Finite Group Databasebeta

← #403300

Subgroups — #403300

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1987C211yesmaximal, characteristic
2C2×#1919C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#13818C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11227C211yesmaximal, characteristic
8#11227C212yesmaximal
9#11208C212yesmaximal
10#10908C212yesmaximal
11#10850C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1987—24no—
22#1919—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—22no—
35#1441—21no—
36#1441—21no—
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×#106—22no—
65C2×#106—22no—
66C2×#106Q814yescomplemented
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—44no—
108C22×Q8—42no—
109C22×Q8—81no—
110C22×Q8—81no—
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—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×Q814yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4C2×Q812yes—
179C2×C4⋊C4C2×Q814yes—
180C2×C4⋊C4#4114yes—
181C2×C4⋊C4#4112yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—82no—
192C2×C42—82no—
193C2×C42C2×D412yes—
194C2×C42C2×Q814yes—
195C2×C42#4112yes—
196C2×C42#4114yes—
197#106—28no—
198C4⋊C8—44no—
199C4⋊C8—48no—
200#95—24no—
201#95—28no—
202#95—22no—
203#95—22no—
204#95—42no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—82no—
211#95—81no—
212#95—82no—
213C24C22×D411yescharacteristic
214C2×Q8—48no—
215C2×Q8—84no—
216C2×Q8—84no—
217C2×Q8—88no—
218C2×Q8—88no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
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—22no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4C22⋊Q814yes—
276C22×C4C22⋊Q814yes—
277C22×C4#13512yes—
278C22×C4#13712yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×Q811yescharacteristic
281C22×C4C22×Q812yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4112yes—
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C42−1+411yescharacteristic
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C4⋊C4C22⋊Q814yes—
330C4⋊C4C4⋊Q814yes—
331C42—22no—
332C42—24no—
333C42—22no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—84no—
339C42—88no—
340C23C2×#12111yescharacteristic
341C23C2×C22⋊Q812yes—
342C23#54611yescharacteristic
343C23#54612yes—
344C23D4×Q811yescharacteristic
345C23C2×#13511yescharacteristic
346C23C2×#13711yescharacteristic
347C23#57512yes—
348C23#57611yescharacteristic
349C23#57711yescharacteristic
350C23#57712yes—
351Q8—816no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4#46414yes—
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4#52511yescharacteristic
414C2×C4#52912yes—
415C2×C4C2×C4⋊Q811yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4#53312yes—
418C2×C4#53312yes—
419C2×C4#53511yescharacteristic
420C2×C4#53711yescharacteristic
421C2×C4#53812yes—
422C2×C4#54212yes—
423C2×C4#55112yes—
424C2×C4#55211yescharacteristic
425C2×C4#55412yes—
426C2×C4#55412yes—
427C2×C4#55411yescharacteristic
428C2×C4#55712yes—
429C2×C4Q8212yes—
430C8—42no—
431C8—82no—
432C8—81no—
433C22#241111yescharacteristic
434C22#241711yescharacteristic
435C22#245012yes—
436C22#245112yes—
437C22#285711yescharacteristic
438C22C2×#46411yescharacteristic
439C22#304011yescharacteristic
440C22#304211yescharacteristic
441C22#304511yescharacteristic
442C22#307612yes—
443C22#307812yes—
444C22#308112yes—
445C22#308412yes—
446C22#330112yes—
447C22#330511yescharacteristic
448C22#331112yes—
449C22#331612yes—
450C22#331912yes—
451C22#332312yes—
452C22#332911yescharacteristic
453C22#338111yescharacteristic
454C22#338511yescharacteristic
455C22#338812yes—
456C4—24no—
457C4—44no—
458C4—44no—
459C4—44no—
460C4—42no—
461C4—44no—
462C4—84no—
463C4—84no—
464C4—82no—
465C4—161no—
466C4#239611yescharacteristic
467C4#252012yes—
468C4#256711yescharacteristic
469C4#258411yescharacteristic
470C4#258612yes—
471C4#266211yescharacteristic
472C2#1487211yescharacteristic
473C2#2448311yescharacteristic
474C2#2449511yescharacteristic
475C2#2481411yescharacteristic
476C2#2483411yescharacteristic
477C2#2641112yes—
478C2#2643012yes—
479C2#2645012yes—
480C2#2645312yes—
481C2#3320211yescharacteristic
482C2#3320511yescharacteristic
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×#106Q8semidirect
75. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19872
C2×#19192
#152302
#138182
#113842
#113222
#112272
#112082
#109082
#108502