Finite Group Databasebeta

← #193396

Subgroups — #193396

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60292C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic, complemented
4C2×#1574C211yesmaximal, characteristic, complemented
5#17700C211yesmaximal, characteristic
6#8936C214yesmaximal, complemented
7#8934C214yesmaximal, complemented
8#7869C211yesmaximal, characteristic, complemented
9#7866C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2991—21no—
12#2990—21no—
13#2989—21no—
14#2988—21no—
15#2986C2212yes—
16#2979C2212yes—
17#2976—22no—
18#2969—22no—
19#2966—22no—
20#2925—22no—
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1994C2212yes—
28#1962C2212yes—
29#1871—22no—
30#1806—22no—
31#1574—22no—
32#1574C2214yes—
33#1574C2214yes—
34#1573—22no—
35#1567—21no—
36#1566—21no—
37C2×#335—21no—
38C2×#334—21no—
39#1544C2212yes—
40#1543C2212yes—
41#1542C411yescharacteristic, complemented
42#1542C411yescharacteristic, complemented
43#1542C412yescomplemented
44#1540C411yescharacteristic, complemented
45#1540C411yescharacteristic, complemented
46#1540C412yescomplemented
47#1276C2214yes—
48#1276C2214yes—
49C22×M16—21no—
50#533C2×C411yescharacteristic
51#533C2×C411yescharacteristic
52#533C2311yescharacteristic
53#532C2×C411yescharacteristic
54#532C2×C411yescharacteristic
55#532C2311yescharacteristic
56#529—21no—
57#529—21no—
58#529—21no—
59#529—21no—
60#529Q812yes—
61#529Q812yes—
62#525—21no—
63#525—21no—
64#525D412yes—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41D412yes—
68C2×#117—21no—
69C22×C42C2311yescharacteristic
70#446—22no—
71#446—22no—
72C8×Q8—22no—
73C8×Q8—22no—
74#436—24no—
75#436—22no—
76#436—22no—
77#435—24no—
78#435—24no—
79C8×D4—24no—
80C8×D4—22no—
81C8×D4—22no—
82#433—24no—
83#433—24no—
84#432—22no—
85#432—21no—
86#432D412yes—
87#431—22no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431C2×C412yes—
96#431C2×C411yescharacteristic
97#431C2×C411yescharacteristic
98#431D412yes—
99#431C2312yes—
100#431C2312yes—
101#431C2312yes—
102#431C2312yes—
103#423—22no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423D411yescharacteristic
108#423D411yescharacteristic
109#423C2312yes—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2×C411yescharacteristic
114C2×C4⋊C8C2×C411yescharacteristic
115C2×C4⋊C8C2×C412yes—
116C2×C4⋊C8D411yescharacteristic
117C2×C4⋊C8D411yescharacteristic
118C2×#104—24no—
119#408—24no—
120#407—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—24no—
125C4×M16—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#376—21no—
134C43C2312yes—
135#339—24no—
136#335—24no—
137#335—24no—
138#334—24no—
139#334—24no—
140#333C2×C412yes—
141#333C2×C412yes—
142#332C2×C412yes—
143#332C2×C412yes—
144#332C2×C412yes—
145#332C2×C412yes—
146#331C2×C412yes—
147#331C2×C412yes—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—24no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—41no—
159C2×M16—82no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—24no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—41no—
167C4⋊Q8C22×C411yescharacteristic
168C4⋊Q8C22×C411yescharacteristic
169C4⋊Q8C22×C411yescharacteristic
170#127C22×C411yescharacteristic
171#126—22no—
172#126—22no—
173#125—21no—
174#125—21no—
175#125C2×Q812yes—
176#124—21no—
177#124—21no—
178#124C2×Q812yes—
179#123—22no—
180#123—22no—
181#123—22no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C4⋊C412yes—
187C22⋊Q8C4⋊C412yes—
188#121—21no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121C4⋊C412yes—
193#121C4⋊C412yes—
194#120—22no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8C4⋊C412yes—
200C4×Q8C4⋊C412yes—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4C4⋊C412yes—
212C4×D4C4⋊C412yes—
213#117—21no—
214#117—21no—
215#117—24no—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—42no—
220#117C2×D412yes—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×Q811yescharacteristic
235C2×C42C2×Q811yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256C4⋊C8C2×D412yes—
257C4⋊C8C2×D412yes—
258C4⋊C8C2×D412yes—
259C4⋊C8C2×D412yes—
260#104—84no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#97—24no—
273#97—22no—
274C4×C8—24no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283#95—41no—
284#95—41no—
285C24#12311yescharacteristic
286#41—42no—
287#41—42no—
288#41—42no—
289#41—42no—
290#41—82no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8C2×C4⋊C412yes—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C4⋊C412yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4C22⋊Q811yescharacteristic
319C22×C4C22⋊Q811yescharacteristic
320C22×C4C22⋊Q811yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—44no—
328M16—81no—
329M16—81no—
330M16—84no—
331M16—82no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—82no—
344C2×C8—81no—
345C2×C8—81no—
346C2×C8—84no—
347C2×C8—82no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—44no—
359C4⋊C4C2×C4⋊C412yes—
360C4⋊C4C2×C4⋊C412yes—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—44no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—24no—
375C42—42no—
376C42—42no—
377C42—44no—
378C42—42no—
379C42C2×C4⋊C411yescharacteristic
380C42C2×C4⋊C411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42C4×D411yescharacteristic
385C42#12011yescharacteristic
386C42#12011yescharacteristic
387C42C22⋊Q811yescharacteristic
388C42C22⋊Q811yescharacteristic
389C42C22⋊Q811yescharacteristic
390C42C22⋊Q811yescharacteristic
391C42C23×C411yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×D411yescharacteristic
394C42C22×Q811yescharacteristic
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23#38511yescharacteristic
400C23#38511yescharacteristic
401C23C2×#12311yescharacteristic
402Q8—42no—
403Q8—41no—
404Q8—41no—
405Q8—42no—
406D4—42no—
407D4—44no—
408D4—41no—
409D4—41no—
410D4—42no—
411D4—82no—
412D4—82no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#38511yescharacteristic
446C2×C4#38511yescharacteristic
447C2×C4#38511yescharacteristic
448C2×C4#38511yescharacteristic
449C2×C4#38511yescharacteristic
450C2×C4#38511yescharacteristic
451C2×C4#44312yes—
452C2×C4#44412yes—
453C2×C4C22×C4⋊C411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×#12011yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4C2×C22⋊Q811yescharacteristic
459C8—42no—
460C8—42no—
461C8—42no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—42no—
466C8—42no—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—82no—
480C22#186612yes—
481C22C2×#38511yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—84no—
494C4#186612yes—
495C4C2×#44311yescharacteristic
496C4C2×#44411yescharacteristic
497C4#295712yes—
498C4#302111yescharacteristic
499C4#302211yescharacteristic
500C2—22no—
501C2—41no—
502C2—41no—
503C2—82no—
504C2#2085811yescharacteristic
505C2C2×#186611yescharacteristic
506C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17817C2semidirect
4. C2×#1574C2semidirect
6. #8936C2semidirect
7. #8934C2semidirect
8. #7869C2semidirect
9. #7866C2semidirect
41. #1542C4semidirect
42. #1542C4semidirect
43. #1542C4semidirect
44. #1540C4semidirect
45. #1540C4semidirect
46. #1540C4semidirect

Maximal subgroups

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

subgroupindex
#602922
#218562
#178172
C2×#15742
#177002
#89362
#89342
#78692
#78662