Finite Group Databasebeta

← #119031

Subgroups — #119031

Order 512.

585 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#17708C211yesmaximal, characteristic
3C2×#1542C211yesmaximal, characteristic
4#10853C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6#7918C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1986—22no—
16#1986—22no—
17C2×#341—21no—
18C2×#341—21no—
19#1733—21no—
20#1733—21no—
21#1550—22no—
22#1549—21no—
23#1548—21no—
24#1542—22no—
25#1542—22no—
26#1542C414yescomplemented
27C2×#333—21no—
28C2×#332—21no—
29#1286—22no—
30#1281—22no—
31#1279—22no—
32C22×M16—21no—
33C22×M16D411yescharacteristic
34C22×M16D411yescharacteristic
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533D412yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431D412yes—
62#423—21no—
63#423—21no—
64#423—21no—
65#423D412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2×C412yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#384—21no—
87#382—21no—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4D412yes—
92#341—42no—
93#341—42no—
94#339—24no—
95#339—24no—
96#333—24no—
97#333—24no—
98#332—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—21no—
106C2×M16—21no—
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—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16SD1612yes—
122C2×M16Q1612yes—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8C22⋊C412yes—
142C4⋊Q8C22⋊C412yes—
143#125—22no—
144#125—22no—
145#125—41no—
146#125—41no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163#117—22no—
164#117—22no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—41no—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#97—41no—
213#97—41no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8—41no—
220#95—22no—
221#95—41no—
222C24#12411yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×Q8—82no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12611yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4C2×SD1611yescharacteristic
253C22×C4C2×Q1611yescharacteristic
254C22×C4#13511yescharacteristic
255C22×C4#13511yescharacteristic
256C22×C4AGL(1,ℤ/8)11yescharacteristic
257C22×C4#13711yescharacteristic
258C22×C4#13711yescharacteristic
259C22×C4#13711yescharacteristic
260M16—22no—
261M16—22no—
262M16—22no—
263M16—22no—
264M16—41no—
265M16—41no—
266M16—41no—
267M16—41no—
268M16—42no—
269M16—42no—
270M16—81no—
271M16—81no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—81no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—81no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42C2×C22⋊C411yescharacteristic
317C42#11711yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12011yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12411yescharacteristic
329C42#12411yescharacteristic
330C42#12411yescharacteristic
331C42#12611yescharacteristic
332C23—41no—
333C23—41no—
334C23#38811yescharacteristic
335C23#39411yescharacteristic
336C23#39711yescharacteristic
337C23#48711yescharacteristic
338C23#48811yescharacteristic
339C23#48911yescharacteristic
340C23#49111yescharacteristic
341Q8—82no—
342Q8—82no—
343Q8—82no—
344Q8—82no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#38111yescharacteristic
365C2×C4#38611yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39411yescharacteristic
373C2×C4#39711yescharacteristic
374C2×C4#39711yescharacteristic
375C2×C4#39711yescharacteristic
376C2×C4C4×SD1611yescharacteristic
377C2×C4C4×Q1611yescharacteristic
378C2×C4#44011yescharacteristic
379C2×C4#44111yescharacteristic
380C2×C4#44811yescharacteristic
381C2×C4#44911yescharacteristic
382C2×C4#45111yescharacteristic
383C2×C4#45211yescharacteristic
384C2×C4#46011yescharacteristic
385C2×C4#46111yescharacteristic
386C2×C4#46211yescharacteristic
387C2×C4#46211yescharacteristic
388C2×C4#46311yescharacteristic
389C2×C4#46411yescharacteristic
390C2×C4#46411yescharacteristic
391C2×C4#46411yescharacteristic
392C2×C4#46511yescharacteristic
393C2×C4#46711yescharacteristic
394C2×C4#46811yescharacteristic
395C2×C4#47011yescharacteristic
396C2×C4#48311yescharacteristic
397C2×C4#48411yescharacteristic
398C2×C4#48511yescharacteristic
399C2×C4#48511yescharacteristic
400C8—42no—
401C8—42no—
402C8—82no—
403C8—82no—
404C8—82no—
405C8—82no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22#144011yescharacteristic
417C22#193311yescharacteristic
418C22#193511yescharacteristic
419C22#204211yescharacteristic
420C22#204311yescharacteristic
421C22#207211yescharacteristic
422C22#207411yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—82no—
432C4—162no—
433C4#187811yescharacteristic
434C4#188011yescharacteristic
435C4#192711yescharacteristic
436C4#192811yescharacteristic
437C4#203511yescharacteristic
438C4#203611yescharacteristic
439C4#205511yescharacteristic
440C4#205611yescharacteristic
441C4#208012yes—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2#1115311yescharacteristic
449C2#1117711yescharacteristic
450C2#1119011yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1440
prime radicalC24#124
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
26. #1542C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#177082
C2×#15422
#108532
#79572
#79182
#78932