Finite Group Databasebeta

← #98231

Subgroups — #98231

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#9898C211yesmaximal, characteristic, complemented
3#9692C211yesmaximal, characteristic, complemented
4#8587C211yesmaximal, characteristic, complemented
5#8563C211yesmaximal, characteristic
6#8563C211yesmaximal, characteristic
7#7571C211yesmaximal, characteristic, complemented
8C2×#533—21no—
9#2868C2211yescharacteristic
10#2857—21no—
11#2552—21no—
12#2542—21no—
13#2538C412yes—
14#2441—22no—
15#2429—22no—
16#2423—22no—
17#2365—21no—
18#2354—21no—
19Q8×C4⋊C4C412yes—
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1842C2211yescharacteristic
23#1842C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#333—21no—
27C2×#332—22no—
28C2×D4⋊C8—21no—
29C2×D4⋊C8—21no—
30#1296—21no—
31#1296—22no—
32#1279—21no—
33#1279—21no—
34#1272—22no—
35#533—42no—
36#533—42no—
37#533—42no—
38#533—42no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D412yes—
43C2×#125—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8C2×C412yes—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×#117—22no—
54C22×C42—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65#399—24no—
66#398—24no—
67#396—24no—
68#391—22no—
69#391D412yes—
70#390—22no—
71#389—22no—
72#389—22no—
73#389—24no—
74#389—22no—
75#389—21no—
76#389—21no—
77#388—24no—
78#387—24no—
79#384—22no—
80#384—21no—
81#384—21no—
82#384C2×C412yes—
83#384D411yescharacteristic
84#384D412yes—
85#380—22no—
86#380—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D411yescharacteristic
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C43C2311yescharacteristic
97#336—21no—
98#336—21no—
99#333—42no—
100#332—44no—
101D4⋊C8—42no—
102D4⋊C8—42no—
103C22×Q8—22no—
104C22×Q8C22⋊C412yes—
105C22×D4C22⋊C411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8D812yes—
126C4⋊Q8SD1612yes—
127#125—44no—
128#125—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C22⋊Q8—82no—
134C22⋊Q8—82no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4SD1612yes—
150C4×D4Q1612yes—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C414yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—24no—
176C2×C42—24no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—24no—
211#95—24no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217C24—21no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8#10214yes—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4#10312yes—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—81no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—81no—
274C2×C8—81no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—84no—
303C4⋊C4#10312yes—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—48no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—84no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—41no—
318C42—48no—
319C42—42no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42C2×D811yescharacteristic
328C42C2×SD1611yescharacteristic
329C42C2×SD1611yescharacteristic
330C42C2×Q1611yescharacteristic
331C42#13511yescharacteristic
332C42#13511yescharacteristic
333C42AGL(1,ℤ/8)11yescharacteristic
334C42AGL(1,ℤ/8)11yescharacteristic
335C42AGL(1,ℤ/8)11yescharacteristic
336C42#13711yescharacteristic
337C42#13711yescharacteristic
338C42#13711yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#37911yescharacteristic
347Q8—44no—
348Q8—44no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353D4—22no—
354D4—22no—
355D4—41no—
356D4—41no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4#41311yescharacteristic
395C2×C4#41311yescharacteristic
396C2×C4C2×#10211yescharacteristic
397C2×C4C2×#10311yescharacteristic
398C2×C4#41611yescharacteristic
399C2×C4#41711yescharacteristic
400C2×C4#41811yescharacteristic
401C2×C4#41911yescharacteristic
402C2×C4#44711yescharacteristic
403C2×C4#44711yescharacteristic
404C2×C4#44811yescharacteristic
405C2×C4#44811yescharacteristic
406C2×C4#44911yescharacteristic
407C2×C4#44911yescharacteristic
408C2×C4#45011yescharacteristic
409C2×C4#45011yescharacteristic
410C2×C4#45111yescharacteristic
411C2×C4#45111yescharacteristic
412C2×C4#45211yescharacteristic
413C2×C4#45211yescharacteristic
414C2×C4#45311yescharacteristic
415C2×C4#45411yescharacteristic
416C2×C4#45511yescharacteristic
417C2×C4#45611yescharacteristic
418C8—82no—
419C8—82no—
420C8—161no—
421C8—161no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#178511yescharacteristic
430C22#178611yescharacteristic
431C22#178711yescharacteristic
432C22#178811yescharacteristic
433C22#179211yescharacteristic
434C22#179311yescharacteristic
435C22#179611yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—44no—
444C4—42no—
445C4—82no—
446C4—81no—
447C4—81no—
448C4—84no—
449C4#154211yescharacteristic
450C4#155011yescharacteristic
451C4#155411yescharacteristic
452C4#155911yescharacteristic
453C4#162111yescharacteristic
454C4#162711yescharacteristic
455C4#163111yescharacteristic
456C4#163711yescharacteristic
457C4#164111yescharacteristic
458C4#164211yescharacteristic
459C4#164911yescharacteristic
460C4#165111yescharacteristic
461C2—82no—
462C2#834311yescharacteristic
463C2#838711yescharacteristic
464C2#842511yescharacteristic
465C2#842711yescharacteristic
466C2#844411yescharacteristic
467C2#844711yescharacteristic
468C2#847011yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9898C2semidirect
3. #9692C2semidirect
4. #8587C2semidirect
7. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#98982
#96922
#85872
#85632
#75712