Finite Group Databasebeta

← #503292

Subgroups — #503292

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2C2×#2255C211yesmaximal, characteristic, complemented
3C2×#2253C211yesmaximal, characteristic, complemented
4#33216C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13453C211yesmaximal, characteristic, complemented
7#13452C211yesmaximal, characteristic, complemented
8#13451C211yesmaximal, characteristic, complemented
9#13284C211yesmaximal, characteristic, complemented
10#13282C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic
12#12921C211yesmaximal, characteristic
13#12920C211yesmaximal, characteristic
14#12919C211yesmaximal, characteristic
15#12600C211yesmaximal, characteristic
16C2×#544—21no—
17#3164—24no—
18#3164—24no—
19C2×#501—21no—
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#499C2211yescharacteristic
24C2×#499C2211yescharacteristic
25C2×#499C2211yescharacteristic
26C2×#428—21no—
27C2×#427C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C2×#405C2211yescharacteristic
30#2629—21no—
31#2255—24no—
32#2253—24no—
33C2×#367C2211yescharacteristic
34C2×#367C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#366C2211yescharacteristic
40C2×#366C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C22⋊C16C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
45#1992—21no—
46#1943—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1845C2211yescharacteristic
50#1809—21no—
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×M16D411yescharacteristic
64C23×C8C2311yescharacteristic
65#544—28no—
66#544—44no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—41no—
70C2×#125—21no—
71C2×#125—21no—
72C2×#125—21no—
73C2×#125—41no—
74C2×C22⋊Q8—41no—
75C2×C22⋊Q8—41no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#501—28no—
84#501—28no—
85#500—22no—
86#500—22no—
87#500—22no—
88#500D414yes—
89#499—22no—
90#499—22no—
91#499—22no—
92#499—22no—
93#499—22no—
94#499D414yes—
95#428—28no—
96#427—22no—
97#427—22no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—21no—
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#106—21no—
113C2×#106—21no—
114C2×#106—21no—
115C2×#106—21no—
116#405—22no—
117#405—22no—
118C2×#97D411yescharacteristic
119C2×#97D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#398—41no—
123#398—41no—
124#396—41no—
125#396—41no—
126#389—21no—
127#389—21no—
128#384—21no—
129#380—21no—
130#380—21no—
131#367—22no—
132#367—22no—
133#367—22no—
134#367—22no—
135#366—22no—
136#366—22no—
137#366—22no—
138#366—22no—
139C22⋊C16Q814yescomplemented
140C22⋊C16Q814yescomplemented
141#346—22no—
142C4×C16—22no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147C22×Q8—41no—
148C22×Q8—41no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C2×M16Q1614yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×Q812yes—
162C22×C8C2×Q812yes—
163C22×C8C2411yescharacteristic
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—42no—
167C4⋊Q8—48no—
168C4⋊Q8—42no—
169#125—24no—
170#125—24no—
171#125—42no—
172#125—48no—
173#125—42no—
174#125—42no—
175#125—42no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178#117—22no—
179#117—22no—
180#117—24no—
181#117—42no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4D811yescharacteristic
206C2×C4⋊C4D811yescharacteristic
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4D811yescharacteristic
211C2×C22⋊C4D811yescharacteristic
212C2×C42—41no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16C2×Q812yes—
224C2×C16C2×Q812yes—
225C2×C16C2×Q812yes—
226C2×C16C2×Q812yes—
227#107—24no—
228#107—24no—
229#107—24no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—42no—
243#107C2×D412yes—
244#107C2×D412yes—
245#107C2×D412yes—
246#107C2×D412yes—
247#106—24no—
248#106—24no—
249#106—24no—
250#106—42no—
251#97—22no—
252#97—22no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8C2×D412yes—
256C4×C8C2×D412yes—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24C2×D811yescharacteristic
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—161no—
265C2×Q8—161no—
266C2×Q8—161no—
267C2×Q8—161no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—44no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—161no—
282C22×C4—161no—
283C22×C4C4⋊Q812yes—
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×Q1612yes—
290C22×C4C22×D411yescharacteristic
291M16—28no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8#12012yes—
301C2×C8C4⋊Q812yes—
302C2×C8C2×Q1612yes—
303C2×C8C2×Q1612yes—
304C2×C8#13712yes—
305C2×C8C22×D411yescharacteristic
306C2×C8C22×D411yescharacteristic
307C2×C8C22×Q811yescharacteristic
308C2×C8C22×Q811yescharacteristic
309C2×C82−1+411yescharacteristic
310C2×C82−1+411yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—161no—
339C4⋊C4—161no—
340C4⋊C4—161no—
341C4⋊C4—161no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—44no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C42—21no—
348C42—21no—
349C42—44no—
350C42#12012yes—
351C42AGL(1,ℤ/8)12yes—
352C16—22no—
353C16—22no—
354C16—22no—
355C16—22no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23#50012yes—
363C23C22×D811yescharacteristic
364Q8—164no—
365Q8—164no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—164no—
407C2×C4#44712yes—
408C2×C4#44712yes—
409C2×C4#50012yes—
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×C4⋊Q811yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4C22×Q1611yescharacteristic
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C8—22no—
417C8—22no—
418C8#45114yes—
419C8D4×Q811yescharacteristic
420C8D4×Q811yescharacteristic
421C8D4×Q811yescharacteristic
422C8D4×Q811yescharacteristic
423C22—22no—
424C22—22no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22C2×#44711yescharacteristic
431C22C2×#50011yescharacteristic
432C22#322611yescharacteristic
433C22#341511yescharacteristic
434C22#341611yescharacteristic
435C22#341611yescharacteristic
436C22#341711yescharacteristic
437C4—22no—
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4#243411yescharacteristic
451C4C2×#45111yescharacteristic
452C4#316511yescharacteristic
453C4#323011yescharacteristic
454C2—42no—
455C2#2581311yescharacteristic
456C2#3352611yescharacteristic
457C2#3352711yescharacteristic
458C2#3372911yescharacteristic
459C2#3373011yescharacteristic
460C2#3380711yescharacteristic
461C2#3380811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2255C2semidirect
3. C2×#2253C2semidirect
5. #13453C2semidirect
6. #13453C2semidirect
7. #13452C2semidirect
8. #13451C2semidirect
9. #13284C2semidirect
10. #13282C2semidirect
139. C22⋊C16Q8semidirect
140. C22⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#22552
C2×#22532
#332162
#134532
#134522
#134512
#132842
#132822
#129212
#129202
#129192
#126002