Finite Group Databasebeta

← #119213

Subgroups — #119213

Order 512.

635 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2#21337C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7860C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1998—22no—
16#1997—21no—
17#1997—21no—
18#1987—21no—
19#1981—21no—
20#1872—22no—
21C4×#103C414yescomplemented
22#1742—21no—
23#1742—21no—
24C2×#339—21no—
25C2×#339—21no—
26#1520—22no—
27#1518—22no—
28#1503—22no—
29#1303—22no—
30#1286—22no—
31#1275—22no—
32C22×M16—21no—
33C22×M16D411yescharacteristic
34C22×M16D411yescharacteristic
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8D412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×#103—24no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2×C412yes—
82#384—21no—
83#382D412yes—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C2×#95—21no—
93#343—22no—
94#343—22no—
95#343—22no—
96#343—22no—
97#340—24no—
98#339—42no—
99#339—42no—
100#328—28no—
101C22×Q8—21no—
102C22×Q8C22⋊C412yes—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16SD1612yes—
124C2×M16Q1612yes—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—41no—
129C22×C8—41no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138#125—41no—
139#125—41no—
140#125—41no—
141#125—41no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208#103—48no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#97—41no—
214#97—41no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—41no—
225C24#12111yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12611yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4C2×Q1611yescharacteristic
262C22×C4#13511yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266C22×C4#13711yescharacteristic
267C22×C4#13711yescharacteristic
268M16—22no—
269M16—22no—
270M16—22no—
271M16—22no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—42no—
277M16—42no—
278M16—81no—
279M16—81no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—82no—
291C2×C8—81no—
292C2×C8—81no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—81no—
321C42—81no—
322C42—81no—
323C42—81no—
324C42#11711yescharacteristic
325C42#11711yescharacteristic
326C42C4×D411yescharacteristic
327C42C4×D411yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42#12111yescharacteristic
331C42#12111yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42#12411yescharacteristic
337C42#12411yescharacteristic
338C42#12611yescharacteristic
339C42#12611yescharacteristic
340C23—41no—
341C23—41no—
342C23#38611yescharacteristic
343C23#39411yescharacteristic
344C23#39711yescharacteristic
345C23#46011yescharacteristic
346C23#46211yescharacteristic
347C23#46411yescharacteristic
348C23#46411yescharacteristic
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38111yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#38811yescharacteristic
383C2×C4#39411yescharacteristic
384C2×C4#39411yescharacteristic
385C2×C4#39411yescharacteristic
386C2×C4#39711yescharacteristic
387C2×C4#39711yescharacteristic
388C2×C4#39711yescharacteristic
389C2×C4C4×SD1611yescharacteristic
390C2×C4C4×Q1611yescharacteristic
391C2×C4#44011yescharacteristic
392C2×C4#44111yescharacteristic
393C2×C4#44811yescharacteristic
394C2×C4#44911yescharacteristic
395C2×C4#45111yescharacteristic
396C2×C4#45211yescharacteristic
397C2×C4#46111yescharacteristic
398C2×C4#46211yescharacteristic
399C2×C4#46311yescharacteristic
400C2×C4#46411yescharacteristic
401C2×C4#46511yescharacteristic
402C2×C4#46711yescharacteristic
403C2×C4#46811yescharacteristic
404C2×C4#47011yescharacteristic
405C2×C4#48311yescharacteristic
406C2×C4#48411yescharacteristic
407C2×C4#48511yescharacteristic
408C2×C4#48511yescharacteristic
409C2×C4#48711yescharacteristic
410C2×C4#48811yescharacteristic
411C2×C4#48911yescharacteristic
412C2×C4#49111yescharacteristic
413C8—42no—
414C8—42no—
415C8—82no—
416C8—82no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22#144011yescharacteristic
429C22#187811yescharacteristic
430C22#188011yescharacteristic
431C22#205511yescharacteristic
432C22#205611yescharacteristic
433C22#207211yescharacteristic
434C22#207411yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—84no—
443C4—82no—
444C4—82no—
445C4—162no—
446C4#182712yes—
447C4#192711yescharacteristic
448C4#192811yescharacteristic
449C4#193311yescharacteristic
450C4#193511yescharacteristic
451C4#203511yescharacteristic
452C4#203611yescharacteristic
453C4#204211yescharacteristic
454C4#204311yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2#1114811yescharacteristic
462C2#1117911yescharacteristic
463C2#1119011yescharacteristic
subgroupG/H
centerC2×C4#388
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1440
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
#213852
#213372
#175752
#108512
C2×#13032
#79572
#78602