Finite Group Databasebeta

← #102166

Subgroups — #102166

Order 512.

545 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#10820C211yesmaximal, characteristic, complemented
3#9884C211yesmaximal, characteristic, complemented
4#9680C211yesmaximal, characteristic
5#9680C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic, complemented
7#8144C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic, complemented
9C22×C4×C8C2211yescharacteristic
10#2857C2211yescharacteristic
11#2856C2211yescharacteristic
12#1990C411yescharacteristic, complemented
13#1957C411yescharacteristic, complemented
14#1934—21no—
15#1934—21no—
16#1893—21no—
17#1893—21no—
18#1834C411yescharacteristic, complemented
19C4×C22⋊C8C2211yescharacteristic
20C4×C22⋊C8C2211yescharacteristic
21C2×#340—21no—
22C2×#340—21no—
23C2×#328—22no—
24C2×#327—22no—
25C2×#327—22no—
26C2×#322—21no—
27C4⋊#95C2211yescharacteristic
28C4⋊#95C2211yescharacteristic
29#1294—22no—
30#1273—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×#127C2×C411yescharacteristic
35C2×C22⋊Q8—21no—
36C2×#121—21no—
37C2×#121—21no—
38C2×#121—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C42C2311yescharacteristic
42#486—42no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×C22⋊C8D412yes—
48C2×C22⋊C8D412yes—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—42no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56#391—21no—
57#390—21no—
58#390—21no—
59#390—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#380—21no—
65#380—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70#340—24no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—42no—
76#336—21no—
77#336—21no—
78#328—44no—
79#327—44no—
80#327—44no—
81#322—42no—
82C22×Q8—21no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108#127—41no—
109#127—41no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112#121—41no—
113#121—41no—
114#121—41no—
115#121—41no—
116#121—42no—
117#121—42no—
118#121—42no—
119#121—42no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151#102—42no—
152#102—42no—
153#102—42no—
154#102—42no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167#95—41no—
168#95—41no—
169#95—42no—
170#95—42no—
171#95—42no—
172#95—42no—
173C24—41no—
174C24#12711yescharacteristic
175C2×Q8—42no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4—81no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4#13511yescharacteristic
228C22×C4#13511yescharacteristic
229C22×C4#13511yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4AGL(1,ℤ/8)12yes—
232C22×C4AGL(1,ℤ/8)12yes—
233C22×C4AGL(1,ℤ/8)12yes—
234C22×C4#13712yes—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8#9912yes—
256C2×C8#10412yes—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—82no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—41no—
288C42—41no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23#39011yescharacteristic
301C23#49511yescharacteristic
302C23#49511yescharacteristic
303C23#49611yescharacteristic
304C23#49611yescharacteristic
305C23#49611yescharacteristic
306C23#49711yescharacteristic
307Q8—162no—
308D4—81no—
309D4—81no—
310D4—81no—
311D4—81no—
312D4—162no—
313D4—162no—
314D4—161no—
315D4—161no—
316D4—161no—
317D4—161no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—161no—
356C2×C4—161no—
357C2×C4—161no—
358C2×C4—161no—
359C2×C4—161no—
360C2×C4—161no—
361C2×C4—161no—
362C2×C4—161no—
363C2×C4C2×#9911yescharacteristic
364C2×C4#41011yescharacteristic
365C2×C4#41611yescharacteristic
366C2×C4#41611yescharacteristic
367C2×C4C2×#10411yescharacteristic
368C2×C4#42111yescharacteristic
369C2×C4#44012yes—
370C2×C4#44212yes—
371C2×C4#46812yes—
372C2×C4#46912yes—
373C2×C4#48811yescharacteristic
374C2×C4#48811yescharacteristic
375C8—22no—
376C8—22no—
377C8—41no—
378C8—41no—
379C8—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22#171711yescharacteristic
401C22#171711yescharacteristic
402C22#171911yescharacteristic
403C22#172011yescharacteristic
404C22#195811yescharacteristic
405C22#196411yescharacteristic
406C22#197211yescharacteristic
407C22#197311yescharacteristic
408C22#197611yescharacteristic
409C22#197611yescharacteristic
410C22#198011yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—162no—
418C4—162no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#150011yescharacteristic
423C4#150111yescharacteristic
424C4#151611yescharacteristic
425C4#152111yescharacteristic
426C2—21no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—321no—
431C2#968111yescharacteristic
432C2#972111yescharacteristic
433C2#972211yescharacteristic
434C2#975711yescharacteristic
435C2#978511yescharacteristic
436C2#978611yescharacteristic
437C2#983811yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10820C2semidirect
3. #9884C2semidirect
6. #9679C2semidirect
7. #8144C2semidirect
8. C2×#486C4semidirect
12. #1990C4semidirect
13. #1957C4semidirect
18. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#330422
#108202
#98842
#96802
#96792
#81442