Finite Group Databasebeta

← #387114

Subgroups — #387114

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic
3#13925C211yesmaximal, characteristic
4#11384C212yesmaximal
5#11228C212yesmaximal
6#11208C212yesmaximal
7#10884C212yesmaximal
8#10880C212yesmaximal
9#10806C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2567—24no—
15#2567C418yescomplemented
16C2×#395—21no—
17C2×#395—21no—
18C2×#391—21no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384—21no—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1984—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1446—21no—
37#1445—21no—
38#1436—21no—
39#1436—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C23×Q8D411yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8C2×C414yes—
52C2×C4×Q8—41no—
53C2×C4×Q8—41no—
54C2×C4×Q8—41no—
55C2×C4×Q8—41no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—41no—
75C2×C4⋊C8—41no—
76C2×C4×C8—22no—
77C2×C4×C8—41no—
78C2×C4×C8—41no—
79#395—28no—
80#395—28no—
81#391—28no—
82#389—24no—
83#389—24no—
84#389C2×C418yes—
85#389C2×C418yes—
86#384—28no—
87#382—41no—
88#382—41no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—41no—
91C4×C4⋊C4—41no—
92C4×C4⋊C4C2×C414yes—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340Q818yescomplemented
111#336—24no—
112#336—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—82no—
121C22×Q8—82no—
122C22×Q8C22⋊C414yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C2×Q814yes—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22⋊C414yes—
191C2×C4⋊C4C22⋊C414yes—
192C2×C4⋊C4C22⋊C414yes—
193C2×C4⋊C4C22×C414yes—
194C2×C4⋊C4C22×C414yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4#4114yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—24no—
211C2×C42—44no—
212C2×C42—42no—
213C2×C42—82no—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C4⋊C8—44no—
221C4×C8—44no—
222#95—24no—
223#95—28no—
224#95—28no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—44no—
229#95—44no—
230#95—82no—
231#95—82no—
232C24C22×D411yescharacteristic
233C2×Q8—28no—
234C2×Q8—28no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8—84no—
242C2×Q8—84no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4#13514yes—
300C22×C4#13714yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4112yes—
305C22×C4H(5,𝔽2)11yescharacteristic
306C22×C4H(5,𝔽2)11yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C42−1+412yes—
310C22×C42−1+412yes—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C4⋊C4C4×Q818yes—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—48no—
360C42—84no—
361C23C22×C22⋊C411yescharacteristic
362C23#53411yescharacteristic
363C23#53411yescharacteristic
364C23#54812yes—
365C23D4×Q812yes—
366C23C2×#13512yes—
367C23C2×#13712yes—
368C23#57612yes—
369C23#57712yes—
370Q8—416no—
371Q8—816no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—28no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#41614yes—
429C2×C4#41914yes—
430C2×C4C2×C4×Q812yes—
431C2×C4#51811yescharacteristic
432C2×C4#51811yescharacteristic
433C2×C4#51912yes—
434C2×C4#53411yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4#54212yes—
437C2×C4#54212yes—
438C2×C4#54412yes—
439C2×C4#54412yes—
440C2×C4#54812yes—
441C2×C4D4×Q812yes—
442C2×C4#55412yes—
443C2×C4#55511yescharacteristic
444C2×C4Q8211yescharacteristic
445C2×C4#56312yes—
446C2×C4#56311yescharacteristic
447C2×C4#56311yescharacteristic
448C8—44no—
449C8—82no—
450C22#230811yescharacteristic
451C22Q8×C22⋊C412yes—
452C22#272912yes—
453C22#273111yescharacteristic
454C22#273411yescharacteristic
455C22C2×#41611yescharacteristic
456C22C2×#41911yescharacteristic
457C22#289712yes—
458C22#319012yes—
459C22#319012yes—
460C22#319812yes—
461C22#319911yescharacteristic
462C22#320011yescharacteristic
463C22#336012yes—
464C22#336112yes—
465C22#336912yes—
466C22#337512yes—
467C22#338012yes—
468C22#338112yes—
469C22#338512yes—
470C22#338712yes—
471C4—24no—
472C4—24no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—48no—
478C4—84no—
479C4—84no—
480C4—162no—
481C4#233112yes—
482C4#236911yescharacteristic
483C4#238111yescharacteristic
484C4#272912yes—
485C4#273111yescharacteristic
486C4#273411yescharacteristic
487C2#1445011yescharacteristic
488C2#2176211yescharacteristic
489C2#2177011yescharacteristic
490C2#2251912yes—
491C2#2252512yes—
492C2#3257312yes—
493C2#3257512yes—
494C2#3263612yes—
495C2#3268712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2567C4semidirect
110. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#19842
#139252
#113842
#112282
#112082
#108842
#108802
#108062