Finite Group Databasebeta

← #193203

Subgroups — #193203

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#8926C214yesmaximal, complemented
7#8922C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2365—22no—
17#2365—22no—
18#2365—22no—
19#2365C2214yes—
20C2×#391—21no—
21C2×#389—21no—
22C2×#387—22no—
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1987—22no—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1947C2212yes—
34#1947C2212yes—
35#1852—21no—
36#1852—21no—
37#1824—22no—
38C4×#107—22no—
39#1736—21no—
40#1549C418yescomplemented
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—21no—
45C22×M16—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C414yes—
51C2×C4⋊Q8D412yes—
52C2×C4⋊Q8Q814yes—
53C2×C4⋊Q8C2312yes—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—48no—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—22no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73#428—44no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78#423—22no—
79#423C2×C414yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#391—24no—
89#391—24no—
90#389—24no—
91#389—24no—
92#387—28no—
93#387—28no—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#382—21no—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4C2312yes—
110C4×C4⋊C4C2312yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115#340—24no—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#332C2×C4116yescomplemented
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8—44no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8C4⋊C418yes—
150C4⋊Q8C22×C414yes—
151C4⋊Q8C2×Q814yes—
152C4×Q8—416no—
153C4×Q8—48no—
154C4×Q8—48no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4C4⋊C418yes—
181C2×C4⋊C4Q1614yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—41no—
188C2×C42—28no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—48no—
194C2×C42—44no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—41no—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q812yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42C2411yescharacteristic
206#107—44no—
207#107—44no—
208#107—44no—
209#106—44no—
210C4⋊C8—24no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214C4⋊C8C22×C414yes—
215C4⋊C8C22×C414yes—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C4×C8—24no—
219C4×C8—22no—
220C4×C8—22no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—41no—
228#95—41no—
229C24#12011yescharacteristic
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—28no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—416no—
236C2×Q8—416no—
237C2×Q8—44no—
238C2×Q8—44no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C4⋊C412yes—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4C22⋊Q812yes—
272C22×C4C22⋊Q812yes—
273C22×C4#12312yes—
274C22×C4C2×Q1612yes—
275C22×C4#13512yes—
276C22×C4AGL(1,ℤ/8)12yes—
277C22×C4#13712yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—48no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—416no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4C2×C4⋊C418yes—
316C4⋊C4C2×Q1614yes—
317C4⋊C4AGL(1,ℤ/8)14yes—
318C22⋊C4—82no—
319C42—44no—
320C42—416no—
321C42—42no—
322C42—42no—
323C42—44no—
324C42—44no—
325C42—82no—
326C42C2×C4⋊C412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12312yes—
332C42C23×C411yescharacteristic
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#38512yes—
341C23#44912yes—
342C23#45112yes—
343C23C2×#12011yescharacteristic
344Q8—416no—
345Q8—48no—
346Q8—48no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—416no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#38512yes—
384C2×C4#38512yes—
385C2×C4#38512yes—
386C2×C4C4×Q1614yes—
387C2×C4#44014yes—
388C2×C4#44912yes—
389C2×C4#45112yes—
390C2×C4#47714yes—
391C2×C4#47914yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C22×Q1611yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)12yes—
402C2×C4C2×#13711yescharacteristic
403C2×C4#57512yes—
404C8—44no—
405C8—44no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#186314yes—
414C22C2×#38511yescharacteristic
415C22C2×#45111yescharacteristic
416C22C2×#44911yescharacteristic
417C22#300311yescharacteristic
418C22#300511yescharacteristic
419C22#302011yescharacteristic
420C22#302011yescharacteristic
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—44no—
427C4—48no—
428C4—82no—
429C4—82no—
430C4—161no—
431C4—161no—
432C4#186314yes—
433C4C2×C4×Q1611yescharacteristic
434C4C2×#44011yescharacteristic
435C4C4×AGL(1,ℤ/8)12yes—
436C4C2×#47711yescharacteristic
437C4C2×#47911yescharacteristic
438C4#307712yes—
439C2—24no—
440C2#1875911yescharacteristic
441C2#1876211yescharacteristic
442C2#1877911yescharacteristic
443C2#1878211yescharacteristic
444C2#2085711yescharacteristic
445C2C2×#186311yescharacteristic
446C2#2087311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8922C2semidirect
40. #1549C4semidirect
121. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
#218342
#213582
#211702
C2×#15492
#89262
#89222
#79012
#78922