Finite Group Databasebeta

← #117646

Subgroups — #117646

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21959C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21306C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#10837C211yesmaximal, characteristic
6#7981C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#1987C412yescomplemented
16#1986—22no—
17#1986—22no—
18#1981C412yescomplemented
19#1948—22no—
20#1948—22no—
21#1892—22no—
22#1807—22no—
23#1803—22no—
24#1733—22no—
25C2×H(3,ℤ/4)—22no—
26#1520—22no—
27#1518—22no—
28#1503—22no—
29#1289—22no—
30#1283—22no—
31C22×M16D412yes—
32C23×C8D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533D412yes—
40#533Q812yes—
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—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57C2×#106—24no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2×C412yes—
64C2×C4⋊C8D412yes—
65C2×#103—24no—
66C2×C22⋊C8D412yes—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#382—21no—
78#382—21no—
79#382—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92#340—24no—
93#340—24no—
94#339—28no—
95H(3,ℤ/4)—44no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16SD1614yes—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—82no—
113C22×C8C22⋊C414yes—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C4⋊C412yes—
121C4⋊Q8C4⋊C412yes—
122#125—22no—
123#125—22no—
124#125—41no—
125#125—41no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142#117—22no—
143#117—22no—
144#117—41no—
145#117—41no—
146#117—42no—
147#117—42no—
148#117—42no—
149#117—42no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—41no—
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187#106—84no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194#103—84no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#97—41no—
198#97—41no—
199C4×C8—41no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8—41no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—41no—
207#95—41no—
208#95—41no—
209C24#12711yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4C2×SD1612yes—
244C22×C4#13512yes—
245C22×C4#13712yes—
246C22×C4#13712yes—
247M16—24no—
248M16—24no—
249M16—42no—
250M16—42no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—44no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—82no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—42no—
293C42—42no—
294C42—44no—
295C42—44no—
296C42—44no—
297C42C2×C4⋊C411yescharacteristic
298C42#11711yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42#12311yescharacteristic
308C42#12311yescharacteristic
309C42#12411yescharacteristic
310C42#12411yescharacteristic
311C42#12611yescharacteristic
312C42#12611yescharacteristic
313C23—41no—
314C23—41no—
315C23#39011yescharacteristic
316C23#39211yescharacteristic
317C23#39911yescharacteristic
318C23#49211yescharacteristic
319C23#49511yescharacteristic
320C23#49711yescharacteristic
321C23#49711yescharacteristic
322Q8—44no—
323Q8—82no—
324Q8—82no—
325Q8—81no—
326Q8—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#38011yescharacteristic
356C2×C4#38511yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38611yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#39311yescharacteristic
362C2×C4#39411yescharacteristic
363C2×C4#39411yescharacteristic
364C2×C4#39611yescharacteristic
365C2×C4#39611yescharacteristic
366C2×C4#39711yescharacteristic
367C2×C4C4×SD1612yes—
368C2×C4#44112yes—
369C2×C4#44812yes—
370C2×C4#45212yes—
371C2×C4#46112yes—
372C2×C4#46412yes—
373C2×C4#46512yes—
374C2×C4#47012yes—
375C2×C4#47512yes—
376C2×C4#47912yes—
377C2×C4#48312yes—
378C2×C4#48512yes—
379C8—44no—
380C8—44no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#143911yescharacteristic
393C22#196811yescharacteristic
394C22#197111yescharacteristic
395C22#201311yescharacteristic
396C22#201511yescharacteristic
397C22#209611yescharacteristic
398C22#209711yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4—162no—
410C4—162no—
411C4#186512yes—
412C4#191111yescharacteristic
413C4#191211yescharacteristic
414C4#192712yes—
415C4#203512yes—
416C4#206612yes—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2#1107211yescharacteristic
424C2#1112211yescharacteristic
425C2#1113211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
15. #1987C4semidirect
18. #1981C4semidirect

Maximal subgroups

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

subgroupindex
#219592
#218852
#213062
#175752
#108372
#79812
C2×#12832