Finite Group Databasebeta

← #117316

Subgroups — #117316

Order 512.

669 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21359C211yesmaximal, characteristic
2#21332C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#10852C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7866C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1996—22no—
16#1996—22no—
17#1948—22no—
18#1853—22no—
19C4×#106—22no—
20#1740—22no—
21C2×#339—22no—
22#1550—22no—
23#1549C412yescomplemented
24#1548C412yescomplemented
25#1542—22no—
26#1503—22no—
27#1503—22no—
28C2×#328—22no—
29#1286—22no—
30#1276—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8D412yes—
45C2×C4×Q8—21no—
46C2×C4×Q8Q812yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#431D412yes—
60#431D412yes—
61C2×#106—24no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423C2×C412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#384—21no—
75#382—21no—
76#380—21no—
77#380D412yes—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88#376—21no—
89C2×#95—21no—
90C43—21no—
91#342—28no—
92#340—24no—
93#339—44no—
94#333—24no—
95#332—24no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8C4⋊C412yes—
99C23×C4—41no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—42no—
106C2×M16—42no—
107C22×C8—22no—
108C22×C8—41no—
109C22×C8—41no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118#125—41no—
119#125—41no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—41no—
142#117—41no—
143#117—41no—
144#117—41no—
145#117—42no—
146#117—42no—
147#117SD1614yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C4⋊C412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185#106—48no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8C22⋊C414yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24#12011yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—82no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12711yescharacteristic
243C22×C4C2×SD1612yes—
244C22×C4#13512yes—
245C22×C4#13712yes—
246C22×C4#13712yes—
247M16—44no—
248M16—81no—
249M16—81no—
250C2×C8—24no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—82no—
255C2×C8—82no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—82no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C22⋊C4—24no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C42—24no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—44no—
287C42—82no—
288C42—82no—
289C42—82no—
290C42—81no—
291C42—81no—
292C42C2×C22⋊C411yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42#12311yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C42#12611yescharacteristic
307C42#12611yescharacteristic
308C23—41no—
309C23—41no—
310C23#38511yescharacteristic
311C23#39211yescharacteristic
312C23#39411yescharacteristic
313C23#44812yes—
314C23#45212yes—
315Q8—82no—
316Q8—82no—
317Q8—82no—
318Q8—82no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#38011yescharacteristic
346C2×C4#38511yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#39011yescharacteristic
351C2×C4#39311yescharacteristic
352C2×C4#39411yescharacteristic
353C2×C4#39611yescharacteristic
354C2×C4#39611yescharacteristic
355C2×C4#39711yescharacteristic
356C2×C4#39911yescharacteristic
357C2×C4C4×SD1612yes—
358C2×C4#44112yes—
359C2×C4#46112yes—
360C2×C4#46412yes—
361C2×C4#46512yes—
362C2×C4#47012yes—
363C2×C4#47512yes—
364C2×C4#47912yes—
365C2×C4#48312yes—
366C2×C4#48512yes—
367C2×C4#49211yescharacteristic
368C2×C4#49511yescharacteristic
369C2×C4#49711yescharacteristic
370C2×C4#49711yescharacteristic
371C8—44no—
372C8—82no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—42no—
381C22—42no—
382C22#143911yescharacteristic
383C22#186512yes—
384C22#201311yescharacteristic
385C22#201511yescharacteristic
386C22#203512yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—84no—
395C4—84no—
396C4—82no—
397C4—82no—
398C4—162no—
399C4#190612yes—
400C4#192712yes—
401C4#196811yescharacteristic
402C4#197111yescharacteristic
403C4#206612yes—
404C4#209611yescharacteristic
405C4#209711yescharacteristic
406C2—21no—
407C2—22no—
408C2—21no—
409C2—22no—
410C2#1106011yescharacteristic
411C2#1112211yescharacteristic
412C2#1113811yescharacteristic
subgroupG/H
centerC2×C4#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#120
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
23. #1549C4semidirect
24. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#213592
#213322
#177082
C2×#15032
#108522
#79572
#78662