Finite Group Databasebeta

← #403306

Subgroups — #403306

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1921C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#13818C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11228C211yesmaximal, characteristic
8#11227C212yesmaximal
9#11208C212yesmaximal
10#10907C212yesmaximal
11#10850C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1988—24no—
22#1921—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1446—21no—
35#1446—21no—
36#1441—22no—
37#1436—22no—
38C4⋊#95—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2312yes—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107Q814yescomplemented
67C2×C4⋊C8—22no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389Q818yescomplemented
76#382—24no—
77#382—42no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—42no—
81C4×C4⋊C4D414yes—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D411yescharacteristic
93C2×#95D411yescharacteristic
94C2×#95D412yes—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—22no—
105#336—24no—
106C22×Q8—44no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—82no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8C2×Q812yes—
139C4⋊Q8—28no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×Q814yes—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4C2×Q814yes—
179C2×C4⋊C4#4114yes—
180C2×C4⋊C4#4112yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42C2×Q814yes—
194C2×C42#4112yes—
195C2×C42#4114yes—
196#107—28no—
197C4⋊C8—44no—
198C4⋊C8—48no—
199#95—24no—
200#95—28no—
201#95—22no—
202#95—22no—
203#95—42no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—42no—
208#95—42no—
209#95—82no—
210#95—82no—
211#95—81no—
212C24C22×D411yescharacteristic
213C2×Q8—48no—
214C2×Q8—88no—
215C2×Q8—88no—
216C2×Q8—84no—
217C2×Q8—84no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4#12112yes—
271C22×C4#12112yes—
272C22×C4C22⋊Q814yes—
273C22×C4C22⋊Q814yes—
274C22×C4#13512yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C22×Q812yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—88no—
325C4⋊C4—84no—
326C4⋊C4C22⋊Q814yes—
327C4⋊C4C4⋊Q814yes—
328C42—22no—
329C42—24no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—84no—
336C42—88no—
337C23C2×#12111yescharacteristic
338C23C2×C22⋊Q812yes—
339C23#54611yescharacteristic
340C23#54612yes—
341C23D4×Q811yescharacteristic
342C23C2×#13511yescharacteristic
343C23C2×#13711yescharacteristic
344C23#57512yes—
345C23#57612yes—
346C23#57711yescharacteristic
347C23#57811yescharacteristic
348Q8—816no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#46414yes—
407C2×C4C2×C22⋊Q811yescharacteristic
408C2×C4#52511yescharacteristic
409C2×C4#52912yes—
410C2×C4C2×C4⋊Q811yescharacteristic
411C2×C4#53211yescharacteristic
412C2×C4#53312yes—
413C2×C4#53312yes—
414C2×C4#53511yescharacteristic
415C2×C4#53711yescharacteristic
416C2×C4#53812yes—
417C2×C4#54212yes—
418C2×C4#55112yes—
419C2×C4#55211yescharacteristic
420C2×C4#55412yes—
421C2×C4#55412yes—
422C2×C4#55411yescharacteristic
423C2×C4#55712yes—
424C2×C4Q8212yes—
425C8—42no—
426C8—82no—
427C8—81no—
428C22#241111yescharacteristic
429C22#241711yescharacteristic
430C22#245012yes—
431C22#245112yes—
432C22#285711yescharacteristic
433C22C2×#46411yescharacteristic
434C22#304011yescharacteristic
435C22#304111yescharacteristic
436C22#304611yescharacteristic
437C22#307612yes—
438C22#307812yes—
439C22#308112yes—
440C22#308312yes—
441C22#329712yes—
442C22#330411yescharacteristic
443C22#331412yes—
444C22#331912yes—
445C22#332112yes—
446C22#332712yes—
447C22#332811yescharacteristic
448C22#338311yescharacteristic
449C22#338711yescharacteristic
450C22#338912yes—
451C4—24no—
452C4—44no—
453C4—44no—
454C4—42no—
455C4—44no—
456C4—44no—
457C4—84no—
458C4—84no—
459C4—82no—
460C4—161no—
461C4#239611yescharacteristic
462C4#252012yes—
463C4#256711yescharacteristic
464C4#258411yescharacteristic
465C4#258612yes—
466C4#266211yescharacteristic
467C2#1487211yescharacteristic
468C2#2447911yescharacteristic
469C2#2449911yescharacteristic
470C2#2481811yescharacteristic
471C2#2483011yescharacteristic
472C2#2640912yes—
473C2#2642812yes—
474C2#2644812yes—
475C2#2645412yes—
476C2#3320211yescharacteristic
477C2#3320311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
66. C2×#107Q8semidirect
75. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19212
#152302
#138182
#113842
#113222
#112282
#112272
#112082
#109072
#108502