Finite Group Databasebeta

← #370363

Subgroups — #370363

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic
4#11220C212yesmaximal
5#11219C212yesmaximal
6#10908C212yesmaximal
7#10907C212yesmaximal
8#10843C211yesmaximal, characteristic
9#10842C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1847C418yescomplemented
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—22no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—42no—
57C2×#107—42no—
58C2×#107C2×C418yes—
59C2×#106—42no—
60C2×#106—42no—
61C2×#106C2×C418yes—
62C2×#97—22no—
63C2×#97—22no—
64C2×#97C2×C414yes—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68#389—42no—
69#389—42no—
70#384—22no—
71#384—22no—
72#384C2×C414yes—
73#384Q814yes—
74#382—22no—
75#382—22no—
76#382—22no—
77#382C2×C414yes—
78C4×C4⋊C4—42no—
79C4×C4⋊C4—42no—
80#376—22no—
81#376—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—24no—
100#336Q818yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
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×C8—22no—
121C22×C8—22no—
122C22×C8C22×C414yes—
123C22×C8C2×Q814yes—
124C22×C8C2×Q814yes—
125C22×C8#4114yes—
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—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C2×Q812yes—
162C2×C4⋊C4C2×Q812yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—82no—
173C2×C42—82no—
174C2×C42C22×C412yes—
175C2×C42C2×Q814yes—
176C2×C42C2×Q812yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181#107—48no—
182#106—48no—
183#97—28no—
184C4×C8—44no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196C24C22×D411yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C4×D414yes—
251C22×C4C22⋊Q814yes—
252C22×C4#12314yes—
253C22×C4C4⋊Q812yes—
254C22×C4C4⋊Q812yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C22×Q812yes—
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4112yes—
262C22×C4C2×#4111yescharacteristic
263C22×C4H(5,𝔽2)12yes—
264C22×C4H(5,𝔽2)12yes—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8C4×Q818yes—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C4⋊C4—88no—
298C42—24no—
299C42—24no—
300C42—42no—
301C42—42no—
302C42—84no—
303C42—84no—
304C42#11714yes—
305C42C4×Q814yes—
306C42#12514yes—
307C42#12614yes—
308C23C2×C4×D411yescharacteristic
309C23C2×C22⋊Q811yescharacteristic
310C23C2×#12311yescharacteristic
311C23C2×C4⋊Q811yescharacteristic
312C23#53211yescharacteristic
313C23#53511yescharacteristic
314C23#53511yescharacteristic
315C23#57612yes—
316C23#57712yes—
317C23#57712yes—
318C23#57812yes—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4#38718yes—
377C2×C4C2×#11711yescharacteristic
378C2×C4C2×C4×Q812yes—
379C2×C4C2×C4×Q811yescharacteristic
380C2×C4#51812yes—
381C2×C4C2×#12511yescharacteristic
382C2×C4C2×#12611yescharacteristic
383C2×C4#53312yes—
384C2×C4#54012yes—
385C2×C4#55312yes—
386C2×C4#55312yes—
387C2×C4#55412yes—
388C2×C4#55412yes—
389C2×C4#55412yes—
390C2×C4#55412yes—
391C2×C4#55512yes—
392C2×C4Q8212yes—
393C8—28no—
394C8—44no—
395C22C2×#38711yescharacteristic
396C22C4×C4⋊Q811yescharacteristic
397C22#232811yescharacteristic
398C22#250011yescharacteristic
399C22#250211yescharacteristic
400C22#256611yescharacteristic
401C22#256711yescharacteristic
402C22#294612yes—
403C22#294712yes—
404C22#308112yes—
405C22#308212yes—
406C22#309812yes—
407C22#309912yes—
408C22#316312yes—
409C22#316412yes—
410C22#317812yes—
411C22#317912yes—
412C22#322011yescharacteristic
413C22#322111yescharacteristic
414C22#322211yescharacteristic
415C22#322311yescharacteristic
416C22#322412yes—
417C22#322512yes—
418C4—28no—
419C4—42no—
420C4—42no—
421C4—82no—
422C4—82no—
423C4—84no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—162no—
428C4#234512yes—
429C4#236912yes—
430C4#249712yes—
431C4#250712yes—
432C2#1404811yescharacteristic
433C2#2103612yes—
434C2#2173012yes—
435C2#2227811yescharacteristic
436C2#2227911yescharacteristic
437C2#2790412yes—
438C2#2795711yescharacteristic
439C2#2795811yescharacteristic
440C2#2920411yescharacteristic
441C2#2920511yescharacteristic
442C2#2921712yes—
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
24. #1847C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18472
#142762
#137792
#112202
#112192
#109082
#109072
#108432
#108422
#108412
#106942