Finite Group Databasebeta

← #120181

Subgroups — #120181

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1850C211yesmaximal, characteristic
2C2×#1810C211yesmaximal, characteristic
3#10723C211yesmaximal, characteristic
4C2×#1293C212yesmaximal
5#7911C212yesmaximal
6C24×C8C2211yescharacteristic
7C22×#108—21no—
8C22×C4⋊C8C411yescharacteristic
9C22×C4⋊C8C411yescharacteristic
10C2×#407C2212yes—
11C2×#407C2212yes—
12C22×C22⋊C8C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14#1850—21no—
15#1850—21no—
16#1850—21no—
17#1850—21no—
18#1810—24no—
19C2×#341—21no—
20C2×#341—21no—
21C2×#341—21no—
22C2×#336C411yescharacteristic
23C2×#336C411yescharacteristic
24#1293—24no—
25#1280—22no—
26#1280—22no—
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C22×M16—22no—
30C23×C8—22no—
31C23×C8—22no—
32C23×C8—42no—
33C23×C8—42no—
34C23×C8C2×C411yescharacteristic
35C23×C8C2×C411yescharacteristic
36C23×C8C2×C411yescharacteristic
37C23×C8C2×C411yescharacteristic
38C23×C8D411yescharacteristic
39C23×C8D411yescharacteristic
40C23×C8D411yescharacteristic
41C23×C8D411yescharacteristic
42C22×C22⋊C4Q811yescharacteristic
43C22×C22⋊C4Q811yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C2×#108—24no—
47C2×#108—42no—
48C2×#108—42no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—42no—
54C2×C4⋊C8—42no—
55#407—22no—
56#407—22no—
57#407—22no—
58#407—22no—
59#407—22no—
60#407—22no—
61#407D414yes—
62#407Q814yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4—21no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#341—24no—
78#341—42no—
79#341—42no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84C25C2×Q811yescharacteristic
85C23×C4—41no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4C4⋊C411yescharacteristic
90C23×C4C4⋊C411yescharacteristic
91C23×C4C4⋊C411yescharacteristic
92C23×C4C4⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q812yes—
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4112yes—
102C23×C4#4112yes—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—48no—
140C22×C8—48no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8C22⋊C411yescharacteristic
148C22×C8C22⋊C411yescharacteristic
149C22×C8C22⋊C411yescharacteristic
150C22×C8C22⋊C411yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—42no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166#108—48no—
167C4⋊C8—42no—
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—42no—
171C22⋊C8—24no—
172C22⋊C8—24no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—41no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C22⋊C8—41no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186C24—21no—
187C24—21no—
188C24—22no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24C2×C4⋊C411yescharacteristic
194C24C22⋊Q811yescharacteristic
195C24C22⋊Q811yescharacteristic
196C24#12512yes—
197C24C4⋊Q812yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
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×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#10812yescomplemented
241C22×C4#10812yescomplemented
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4C4×D412yes—
245C22×C4C4×Q812yes—
246C22×C4#12011yescharacteristic
247C22×C4#12112yes—
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12112yes—
251C22×C4C22⋊Q812yes—
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12311yescharacteristic
255C22×C4#12312yes—
256C22×C4#12411yescharacteristic
257C22×C4#12512yes—
258C22×C4#12612yes—
259C22×C4C4⋊Q811yescharacteristic
260M16—44no—
261M16—44no—
262M16—44no—
263M16—44no—
264M16—82no—
265M16—82no—
266M16—82no—
267M16—82no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C2×C8—82no—
298C2×C8—82no—
299C2×C8—82no—
300C2×C8#9912yes—
301C2×C8#9912yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C42—42no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—44no—
341C23—44no—
342C23#38011yescharacteristic
343C23#38912yes—
344C23#39511yescharacteristic
345C23#39812yes—
346C23#40011yescharacteristic
347C23C2×#10811yescharacteristic
348C23C2×#10811yescharacteristic
349C23#43012yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#38611yescharacteristic
397C2×C4#38712yes—
398C2×C4#39111yescharacteristic
399C2×C4#39411yescharacteristic
400C2×C4#39612yes—
401C2×C4#39811yescharacteristic
402C2×C4C2×#9911yescharacteristic
403C2×C4C2×#9911yescharacteristic
404C2×C4#41011yescharacteristic
405C2×C4#41011yescharacteristic
406C8—22no—
407C8—22no—
408C8—41no—
409C8—41no—
410C8—84no—
411C8—84no—
412C8—81no—
413C8—81no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—21no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22#144111yescharacteristic
444C22#179911yescharacteristic
445C22#179911yescharacteristic
446C22#181011yescharacteristic
447C22#181111yescharacteristic
448C22#181311yescharacteristic
449C22#181311yescharacteristic
450C22#195012yes—
451C22#195012yes—
452C4—21no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4#189811yescharacteristic
461C4#189911yescharacteristic
462C4#199611yescharacteristic
463C4#199611yescharacteristic
464C2—22no—
465C2—22no—
466C2—41no—
467C2—42no—
468C2—41no—
469C2#861111yescharacteristic
470C2#861211yescharacteristic
471C2#861311yescharacteristic
472C2#861311yescharacteristic
473C2#1120311yescharacteristic
474C2#1123011yescharacteristic
475C2#1123311yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
240. C22×C4#108semidirect
241. C22×C4#108semidirect

Maximal subgroups

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

subgroupindex
C2×#18502
C2×#18102
#107232
C2×#12932
#79112