Finite Group Databasebeta

← #119860

Subgroups — #119860

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21318C211yesmaximal, characteristic
2#17708C211yesmaximal, characteristic
3#17486C211yesmaximal, characteristic
4#10844C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7849C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15#1991—22no—
16#1990—21no—
17#1990—21no—
18C2×#341—21no—
19C2×#341—21no—
20#1735—21no—
21#1735—21no—
22#1550C414yescomplemented
23#1549—21no—
24#1548—21no—
25#1542—22no—
26#1493—22no—
27#1487—22no—
28#1481—22no—
29#1303—22no—
30#1286—22no—
31#1273—22no—
32C22×M16—21no—
33C22×M16—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
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#431—21no—
61#431—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423D412yes—
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—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8D412yes—
86#384—21no—
87#382D412yes—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92#341—42no—
93#341—42no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#333—24no—
99#332—24no—
100Q8⋊C8—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8SD1612yes—
135C22×C8Q1612yes—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C22⋊C412yes—
143C4⋊Q8C22⋊C412yes—
144#125—22no—
145#125—22no—
146#125—41no—
147#125—41no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164#117—22no—
165#117—22no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221#95—22no—
222#95—22no—
223C24#12411yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—82no—
229C2×Q8—82no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4C2×SD1611yescharacteristic
254C22×C4C2×Q1611yescharacteristic
255C22×C4#13511yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4AGL(1,ℤ/8)11yescharacteristic
258C22×C4#13711yescharacteristic
259C22×C4#13711yescharacteristic
260C22×C4#13711yescharacteristic
261M16—42no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—81no—
266M16—81no—
267M16—81no—
268M16—81no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—81no—
292C2×C8—81no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C42—42no—
314C42—42no—
315C42—81no—
316C42—81no—
317C42C2×C22⋊C411yescharacteristic
318C42#11711yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12011yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12411yescharacteristic
330C42#12411yescharacteristic
331C42#12411yescharacteristic
332C42#12611yescharacteristic
333C23—41no—
334C23—41no—
335C23#38811yescharacteristic
336C23#39411yescharacteristic
337C23#39711yescharacteristic
338C23#48711yescharacteristic
339C23#48811yescharacteristic
340C23#48911yescharacteristic
341C23#49111yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#38111yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39411yescharacteristic
373C2×C4#39411yescharacteristic
374C2×C4#39711yescharacteristic
375C2×C4#39711yescharacteristic
376C2×C4#39711yescharacteristic
377C2×C4C4×SD1611yescharacteristic
378C2×C4C4×Q1611yescharacteristic
379C2×C4#44011yescharacteristic
380C2×C4#44111yescharacteristic
381C2×C4#44811yescharacteristic
382C2×C4#44911yescharacteristic
383C2×C4#45111yescharacteristic
384C2×C4#45211yescharacteristic
385C2×C4#46011yescharacteristic
386C2×C4#46111yescharacteristic
387C2×C4#46211yescharacteristic
388C2×C4#46211yescharacteristic
389C2×C4#46311yescharacteristic
390C2×C4#46411yescharacteristic
391C2×C4#46411yescharacteristic
392C2×C4#46411yescharacteristic
393C2×C4#46511yescharacteristic
394C2×C4#46711yescharacteristic
395C2×C4#46811yescharacteristic
396C2×C4#47011yescharacteristic
397C2×C4#48311yescharacteristic
398C2×C4#48411yescharacteristic
399C2×C4#48511yescharacteristic
400C2×C4#48511yescharacteristic
401C8—42no—
402C8—42no—
403C8—82no—
404C8—82no—
405C8—82no—
406C8—82no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22#144011yescharacteristic
418C22#193311yescharacteristic
419C22#193511yescharacteristic
420C22#204211yescharacteristic
421C22#204311yescharacteristic
422C22#207211yescharacteristic
423C22#207411yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—82no—
432C4—162no—
433C4#187811yescharacteristic
434C4#188011yescharacteristic
435C4#188112yes—
436C4#192711yescharacteristic
437C4#192811yescharacteristic
438C4#203511yescharacteristic
439C4#203611yescharacteristic
440C4#205511yescharacteristic
441C4#205611yescharacteristic
442C4#208012yes—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2#1115011yescharacteristic
450C2#1119011yescharacteristic
451C2#1119411yescharacteristic
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
22. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#213182
#177082
#174862
#108442
#80952
#79572
#78492