Finite Group Databasebeta

← #97578

Subgroups — #97578

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3#8311C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7575C212yesmaximal, complemented
6C24×Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9Q8×C22⋊C4—24no—
10Q8×C22⋊C4—24no—
11C2×#391—21no—
12C2×#391—21no—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#380—21no—
16C2×#380C411yescharacteristic
17C2×C4×C22⋊C4—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#323—21no—
23C2×#323—21no—
24#1280—21no—
25#1280—21no—
26#1272—24no—
27C23×Q8—22no—
28C23×Q8—22no—
29C23×Q8—22no—
30C23×Q8—22no—
31C23×Q8—21no—
32C23×Q8—22no—
33C23×Q8—44no—
34C23×Q8—44no—
35C23×Q8C814yescomplemented
36C23×Q8C2×C411yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C24×C4—21no—
39C24×C4C2×C411yescharacteristic
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—44no—
44C2×C4×Q8—44no—
45C2×C4×Q8D414yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C4×C8—22no—
61#391—28no—
62#391—28no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391D414yes—
67#380—28no—
68#380—22no—
69#380—22no—
70C4×C22⋊C4—28no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95D411yescharacteristic
79C2×#95D411yescharacteristic
80#336—22no—
81#336—22no—
82Q8⋊C8—48no—
83#323—24no—
84#323—42no—
85C25C22⋊C411yescharacteristic
86C22×Q8—24no—
87C22×Q8—24no—
88C22×Q8—24no—
89C22×Q8—28no—
90C22×Q8—28no—
91C22×Q8—24no—
92C22×Q8—28no—
93C22×Q8—28no—
94C22×Q8—24no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8—416no—
106C22×Q8—48no—
107C22×Q8—48no—
108C22×Q8—416no—
109C22×Q8—416no—
110C22×Q8—416no—
111C22×Q8—44no—
112C22×Q8—44no—
113C22×Q8—44no—
114C22×Q8—44no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22⋊C412yes—
117C22×Q8C2×C812yes—
118C22×Q8C2×C812yes—
119C22×Q8M1612yes—
120C22×Q8M1612yes—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C2×C812yes—
140C23×C4M1612yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—42no—
145C22×C8—42no—
146C4×Q8—28no—
147C4×Q8—44no—
148C4×Q8—48no—
149C4×Q8—48no—
150C4×Q8—44no—
151C4×Q8—44no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4SD1612yes—
171C2×C22⋊C4Q1612yes—
172C2×C42—24no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—24no—
176C2×C42—42no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—42no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C4⋊C8—44no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C4×C8—44no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24C22⋊C812yes—
204C24#10312yes—
205C24#10412yes—
206C24C2×C22⋊C411yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—28no—
210C2×Q8—28no—
211C2×Q8—28no—
212C2×Q8—28no—
213C2×Q8—216no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—416no—
225C2×Q8—416no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—432no—
233C2×Q8—432no—
234C2×Q8—416no—
235C2×Q8—416no—
236C2×Q8—416no—
237C2×Q8—416no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8C22⋊C814yes—
241C2×Q8C22⋊C814yes—
242C2×Q8#9914yes—
243C2×Q8#10014yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—216no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—416no—
275C22×C4—416no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C22⋊C812yes—
303C22×C4#10312yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4#12012yes—
307C22×C4#12012yes—
308C22×C4C22×C811yescharacteristic
309C22×C4C2×M1611yescharacteristic
310C22×C4C2×M1611yescharacteristic
311C22×C4C2×M1611yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C2×Q1611yescharacteristic
314C22×C4#13712yes—
315C2×C8—44no—
316C2×C8—82no—
317C2×C8—82no—
318C2×C8—82no—
319C2×C8—82no—
320C2×C8—82no—
321C4⋊C4—24no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—48no—
331C42—24no—
332C42—42no—
333C42—48no—
334C42—48no—
335C42—42no—
336C42—42no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23Q8⋊C814yescomplemented
364C23#37911yescharacteristic
365C23C2×C22⋊C811yescharacteristic
366C23#40711yescharacteristic
367C23C2×#10311yescharacteristic
368C23#41911yescharacteristic
369C23C2×#10411yescharacteristic
370C23#42111yescharacteristic
371Q8—28no—
372Q8—28no—
373Q8—24no—
374Q8—24no—
375Q8—24no—
376Q8—44no—
377Q8—432no—
378Q8—44no—
379Q8—44no—
380Q8—44no—
381Q8#32318yescomplemented
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—432no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4C2×C22⋊C811yescharacteristic
439C2×C4C2×C22⋊C811yescharacteristic
440C2×C4#40711yescharacteristic
441C2×C4#40711yescharacteristic
442C2×C4C2×#9912yes—
443C2×C4C2×#10012yes—
444C2×C4#44812yes—
445C2×C4#45112yes—
446C8—84no—
447C8—162no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—24no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×Q8⋊C811yescharacteristic
472C22#147911yescharacteristic
473C22#148711yescharacteristic
474C22#149111yescharacteristic
475C22#177911yescharacteristic
476C22#178111yescharacteristic
477C22#178311yescharacteristic
478C22#178711yescharacteristic
479C22#178911yescharacteristic
480C22#179611yescharacteristic
481C22#179611yescharacteristic
482C4—28no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—82no—
492C4—81no—
493C4—82no—
494C4C2×#32312yes—
495C4#146212yes—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2—44no—
501C2#829811yescharacteristic
502C2#831111yescharacteristic
503C2#832311yescharacteristic
504C2#836611yescharacteristic
505C2#836911yescharacteristic
506C2#837911yescharacteristic
507C2#839111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8311C2semidirect
5. #7575C2semidirect
35. C23×Q8C8semidirect
363. C23Q8⋊C8semidirect
381. Q8#323semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
#85572
#83112
C4×#3232
#75752