Finite Group Databasebeta

← #192471

Subgroups — #192471

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60292C211yesmaximal, characteristic
2#21382C211yesmaximal, characteristic
3#21370C211yesmaximal, characteristic, complemented
4#21354C211yesmaximal, characteristic, complemented
5#17573C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8052C211yesmaximal, characteristic, complemented
9#7924C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2991—21no—
12#2990—21no—
13#2989—21no—
14#2988—21no—
15#2986—21no—
16#2979—21no—
17#2976C2214yes—
18#2969—22no—
19#2966—22no—
20#2925—22no—
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2283C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1986—21no—
28#1980—21no—
29#1950—21no—
30#1950—21no—
31#1939—21no—
32#1939—21no—
33#1871C2212yes—
34#1871C2212yes—
35#1855C2212yes—
36#1855C2212yes—
37#1851C2212yes—
38#1851C2212yes—
39C4×#108—22no—
40#1768—22no—
41C4×#104—22no—
42#1521C414yescomplemented
43#1518C411yescharacteristic, complemented
44#1518C411yescharacteristic, complemented
45#1517C411yescharacteristic, complemented
46#1517C411yescharacteristic, complemented
47#1501C2214yes—
48#1297C2214yes—
49#1282C2214yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52#533C2×C411yescharacteristic
53#533D411yescharacteristic
54#533Q811yescharacteristic
55#532C2×C411yescharacteristic
56#532D411yescharacteristic
57#532Q811yescharacteristic
58#529—21no—
59#529—21no—
60#529—21no—
61#529—21no—
62#529C2×C412yes—
63#529Q812yes—
64#525—21no—
65#525—21no—
66#525C2312yes—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41D412yes—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#446—22no—
73#446—22no—
74C8×Q8—22no—
75C8×Q8—22no—
76#436—22no—
77#436—22no—
78#436—24no—
79#435—24no—
80#435—24no—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4—24no—
84#433—24no—
85#433—24no—
86#432—22no—
87#432—21no—
88#432—21no—
89#431—22no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431C2×C411yescharacteristic
99#431C2×C411yescharacteristic
100#431C2×C412yes—
101C2×#108—22no—
102C2×#108—22no—
103#423—22no—
104#423D411yescharacteristic
105#423D411yescharacteristic
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×#104—22no—
113C2×#104—22no—
114#408—24no—
115#407C2312yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118#405—24no—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16D412yes—
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8D412yes—
128#383C2×C412yes—
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#381—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4C2312yes—
138C4×C22⋊C4C2312yes—
139#376—21no—
140#376—21no—
141C43—21no—
142#341—22no—
143#341—22no—
144#339—22no—
145#339—22no—
146#336—22no—
147#336—22no—
148#328C2×C414yes—
149#328C2×C414yes—
150#327C2×C414yes—
151#327C2×C414yes—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41C2×D412yes—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—24no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16C2×D412yes—
166C2×M16C2×D412yes—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—24no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8C4⋊C411yescharacteristic
174C4⋊Q8C4⋊C411yescharacteristic
175C4⋊Q8C4⋊C411yescharacteristic
176#127C4⋊C411yescharacteristic
177#126—22no—
178#126—22no—
179#125—21no—
180#125—21no—
181#125C4⋊C412yes—
182#124—21no—
183#124—21no—
184#124C4⋊C412yes—
185#123—22no—
186#123—22no—
187#123—22no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8C22×C412yes—
193C22⋊Q8C2×Q812yes—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121C22×C412yes—
199#121C2×Q812yes—
200#120—22no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8C4⋊C412yes—
206C4×Q8C4⋊C412yes—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4C4⋊C412yes—
218C4×D4C4⋊C412yes—
219#117—21no—
220#117—21no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—24no—
225#117—42no—
226#117C2×D412yes—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×Q811yescharacteristic
246C2×C42C2×Q811yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249#108—44no—
250#108—44no—
251C4⋊C8—24no—
252C4⋊C8—24no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266#104—44no—
267#104—44no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8C22×C412yes—
276C22⋊C8C22×C412yes—
277#97—24no—
278#97—22no—
279#97—41no—
280#97—41no—
281C4×C8—24no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—22no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—41no—
289C4×C8—41no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95—22no—
295#95—41no—
296#95C22×C412yes—
297#95C22×C412yes—
298C24C2×#4111yescharacteristic
299#41—42no—
300#41—42no—
301#41—42no—
302#41—42no—
303#41—82no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8C2×C4⋊C412yes—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4C2×C4⋊C412yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—81no—
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C22⋊Q811yescharacteristic
329C22×C4C22⋊Q811yescharacteristic
330C22×C4C22⋊Q811yescharacteristic
331C22×C4C22⋊Q811yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—42no—
351M16—42no—
352M16—84no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—84no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4C2×C4⋊C412yes—
384C4⋊C4C2×C4⋊C412yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—82no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—42no—
404C42—42no—
405C42—82no—
406C42C2×C4⋊C411yescharacteristic
407C42C2×C4⋊C411yescharacteristic
408C42C2×C4⋊C411yescharacteristic
409C42C2×C4⋊C411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42C4×D411yescharacteristic
413C42C4×D411yescharacteristic
414C42#12011yescharacteristic
415C42#12011yescharacteristic
416C42#12011yescharacteristic
417C42#12011yescharacteristic
418C42C22⋊Q811yescharacteristic
419C42C22⋊Q811yescharacteristic
420C42C22⋊Q811yescharacteristic
421C42C22⋊Q811yescharacteristic
422C23—21no—
423C23—21no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23C2×C4×D411yescharacteristic
428C23C2×C22⋊Q811yescharacteristic
429C23C2×#12311yescharacteristic
430Q8—42no—
431Q8—42no—
432Q8—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#38511yescharacteristic
478C2×C4#38511yescharacteristic
479C2×C4#38511yescharacteristic
480C2×C4#38511yescharacteristic
481C2×C4#38511yescharacteristic
482C2×C4#38511yescharacteristic
483C2×C4#38511yescharacteristic
484C2×C4#38511yescharacteristic
485C2×C4#44312yes—
486C2×C4#44412yes—
487C2×C4C22×C4⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×C22⋊Q811yescharacteristic
491C8—42no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—42no—
496C8—42no—
497C8—42no—
498C8—42no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22C2×#38511yescharacteristic
513C22C2×#44311yescharacteristic
514C22C2×#44411yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4#186612yes—
531C4#186612yes—
532C4#302511yescharacteristic
533C4#302511yescharacteristic
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—82no—
539C2#2086711yescharacteristic
540C2C2×#186611yescharacteristic
541C2#2088811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21370C2semidirect
4. #21354C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8052C2semidirect
9. #7924C2semidirect
42. #1521C4semidirect
43. #1518C4semidirect
44. #1518C4semidirect
45. #1517C4semidirect
46. #1517C4semidirect

Maximal subgroups

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

subgroupindex
#602922
#213822
#213702
#213542
#175732
#88752
#88742
#80522
#79242