Finite Group Databasebeta

← #118887

Subgroups — #118887

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1995C211yesmaximal, characteristic
2#17708C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#10840C211yesmaximal, characteristic
5#8001C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7906C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1995—22no—
16#1995—22no—
17C2×#339—21no—
18C2×#339—21no—
19C2×#339—21no—
20C2×#339—21no—
21#1550—22no—
22#1549C412yescomplemented
23#1548C412yescomplemented
24#1542—22no—
25#1520—22no—
26#1518—22no—
27#1503—22no—
28#1292—22no—
29#1286—22no—
30#1280—22no—
31C22×M16—21no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533D412yes—
40#533D412yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68#423D412yes—
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×C22⋊C8—21no—
76C2×C22⋊C8D412yes—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#384—21no—
80#382D412yes—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85#376—21no—
86#376—21no—
87#376—21no—
88#376—21no—
89C2×#95—21no—
90C2×#95—21no—
91#339—24no—
92#339—24no—
93#339—42no—
94#339—42no—
95#333—24no—
96#332—24no—
97#328—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
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—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C22⋊C412yes—
128C4⋊Q8C22⋊C412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42SD1612yes—
181C2×C42Q1612yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210C4×C8—22no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214#95—22no—
215#95—22no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24#12111yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
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—42no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16—81no—
273M16—81no—
274M16—81no—
275M16—81no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—82no—
281C2×C8—81no—
282C2×C8—81no—
283C2×C8—81no—
284C2×C8—81no—
285C2×C8—82no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42—42no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—81no—
317C42—81no—
318C42C2×C22⋊C411yescharacteristic
319C42#11711yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12411yescharacteristic
331C42#12611yescharacteristic
332C42#12611yescharacteristic
333C42#12611yescharacteristic
334C23—41no—
335C23—41no—
336C23#38811yescharacteristic
337C23#39411yescharacteristic
338C23#39411yescharacteristic
339C23#46511yescharacteristic
340C23#46711yescharacteristic
341C23#46811yescharacteristic
342C23#47011yescharacteristic
343Q8—82no—
344Q8—82no—
345Q8—82no—
346Q8—82no—
347C2×C4—41no—
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—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4#38111yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39411yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39711yescharacteristic
386C2×C4#39711yescharacteristic
387C2×C4#39711yescharacteristic
388C2×C4C4×SD1611yescharacteristic
389C2×C4C4×Q1611yescharacteristic
390C2×C4#44011yescharacteristic
391C2×C4#44111yescharacteristic
392C2×C4#44811yescharacteristic
393C2×C4#44911yescharacteristic
394C2×C4#45111yescharacteristic
395C2×C4#45211yescharacteristic
396C2×C4#46011yescharacteristic
397C2×C4#46111yescharacteristic
398C2×C4#46211yescharacteristic
399C2×C4#46211yescharacteristic
400C2×C4#46311yescharacteristic
401C2×C4#46411yescharacteristic
402C2×C4#46411yescharacteristic
403C2×C4#46411yescharacteristic
404C2×C4#48311yescharacteristic
405C2×C4#48411yescharacteristic
406C2×C4#48511yescharacteristic
407C2×C4#48511yescharacteristic
408C2×C4#48711yescharacteristic
409C2×C4#48811yescharacteristic
410C2×C4#48911yescharacteristic
411C2×C4#49111yescharacteristic
412C8—82no—
413C8—82no—
414C8—82no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22#144011yescharacteristic
427C22#193311yescharacteristic
428C22#193511yescharacteristic
429C22#203511yescharacteristic
430C22#203611yescharacteristic
431C22#205511yescharacteristic
432C22#205611yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—162no—
446C4#187811yescharacteristic
447C4#188011yescharacteristic
448C4#192711yescharacteristic
449C4#192811yescharacteristic
450C4#204211yescharacteristic
451C4#204311yescharacteristic
452C4#207211yescharacteristic
453C4#207411yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#1115911yescharacteristic
461C2#1118011yescharacteristic
462C2#1119011yescharacteristic
subgroupG/H
centerC2×C4#397
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
22. #1549C4semidirect
23. #1548C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19952
#177082
#175752
#108402
#80012
#79572
#79062