Finite Group Databasebeta

← #119784

Subgroups — #119784

Order 512.

593 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
1#21329C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3#17486C211yesmaximal, characteristic
4#10856C211yesmaximal, characteristic
5#8001C211yesmaximal, characteristic
6C2×#1292C211yesmaximal, characteristic
7#7891C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1995—22no—
16#1994—22no—
17C2×#344—21no—
18C2×#344—21no—
19#1733—21no—
20#1733—21no—
21#1550—22no—
22#1550—22no—
23C2×#333C412yescomplemented
24C2×#332C412yescomplemented
25#1493—22no—
26#1487—22no—
27#1481—22no—
28#1292—22no—
29#1292—22no—
30#1279—22no—
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533D412yes—
42#533D412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423D412yes—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2×C412yes—
82C2×C4⋊C8C2×C412yes—
83C2×C22⋊C8—21no—
84C2×C4×C8D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90C43—21no—
91#344—42no—
92#344—42no—
93#339—24no—
94#339—24no—
95#333—24no—
96#332—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—41no—
106C2×M16—41no—
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—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—82no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8C22⋊C412yes—
136C4⋊Q8C22⋊C412yes—
137#125—22no—
138#125—22no—
139#125—41no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—22no—
162#117—22no—
163#117—21no—
164#117—21no—
165#117—41no—
166#117—41no—
167#117SD1612yes—
168#117Q1612yes—
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—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—41no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—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—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C4×C8—22no—
218C4×C8—22no—
219#95—41no—
220#95—41no—
221C24#12311yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—81no—
227C2×Q8—81no—
228C2×Q8—82no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4#11711yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4C2×SD1611yescharacteristic
255C22×C4C2×Q1611yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259C22×C4#13711yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262M16—42no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—81no—
267M16—81no—
268M16—81no—
269M16—81no—
270M16—81no—
271M16—81no—
272M16—81no—
273M16—81no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—81no—
284C2×C8—81no—
285C2×C8—81no—
286C2×C8—81no—
287C2×C8—82no—
288C2×C8—82no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42—41no—
316C42—41no—
317C42—81no—
318C42—81no—
319C42—82no—
320C42C2×C22⋊C411yescharacteristic
321C42#11711yescharacteristic
322C42C4×D411yescharacteristic
323C42C4×D411yescharacteristic
324C42#12011yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12411yescharacteristic
335C42#12611yescharacteristic
336C23—41no—
337C23—41no—
338C23#38811yescharacteristic
339C23#39411yescharacteristic
340C23#39711yescharacteristic
341C23#48311yescharacteristic
342C23#48411yescharacteristic
343C23#48511yescharacteristic
344C23#48511yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
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—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#38111yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#39411yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39411yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39711yescharacteristic
386C2×C4#39711yescharacteristic
387C2×C4C4×SD1611yescharacteristic
388C2×C4C4×Q1611yescharacteristic
389C2×C4#44011yescharacteristic
390C2×C4#44111yescharacteristic
391C2×C4#44811yescharacteristic
392C2×C4#44911yescharacteristic
393C2×C4#45111yescharacteristic
394C2×C4#45211yescharacteristic
395C2×C4#46011yescharacteristic
396C2×C4#46111yescharacteristic
397C2×C4#46211yescharacteristic
398C2×C4#46211yescharacteristic
399C2×C4#46311yescharacteristic
400C2×C4#46411yescharacteristic
401C2×C4#46411yescharacteristic
402C2×C4#46411yescharacteristic
403C2×C4#46511yescharacteristic
404C2×C4#46711yescharacteristic
405C2×C4#46811yescharacteristic
406C2×C4#47011yescharacteristic
407C2×C4#48711yescharacteristic
408C2×C4#48811yescharacteristic
409C2×C4#48911yescharacteristic
410C2×C4#49111yescharacteristic
411C8—82no—
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#192711yescharacteristic
428C22#192811yescharacteristic
429C22#203511yescharacteristic
430C22#203611yescharacteristic
431C22#207211yescharacteristic
432C22#207411yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4#187811yescharacteristic
445C4#188011yescharacteristic
446C4#190211yescharacteristic
447C4#190311yescharacteristic
448C4#193311yescharacteristic
449C4#193511yescharacteristic
450C4#204211yescharacteristic
451C4#204311yescharacteristic
452C4#205511yescharacteristic
453C4#205611yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#1115911yescharacteristic
461C2#1118511yescharacteristic
462C2#1119011yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ 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
23. C2×#333C4semidirect
24. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#213292
C2×#15502
#174862
#108562
#80012
C2×#12922
#78912