Finite Group Databasebeta

← #192469

Subgroups — #192469

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60292C211yesmaximal, characteristic
2#21383C211yesmaximal, characteristic
3#21371C211yesmaximal, characteristic, complemented
4#21350C211yesmaximal, characteristic, complemented
5#17543C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8052C211yesmaximal, characteristic, complemented
9#7928C211yesmaximal, 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×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2283C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1986—21no—
28#1980—21no—
29#1950—21no—
30#1950—21no—
31#1940—21no—
32#1940—21no—
33#1871C2212yes—
34#1871C2212yes—
35#1855C2212yes—
36#1855C2212yes—
37#1851C2212yes—
38#1851C2212yes—
39#1778—22no—
40C4×C4⋊C8—22no—
41#1759—22no—
42#1509C414yescomplemented
43#1506C411yescharacteristic, complemented
44#1506C411yescharacteristic, complemented
45#1505C411yescharacteristic, complemented
46#1505C411yescharacteristic, 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#432—21no—
90#432—21no—
91#432C2×C411yescharacteristic
92#432C2×C411yescharacteristic
93#432C2×C412yes—
94#431—22no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
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×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8D412yes—
128#383—21no—
129#382—21no—
130#381—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C412yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4C2312yes—
137C4×C22⋊C4C2312yes—
138#376—21no—
139#376C2×C411yescharacteristic
140#376C2×C411yescharacteristic
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—44no—
225#117C2×D412yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×Q811yescharacteristic
245C2×C42C2×Q811yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248#108—44no—
249#108—44no—
250C4⋊C8—24no—
251C4⋊C8—24no—
252C4⋊C8—22no—
253C4⋊C8—21no—
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—44no—
264C4⋊C8—44no—
265#104—44no—
266#104—44no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8C22×C412yes—
275C22⋊C8C22×C412yes—
276#97—24no—
277#97—21no—
278#97—21no—
279#97—22no—
280#97—22no—
281#97—41no—
282#97—41no—
283C4×C8—24no—
284C4×C8—22no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—41no—
288C4×C8—41no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—41no—
293#95—41no—
294#95—41no—
295#95C22×C412yes—
296#95C22×C412yes—
297C24C2×#4111yescharacteristic
298#41—44no—
299#41—42no—
300#41—42no—
301#41—82no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8C2×C4⋊C412yes—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C2×C4⋊C412yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C22⋊Q811yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C22⋊Q811yescharacteristic
328C22×C4C22⋊Q811yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—41no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—42no—
348M16—42no—
349M16—84no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—84no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4C2×C4⋊C412yes—
380C4⋊C4C2×C4⋊C412yes—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C42—24no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—44no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42C2×C4⋊C411yescharacteristic
403C42C2×C4⋊C411yescharacteristic
404C42C2×C4⋊C411yescharacteristic
405C42C2×C4⋊C411yescharacteristic
406C42C4×D411yescharacteristic
407C42C4×D411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42#12011yescharacteristic
411C42#12011yescharacteristic
412C42#12011yescharacteristic
413C42#12011yescharacteristic
414C42C22⋊Q811yescharacteristic
415C42C22⋊Q811yescharacteristic
416C42C22⋊Q811yescharacteristic
417C42C22⋊Q811yescharacteristic
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23C2×C4×D411yescharacteristic
423C23C2×C22⋊Q811yescharacteristic
424C23C2×#12311yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—42no—
428D4—42no—
429D4—42no—
430D4—44no—
431D4—42no—
432D4—82no—
433D4—82no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#38511yescharacteristic
472C2×C4#38511yescharacteristic
473C2×C4#38511yescharacteristic
474C2×C4#38511yescharacteristic
475C2×C4#38511yescharacteristic
476C2×C4#38511yescharacteristic
477C2×C4#38511yescharacteristic
478C2×C4#38511yescharacteristic
479C2×C4C22×C4⋊C411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×#12011yescharacteristic
482C2×C4C2×C22⋊Q811yescharacteristic
483C8—42no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—42no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22C2×#38511yescharacteristic
504C22#295512yes—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—162no—
521C4#186112yes—
522C4#302511yescharacteristic
523C4#302511yescharacteristic
524C2—22no—
525C2—41no—
526C2—41no—
527C2—82no—
528C2C2×#186111yescharacteristic
529C2#2088411yescharacteristic
530C2#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. #21371C2semidirect
4. #21350C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8052C2semidirect
9. #7928C2semidirect
42. #1509C4semidirect
43. #1506C4semidirect
44. #1506C4semidirect
45. #1505C4semidirect
46. #1505C4semidirect

Maximal subgroups

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

subgroupindex
#602922
#213832
#213712
#213502
#175432
#88752
#88742
#80522
#79282