristic
407C22#322111yescharacteristic
408C22#322211yescharacteristic
409C22#322311yescharacteristic
410C22#324112yes—
411C22#324212yes—
412C4—28no—
413C4—44no—
414C4—84no—
415C4—82no—
416C4—82no—
417C4—84no—
418C4—81no—
419C4—81no—
420C4—161no—
421C4—161no—
422C4#234612yes—
423C4#237012yes—
424C4#249512yes—
425C4#250812yes—
426C2#1405111yescharacteristic
427C2#2103612yes—
428C2#2165412yes—
429C2#2223111yescharacteristic
430C2#2223211yescharacteristic
431C2#2789811yescharacteristic
432C2#2789911yescharacteristic
433C2#2797012yes—
434C2#2915211yescharacteristic
435C2#2915311yescharacteristic
436C2#2918712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1836C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18362
#142762
#137792
#112932
#112922
#108432
#108422
#108412
#108072
#108062
#106942