Finite Group Databasebeta

← #107170

Subgroups — #107170

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34247C211yesmaximal, characteristic
2#17398C211yesmaximal, characteristic
3C2×#1473C211yesmaximal, characteristic
4#8092C211yesmaximal, characteristic, complemented
5#8071C211yesmaximal, characteristic, complemented
6#7978C211yesmaximal, characteristic, complemented
7#7958C211yesmaximal, characteristic, complemented
8C22×#126C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2373—22no—
14#2373—22no—
15#2345—21no—
16#2344—21no—
17#2342—21no—
18#2342—21no—
19#2334—22no—
20#2313C411yescharacteristic
21#2312C411yescharacteristic
22C4×#126—22no—
23#2283C2211yescharacteristic
24#2274C411yescharacteristic
25#2273C411yescharacteristic
26C4×#117C2211yescharacteristic
27#1473—22no—
28#1473—22no—
29#1473C414yescomplemented
30#1473C414yescomplemented
31#1463—21no—
32#1462—21no—
33C2×#324—21no—
34C2×#323—21no—
35#1303—22no—
36#1300—22no—
37#1288—22no—
38#1286—22no—
39C2×#126—21no—
40C2×#126—21no—
41C2×#126—21no—
42C2×#126—21no—
43C2×#126—21no—
44C2×#126—21no—
45C2×#126—21no—
46C2×#126—21no—
47C2×#126C2×C412yes—
48C2×#126C2×C412yes—
49C2×#126D412yes—
50C2×#126Q812yes—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432C2×C412yescomplemented
68#432C2×C412yescomplemented
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431C2×C412yescomplemented
76#431C2×C412yescomplemented
77#423—21no—
78#423—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81#407—21no—
82#407—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89#389—22no—
90#389—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#383D412yes—
106#382D412yes—
107#381—21no—
108#381—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#379—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C411yescharacteristic
128C4×C22⋊C4C2×C411yescharacteristic
129C4×C22⋊C4C2×C411yescharacteristic
130C4×C22⋊C4C2×C411yescharacteristic
131#376—21no—
132#376D412yes—
133C43—21no—
134#324—24no—
135#324—24no—
136#323—24no—
137#323—24no—
138C25C22⋊C411yescharacteristic
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—41no—
147C2×M16—41no—
148C22×C8—41no—
149C22×C8—41no—
150#126—22no—
151#126—22no—
152#126—22no—
153#126—22no—
154#126—22no—
155#126—22no—
156#126—22no—
157#126—22no—
158#126—22no—
159#126—22no—
160#126—22no—
161#126—22no—
162#126—22no—
163#126—22no—
164#126—22no—
165#126—22no—
166#117—44no—
167#117—44no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C4212yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C4⋊C412yes—
189C2×C4⋊C4C4⋊C412yes—
190C2×C4⋊C4C4⋊C412yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4C4212yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C4⋊C412yes—
215C2×C22⋊C4C4⋊C412yes—
216C2×C22⋊C4C4⋊C412yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×Q811yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—41no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254#97—22no—
255#97—22no—
256#97—41no—
257#97—41no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—41no—
261C4×C8—41no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—42no—
280C24—81no—
281C24#9512yes—
282C24C2×C22⋊C411yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4#9512yes—
303C22×C4#9512yes—
304C22×C4#9512yes—
305C22×C4C2×C4211yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313C22×C4#11711yescharacteristic
314C22×C4#11711yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323M16—82no—
324M16—82no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—82no—
330C2×C8—82no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—44no—
368C42—44no—
369C42—82no—
370C42—81no—
371C42—81no—
372C42—81no—
373C42—81no—
374C42—82no—
375C42—82no—
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42C22⋊Q811yescharacteristic
389C42C22⋊Q811yescharacteristic
390C42C22⋊Q811yescharacteristic
391C42C22⋊Q811yescharacteristic
392C23—22no—
393C23—22no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—81no—
405C23—81no—
406C23—164no—
407C23C2×#9511yescharacteristic
408C23#37911yescharacteristic
409C23#38111yescharacteristic
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
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—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—164no—
454C2×C4C4×C22⋊C411yescharacteristic
455C2×C4C4×C22⋊C411yescharacteristic
456C2×C4#38011yescharacteristic
457C2×C4#38011yescharacteristic
458C2×C4#38511yescharacteristic
459C2×C4#38511yescharacteristic
460C2×C4#38511yescharacteristic
461C2×C4#38511yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C8—82no—
467C8—82no—
468C8—82no—
469C8—82no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—82no—
482C22—82no—
483C22—162no—
484C22—162no—
485C22C22⋊#9511yescharacteristic
486C22#179412yes—
487C22#182712yes—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—82no—
507C2—22no—
508C2—22no—
509C2—41no—
510C2—82no—
511C2#1075211yescharacteristic
512C2#1075611yescharacteristic
513C2#1075811yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8092C2semidirect
5. #8071C2semidirect
6. #7978C2semidirect
7. #7958C2semidirect
29. #1473C4semidirect
30. #1473C4semidirect
67. #432C2×C4semidirect
68. #432C2×C4semidirect
75. #431C2×C4semidirect
76. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342472
#173982
C2×#14732
#80922
#80712
#79782
#79582