Finite Group Databasebeta

← #220014

Subgroups — #220014

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3#22795C211yesmaximal, characteristic
4C2×#1544C212yesmaximal, complemented
5#9694C211yesmaximal, characteristic, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9132C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8345C212yesmaximal, complemented
11C4×#332C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#431C2212yes—
14C2×#431C2212yes—
15#2868C2211yescharacteristic
16#2857C2212yes—
17#2552C412yes—
18#2542—21no—
19#2538—21no—
20#2441C414yes—
21#2429—22no—
22#2423—22no—
23#2365—21no—
24#2354C412yes—
25Q8×C4⋊C4—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1923—22no—
31#1923—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1544—28no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51#533—24no—
52#533—24no—
53#533—42no—
54#533—42no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2312yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125C2×C414yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C414yes—
70C22×C42C2312yes—
71#431—24no—
72#431—24no—
73#431—42no—
74#431—42no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#399—24no—
94#398—24no—
95#396—24no—
96#391—21no—
97#391—22no—
98#390—22no—
99#389—22no—
100#389—21no—
101#389—22no—
102#389—24no—
103#389C2×C412yes—
104#389C2×C414yes—
105#388—24no—
106#387—24no—
107#384—22no—
108#384—21no—
109#384—21no—
110#384C2×C412yes—
111#384C2×C412yes—
112#384C2311yescharacteristic
113#380—22no—
114#380C2×C414yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123C4×C22⋊C4D412yes—
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—21no—
128#336—21no—
129#332—24no—
130#332—22no—
131#332—24no—
132#332—24no—
133#332—42no—
134#332—42no—
135#332D414yescomplemented
136C22×Q8—21no—
137C22×Q8—22no—
138C22×D4C22⋊C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—81no—
157C4⋊Q8—81no—
158C4⋊Q8C2×D412yes—
159C4⋊Q8#4112yes—
160#125—28no—
161#125—44no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C22⋊Q8—44no—
166C22⋊Q8—82no—
167C22⋊Q8—82no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179#117—28no—
180#117—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8C2×D412yes—
235C4⋊C8C2×D412yes—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C22⋊C8—82no—
239C22⋊C8—82no—
240C4×C8—44no—
241C4×C8—44no—
242C4×C8—42no—
243C4×C8—42no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—24no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24C2×C22⋊C412yes—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—42no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—84no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4#10114yes—
337C4⋊C4C4×D414yes—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—48no—
342C22⋊C4—84no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—22no—
354C42—41no—
355C42—48no—
356C42—44no—
357C42#13512yes—
358C42#13512yes—
359C42#13512yes—
360C42#13512yes—
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C42H(5,𝔽2)11yescharacteristic
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23C22×C22⋊C411yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—82no—
378Q8—82no—
379Q8—82no—
380Q8—82no—
381D4—44no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4C2×#10114yes—
419C2×C4C2×#10112yes—
420C2×C4#41612yes—
421C2×C4#41612yes—
422C2×C4#41612yes—
423C2×C4#41612yes—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4#51811yescharacteristic
426C2×C4D4211yescharacteristic
427C2×C4D4211yescharacteristic
428C2×C4#54611yescharacteristic
429C2×C4#54611yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57811yescharacteristic
438C8—42no—
439C8—42no—
440C8—82no—
441C8—82no—
442C8—81no—
443C8—81no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22D4×C22⋊C411yescharacteristic
452C22C22×#10111yescharacteristic
453C22#288911yescharacteristic
454C22C2×#41611yescharacteristic
455C22C2×#41611yescharacteristic
456C22#289711yescharacteristic
457C22#289811yescharacteristic
458C4—22no—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—24no—
463C4—41no—
464C4—42no—
465C4—44no—
466C4—42no—
467C4—82no—
468C4—84no—
469C4#154414yescomplemented
470C4#328311yescharacteristic
471C4#328411yescharacteristic
472C4#328411yescharacteristic
473C4#328711yescharacteristic
474C4#330211yescharacteristic
475C4#330311yescharacteristic
476C4#330411yescharacteristic
477C4#330511yescharacteristic
478C2—82no—
479C2C2×#154411yescharacteristic
480C2#1772511yescharacteristic
481C2#1772711yescharacteristic
482C2#1777711yescharacteristic
483C2D4×#10111yescharacteristic
484C2#2241711yescharacteristic
485C2#2241811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1544C2semidirect
5. #9694C2semidirect
6. #9692C2semidirect
7. #9132C2semidirect
8. #9130C2semidirect
9. #8586C2semidirect
10. #8345C2semidirect
11. C4×#332C2semidirect
135. #332D4semidirect
469. C4#1544semidirect

Maximal subgroups

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

subgroupindex
#361342
#227952
C2×#15442
#96942
#96922
#91322
#91302
#85862
#83452
C4×#3322