Finite Group Databasebeta

← #127624

Subgroups — #127624

Order 512.

675 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20632C211yesmaximal, characteristic
2#20629C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#7252C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#431—21no—
7C2×#423C411yescharacteristic, complemented
8C22×C4⋊C8C411yescharacteristic, complemented
9C2×#410—21no—
10C2×#408—21no—
11#2285—21no—
12#1799—21no—
13#1799—21no—
14#1799C412yes—
15#1799C412yes—
16#1798C412yescomplemented
17#1798C2212yes—
18#1797—21no—
19#1797—21no—
20#1740C2212yes—
21#1729—21no—
22#1461C412yescomplemented
23#1460—21no—
24#1459—21no—
25#1459C412yescomplemented
26#1458—21no—
27#1458C2212yes—
28C22×#41D411yescharacteristic
29C22×M16—21no—
30C23×C8—21no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8Q811yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117—22no—
44C2×#117C2×C411yescharacteristic
45C2×#117D411yescharacteristic
46C2×#117C2311yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50#431—22no—
51#431—22no—
52#431—41no—
53#431—41no—
54#423—41no—
55#423—41no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8C2×C412yes—
63#410—42no—
64#410—42no—
65C2×#99—22no—
66C2×#99—22no—
67#408—42no—
68#408—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C4×C8—41no—
73#384—41no—
74#382—41no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—41no—
78C4×C4⋊C4—41no—
79#342—22no—
80#342C2×C414yes—
81#336—22no—
82#336—22no—
83#324—22no—
84#324C2×C414yes—
85#323—22no—
86#323—22no—
87C2×#41—41no—
88C2×#41—41no—
89C2×#41—41no—
90C2×#41—41no—
91C2×#41D812yes—
92C2×#41SD1612yes—
93C2×#41SD1612yes—
94C2×#41Q1612yes—
95C22×Q8C4⋊C411yescharacteristic
96C22×D4C22⋊C411yescharacteristic
97C22×D4C22⋊C411yescharacteristic
98C22×D4C4⋊C411yescharacteristic
99C23×C4C4211yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C4⋊C411yescharacteristic
106C23×C4C4⋊C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—42no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—41no—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C4211yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C4⋊C411yescharacteristic
149C2×C4⋊C4C4⋊C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C2×Q811yescharacteristic
152C2×C22⋊C4—24no—
153C2×C22⋊C4C4211yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C22×C411yescharacteristic
158C2×C22⋊C4C2×D411yescharacteristic
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C4211yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C4⋊C411yescharacteristic
177C2×C42C4⋊C411yescharacteristic
178C2×C42C4⋊C411yescharacteristic
179C2×C42C4⋊C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C4⋊C8—42no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8—42no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190#99—44no—
191#99—44no—
192C22⋊C8—24no—
193C22⋊C8—42no—
194C22⋊C8—44no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C4×C8—42no—
198C4×C8—42no—
199#95—42no—
200#95—42no—
201C24#9511yescharacteristic
202C24#9511yescharacteristic
203C24C2×C22⋊C411yescharacteristic
204#41—42no—
205#41—42no—
206#41—42no—
207#41—42no—
208#41—42no—
209#41—42no—
210#41—42no—
211#41—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8#10612yes—
215C2×Q8#10712yes—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4#10212yes—
223C2×D4#10212yes—
224C2×D4#10312yes—
225C2×D4#10312yes—
226C2×D4#10612yes—
227C2×D4#10712yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4#9511yescharacteristic
260C22×C4#9511yescharacteristic
261C22×C4#9511yescharacteristic
262C22×C4#9511yescharacteristic
263C22×C4#9511yescharacteristic
264C22×C4#9511yescharacteristic
265C22×C4#10212yes—
266C22×C4#10212yes—
267C22×C4#10312yes—
268C22×C4#10312yes—
269C22×C4C2×C4211yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C4⋊C411yescharacteristic
273C22×C4C2×C4⋊C411yescharacteristic
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×SD1611yescharacteristic
277C22×C4C2×SD1611yescharacteristic
278C22×C4C2×Q1611yescharacteristic
279M16—84no—
280C2×C8—44no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—84no—
288C2×C8—84no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—82no—
326C42—82no—
327C42#9911yescharacteristic
328C42#9911yescharacteristic
329C42#9911yescharacteristic
330C42#9911yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23#34012yes—
344C23#34012yes—
345C23C2×#9511yescharacteristic
346C23C2×#10211yescharacteristic
347C23C2×#10311yescharacteristic
348Q8—42no—
349Q8—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—42no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4#34012yes—
408C2×C4#34012yes—
409C2×C4#34211yescharacteristic
410C2×C4#34211yescharacteristic
411C2×C4#34211yescharacteristic
412C2×C4#34211yescharacteristic
413C2×C4C2×#9911yescharacteristic
414C2×C4C2×#9911yescharacteristic
415C2×C4C2×#10211yescharacteristic
416C2×C4C2×#10311yescharacteristic
417C2×C4#42111yescharacteristic
418C2×C4#42111yescharacteristic
419C2×C4C2×#10611yescharacteristic
420C2×C4C2×#10711yescharacteristic
421C8—84no—
422C8—82no—
423C8—82no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—81no—
442C22—81no—
443C22#173311yescharacteristic
444C22C2×#34011yescharacteristic
445C22C2×#34211yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—164no—
460C4#139512yes—
461C4#151911yescharacteristic
462C4#151911yescharacteristic
463C4#152011yescharacteristic
464C4#152011yescharacteristic
465C2—21no—
466C2—21no—
467C2—41no—
468C2—42no—
469C2—42no—
470C2—41no—
471C2#802311yescharacteristic
472C2#1257711yescharacteristic
473C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7252C2semidirect
7. C2×#423C4semidirect
8. C22×C4⋊C8C4semidirect
16. #1798C4semidirect
22. #1461C4semidirect
25. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#206322
#206292
#173492
#72522