Finite Group Databasebeta

← #102155

Subgroups — #102155

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#10861C211yesmaximal, characteristic, complemented
3#9884C211yesmaximal, characteristic, complemented
4#9680C211yesmaximal, characteristic
5#9680C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic, complemented
7#8145C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic, complemented
9C22×#106—21no—
10C22×C4×C8C2211yescharacteristic
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#1990C411yescharacteristic, complemented
14#1957C411yescharacteristic, complemented
15#1934—21no—
16#1934—21no—
17#1893—21no—
18#1893—21no—
19#1834C411yescharacteristic, complemented
20C4×C22⋊C8C2211yescharacteristic
21C4×C22⋊C8C2211yescharacteristic
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×#106—24no—
44C2×#106—42no—
45C2×#106—42no—
46C2×#102—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8D412yes—
51C2×C22⋊C8D412yes—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—42no—
55C2×C4×C8C2×C411yescharacteristic
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59#391—21no—
60#390—21no—
61#390—21no—
62#390—21no—
63#389—21no—
64#389—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#380—21no—
68#380—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—42no—
78#336—21no—
79#336—21no—
80#328—44no—
81#327—44no—
82#327—44no—
83#322—42no—
84C22×Q8—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110#127—41no—
111#127—41no—
112C22⋊Q8—42no—
113C22⋊Q8—42no—
114#121—42no—
115#121—42no—
116#121—42no—
117#121—41no—
118#121—41no—
119#121—42no—
120#121—41no—
121#121—41no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
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—21no—
131C2×C4⋊C4—21no—
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×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42C22×C411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153#106—48no—
154#102—42no—
155#102—42no—
156#102—42no—
157#102—42no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8SD1614yes—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—41no—
169C4×C8—41no—
170#95—41no—
171#95—41no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—42no—
176C24—41no—
177C24#12711yescharacteristic
178C2×Q8—42no—
179C2×Q8—81no—
180C2×Q8—81no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—81no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
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×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4C2×SD1612yes—
231C22×C4#13512yes—
232C22×C4AGL(1,ℤ/8)11yescharacteristic
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4AGL(1,ℤ/8)12yes—
235C22×C4AGL(1,ℤ/8)12yes—
236C22×C4#13711yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8#9912yes—
259C2×C8#10412yes—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—41no—
291C42—41no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23#39011yescharacteristic
304C23#49211yescharacteristic
305C23#49511yescharacteristic
306C23#49611yescharacteristic
307C23#49611yescharacteristic
308C23#49611yescharacteristic
309C23#49711yescharacteristic
310Q8—162no—
311D4—81no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—162no—
316D4—162no—
317D4—161no—
318D4—161no—
319D4—161no—
320D4—161no—
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—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—161no—
359C2×C4—161no—
360C2×C4—161no—
361C2×C4—161no—
362C2×C4—161no—
363C2×C4—161no—
364C2×C4—161no—
365C2×C4—161no—
366C2×C4C2×#9911yescharacteristic
367C2×C4#41011yescharacteristic
368C2×C4#41711yescharacteristic
369C2×C4#41711yescharacteristic
370C2×C4C2×#10411yescharacteristic
371C2×C4#42111yescharacteristic
372C2×C4C4×SD1612yes—
373C2×C4#44212yes—
374C2×C4#46512yes—
375C2×C4#46912yes—
376C2×C4#48911yescharacteristic
377C2×C4#48911yescharacteristic
378C8—22no—
379C8—22no—
380C8—44no—
381C8—41no—
382C8—41no—
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22#171411yescharacteristic
404C22#171511yescharacteristic
405C22#171711yescharacteristic
406C22#172011yescharacteristic
407C22#195911yescharacteristic
408C22#196411yescharacteristic
409C22#196811yescharacteristic
410C22#197211yescharacteristic
411C22#197611yescharacteristic
412C22#197611yescharacteristic
413C22#198011yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—162no—
421C4—162no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#150411yescharacteristic
426C4#150911yescharacteristic
427C4#151211yescharacteristic
428C4#151311yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—321no—
434C2#968211yescharacteristic
435C2#971711yescharacteristic
436C2#972111yescharacteristic
437C2#975811yescharacteristic
438C2#978111yescharacteristic
439C2#978511yescharacteristic
440C2#982711yescharacteristic
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. #10861C2semidirect
3. #9884C2semidirect
6. #9679C2semidirect
7. #8145C2semidirect
8. C2×#486C4semidirect
13. #1990C4semidirect
14. #1957C4semidirect
19. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#330422
#108612
#98842
#96802
#96792
#81452