Finite Group Databasebeta

← #188855

Subgroups — #188855

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#21969C212yesmaximal, complemented
3#17366C211yesmaximal, characteristic
4#17361C211yesmaximal, characteristic
5#8613C214yesmaximal, complemented
6#8612C212yesmaximal, complemented
7#8611C212yesmaximal, complemented
8#8010C212yesmaximal, complemented
9C2×C4×#41C2211yescharacteristic
10#2990—24no—
11#2990—22no—
12#2990—22no—
13C2×#446C412yes—
14C2×#435C412yes—
15C2×#435C2212yes—
16C2×C8×D4C2212yes—
17#2920C2212yes—
18#2920C2212yes—
19C2×#432C411yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C411yescharacteristic
22C2×#423C411yescharacteristic
23C22×C4⋊C8C411yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25#1950—22no—
26#1950—24no—
27#1950—22no—
28#1810C2214yes—
29#1810C2214yes—
30#1467—21no—
31#1466—21no—
32#1465C2211yescharacteristic
33#1465C2212yes—
34#1465C2211yescharacteristic
35#1465C2212yes—
36#1465C2212yes—
37#1464—21no—
38#1464—21no—
39#1461—21no—
40#1460—21no—
41#1458—21no—
42#1458—21no—
43#1293C2214yes—
44#1293C2214yes—
45C22×#41Q811yescharacteristic
46C22×M16C2×C412yes—
47C23×C8C2×C412yes—
48C23×C8C2312yes—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4Q811yescharacteristic
58C2×C4×D4Q811yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117Q811yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67#446—24no—
68#446—24no—
69#435—24no—
70#435—24no—
71#435—24no—
72#435—24no—
73C8×D4—24no—
74C8×D4—24no—
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432—22no—
82#432—22no—
83#431—22no—
84#431—21no—
85#431—21no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90C2×#108—24no—
91C2×#108—24no—
92#423—22no—
93#423—22no—
94#423—22no—
95#423—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2×C412yes—
101C2×C4⋊C8C2×C412yes—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C412yes—
106C2×C4⋊C8D414yes—
107C2×C4⋊C8D412yes—
108C2×C4⋊C8D412yes—
109#407C2312yes—
110#407C2314yes—
111#407C2312yes—
112C2×C22⋊C8C2×C412yes—
113C2×C22⋊C8C2×C412yes—
114C2×C22⋊C8C2312yes—
115C4×M16—22no—
116C4×M16—22no—
117C2×#97C2×C412yes—
118C2×#97C2×C412yes—
119C2×C4×C8C2×C412yes—
120#341—24no—
121#341—24no—
122#324—22no—
123#324—22no—
124#324—21no—
125#324—21no—
126#324—22no—
127#323—22no—
128#323—22no—
129#323—21no—
130#323—21no—
131#323—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—42no—
135C2×#41—42no—
136C22×Q8C4⋊C411yescharacteristic
137C22×D4C4⋊C411yescharacteristic
138C22×D4C2×Q811yescharacteristic
139C22×D4C2×Q811yescharacteristic
140C23×C4C4⋊C411yescharacteristic
141C23×C4C4⋊C411yescharacteristic
142C23×C4C4⋊C411yescharacteristic
143C23×C4C4⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×Q811yescharacteristic
147C23×C4C2×Q811yescharacteristic
148C23×C4C2411yescharacteristic
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—24no—
152C2×M16—22no—
153C2×M16—82no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—82no—
157C22×C8—82no—
158C22×C8C22×C412yes—
159C22×C8C22×C412yes—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C22×C8C2×D414yes—
163C22×C8C2×D414yes—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4C4⋊C411yescharacteristic
187C2×C4⋊C4C4⋊C411yescharacteristic
188C2×C4⋊C4C4⋊C411yescharacteristic
189C2×C4⋊C4C4⋊C411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C22⋊C4C4⋊C411yescharacteristic
194C2×C22⋊C4C2×D411yescharacteristic
195C2×C22⋊C4C2×D411yescharacteristic
196C2×C22⋊C4C2×Q811yescharacteristic
197C2×C22⋊C4C2×Q811yescharacteristic
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×C42C4⋊C411yescharacteristic
207C2×C42C4⋊C411yescharacteristic
208C2×C42C4⋊C411yescharacteristic
209C2×C42C4⋊C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220#108—84no—
221#108—84no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—24no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8C22⋊C412yes—
234C4⋊C8C22⋊C411yescharacteristic
235C4⋊C8C22⋊C411yescharacteristic
236C4⋊C8C22⋊C412yes—
237C4⋊C8C22⋊C412yes—
238C4⋊C8C22⋊C414yes—
239C4⋊C8C22⋊C414yes—
240C22⋊C8—24no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—24no—
249C22⋊C8—22no—
250#97—24no—
251#97—24no—
252#97—42no—
253#97—42no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—42no—
257C4×C8—42no—
258C24C2×C4⋊C411yescharacteristic
259C24C2×C4⋊C411yescharacteristic
260C24C22×Q811yescharacteristic
261#41—48no—
262#41—48no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4C2×C4⋊C411yescharacteristic
304C22×C4C2×C4⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D412yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4112yes—
320M16—42no—
321M16—44no—
322M16—42no—
323M16—42no—
324M16—44no—
325M16—42no—
326M16—84no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—82no—
339C2×C8—82no—
340C2×C8C2×C22⋊C412yes—
341C2×C8C2×C22⋊C412yes—
342C2×C8C2×C22⋊C414yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42C2×C22⋊C412yes—
368C42C2×C22⋊C411yescharacteristic
369C42C2×C22⋊C411yescharacteristic
370C42#11711yescharacteristic
371C42#11711yescharacteristic
372C42#11712yes—
373C42#12112yes—
374C42#12112yes—
375C42#12112yes—
376C42#12112yes—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23#43012yes—
391C23C22×C4⋊C411yescharacteristic
392C23C2×C22⋊Q812yes—
393Q8—44no—
394D4—28no—
395D4—44no—
396D4—44no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4#38011yescharacteristic
441C2×C4#38011yescharacteristic
442C2×C4#38011yescharacteristic
443C2×C4#38012yes—
444C2×C4#38012yes—
445C2×C4#38011yescharacteristic
446C2×C4#43012yes—
447C2×C4C22×C22⋊C411yescharacteristic
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×#12112yes—
450C8—22no—
451C8—22no—
452C8—24no—
453C8—42no—
454C8—42no—
455C8—84no—
456C8—82no—
457C8—82no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22C2×#38011yescharacteristic
476C22C2×#43011yescharacteristic
477C22C2×#43011yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—21no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4#181312yes—
492C4#288311yescharacteristic
493C4#288411yescharacteristic
494C2—21no—
495C2—21no—
496C2—42no—
497C2—42no—
498C2—81no—
499C2#2063411yescharacteristic
500C2#2066711yescharacteristic
501C2C2×#181311yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21969C2semidirect
5. #8613C2semidirect
6. #8612C2semidirect
7. #8611C2semidirect
8. #8010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#219692
#173662
#173612
#86132
#86122
#86112
#80102