Finite Group Databasebeta

← #190081

Subgroups — #190081

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#17363C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8745C214yesmaximal, complemented
5#8744C214yesmaximal, complemented
6#7952C212yesmaximal, complemented
7C2×#1285C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#432C2212yes—
10C2×#432C2212yes—
11#2357—21no—
12#2356—21no—
13#2355—21no—
14#2354—21no—
15#2353—21no—
16#2353—21no—
17#2352—21no—
18#2352—21no—
19#2298C2212yes—
20#2298C2212yes—
21#2297C412yes—
22#2285C2211yescharacteristic
23#2283C411yescharacteristic
24#2283C411yescharacteristic
25C4×#117C411yescharacteristic
26C4×#117C411yescharacteristic
27C4×#117C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1467—21no—
30#1467—21no—
31#1466—21no—
32#1466—21no—
33#1464C2212yes—
34#1464C2212yes—
35#1464C2212yes—
36#1464C2212yes—
37#1459—21no—
38#1459—21no—
39#1459—21no—
40#1459—21no—
41#1285C2218yes—
42#1285C2218yes—
43C22×#41C2×C411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×#117—21no—
57C2×#117—21no—
58C2×#117C2×C412yes—
59C2×#117C2×C411yescharacteristic
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432C2314yes—
74#432C2314yes—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×#97—22no—
81#391—22no—
82#390—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#384—21no—
94#383—21no—
95#383—21no—
96#382—21no—
97#382—21no—
98#382—21no—
99#381—21no—
100#381—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4C2×C412yes—
125#376—21no—
126#376—21no—
127#376C2×C412yes—
128#376C2×C412yes—
129C2×#95—21no—
130C43C2×C412yes—
131C43C2×C412yes—
132#324—22no—
133#324—22no—
134#324—22no—
135#324—22no—
136#323—22no—
137#323—22no—
138#323—22no—
139#323—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
147C22×D4C22⋊C411yescharacteristic
148C23×C4—22no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C22×C8—42no—
159C22×C8—42no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—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—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—44no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C22⋊C411yescharacteristic
206C2×C22⋊C4C2×D411yescharacteristic
207C2×C22⋊C4C2×D411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—24no—
243C4⋊C8—24no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250#97—24no—
251#97—24no—
252#97—42no—
253#97—42no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271#41—48no—
272#41—48no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4#11712yes—
326C22×C4#11712yes—
327C22×C4#12312yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4#12312yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—84no—
337C2×C8—84no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—82no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—44no—
379C42—44no—
380C42—44no—
381C42—82no—
382C42—82no—
383C42#11711yescharacteristic
384C42#11711yescharacteristic
385C42#11711yescharacteristic
386C42#11711yescharacteristic
387C42#12312yes—
388C42#12312yes—
389C42#12312yes—
390C42#12312yes—
391C42C2×#4112yes—
392C42C2×#4112yes—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23#38112yes—
406C23#41312yes—
407C23#41312yes—
408C23C22×C22⋊C411yescharacteristic
409Q8—44no—
410D4—44no—
411D4—44no—
412D4—44no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—88no—
466C2×C4#38111yescharacteristic
467C2×C4#38111yescharacteristic
468C2×C4#38111yescharacteristic
469C2×C4#38112yes—
470C2×C4#38111yescharacteristic
471C2×C4C2×#11711yescharacteristic
472C2×C4C2×#11711yescharacteristic
473C2×C4C2×#12312yes—
474C2×C4C2×#12312yes—
475C8—84no—
476C8—84no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#182214yes—
494C22C2×#38111yescharacteristic
495C22C2×#41311yescharacteristic
496C22C2×#41311yescharacteristic
497C4—21no—
498C4—21no—
499C4—22no—
500C4—22no—
501C4—21no—
502C4—22no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4#182712yes—
516C2—21no—
517C2—21no—
518C2—24no—
519C2—81no—
520C2—81no—
521C2C2×#182211yescharacteristic
522C2C2×#182711yescharacteristic
523C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8745C2semidirect
5. #8744C2semidirect
6. #7952C2semidirect
7. C2×#1285C2semidirect

Maximal subgroups

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

subgroupindex
#341212
#173632
#87452
#87442
#79522
C2×#12852