Finite Group Databasebeta

← #190208

Subgroups — #190208

Order 512.

1317 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#8760C218yesmaximal, complemented
5#8023C212yesmaximal, complemented
6C2×#1289C212yesmaximal, complemented
7C2×#518—21no—
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C22×C22⋊C8C2212yes—
11#2349C414yescomplemented
12#2349C2214yes—
13D4×C4⋊C4—22no—
14D4×C4⋊C4—22no—
15#2316—21no—
16#2315—21no—
17#2315—21no—
18#2285C412yescomplemented
19C2×#380C412yescomplemented
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1799—22no—
24#1799C2214yes—
25C2×#342—21no—
26#1459—22no—
27#1459C2214yes—
28C2×#324—21no—
29C2×#323—21no—
30#1294C2218yescomplemented
31#1289C2218yescomplemented
32C23×D4D411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36#518—44no—
37#518—44no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C412yes—
45C2×C4×D4D412yes—
46C2×C4×D4D412yes—
47C2×C4×D4C2312yes—
48C2×#117—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C2×#99—24no—
62C2×#99—44no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2314yes—
69#389—24no—
70#389—24no—
71#385—24no—
72#385—24no—
73#385—24no—
74#385—24no—
75#384—22no—
76#382C2×C414yescomplemented
77#380—22no—
78#380—22no—
79#380—22no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C412yes—
90#380C2×C414yescomplemented
91#380C2314yes—
92#380C2312yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yescomplemented
95C2×#95—22no—
96C2×#95C2×C412yes—
97C2×#95C2×C414yescomplemented
98C2×#95C2312yes—
99#342—24no—
100#324—24no—
101#323—24no—
102C25C22⋊C411yescharacteristic
103C25C2×D411yescharacteristic
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—24no—
107C22×D4C22⋊C412yes—
108C22×D4D814yes—
109C22×D4SD1614yes—
110C22×D4C2×D412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22×C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136C22×C8—24no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—84no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—416no—
149C4×D4—48no—
150C4×D4—48no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—24no—
196C2×C42C22⋊C412yes—
197C2×C42C22×C412yes—
198#99—48no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203#95—22no—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—24no—
208#95—42no—
209#95C22×C414yes—
210#95C22×C414yes—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—81no—
215C24—81no—
216C24—81no—
217C24—81no—
218C24#10214yes—
219C24C2×C22⋊C412yes—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24#12312yes—
223C24#12312yes—
224C24C2×D812yes—
225C24C2×SD1612yes—
226C24#13714yes—
227C24C22×D411yescharacteristic
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—28no—
235C2×D4—28no—
236C2×D4#10218yes—
237C2×D4C2×D814yes—
238C2×D4C2×SD1614yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—81no—
273C22×C4—82no—
274C22×C4—81no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—84no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4#10214yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4#11711yescharacteristic
291C22×C4#11711yescharacteristic
292C22×C4#11711yescharacteristic
293C22×C4#11711yescharacteristic
294C22×C4#11712yes—
295C22×C4#11712yes—
296C22×C4#12312yes—
297C22×C4#12312yes—
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4112yes—
306C2×C8—44no—
307C2×C8—84no—
308C2×C8—84no—
309C2×C8—84no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—84no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—48no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—82no—
360C23—82no—
361C23—84no—
362C23—84no—
363C23—84no—
364C23#38112yes—
365C23#38111yescharacteristic
366C23#38111yescharacteristic
367C23C2×#10214yes—
368C23C2×#10212yes—
369C23C2×#10212yes—
370C23#41912yes—
371C23#41912yes—
372C23#48014yes—
373C23#48014yes—
374C23#48314yes—
375C23#48314yes—
376C23C22×C22⋊C411yescharacteristic
377C23C2×#12312yes—
378C23C22×D811yescharacteristic
379C23C22×SD1611yescharacteristic
380C23C2×#13712yes—
381D4—28no—
382D4—28no—
383D4—216no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—162no—
428C2×C4—162no—
429C2×C4—164no—
430C2×C4—168no—
431C2×C4—168no—
432C2×C4#38111yescharacteristic
433C2×C4#38112yes—
434C2×C4#38111yescharacteristic
435C2×C4C2×#10214yes—
436C2×C4C2×#11711yescharacteristic
437C2×C4C2×#11711yescharacteristic
438C2×C4C2×#12312yes—
439C8—84no—
440C22—21no—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—24no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—84no—
467C22—84no—
468C22#182312yes—
469C22#182312yes—
470C22#182314yes—
471C22C2×#38111yescharacteristic
472C22C22×#10211yescharacteristic
473C22C2×#41911yescharacteristic
474C22C2×#48012yes—
475C22C2×#48312yes—
476C4—24no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—84no—
485C4—84no—
486C4—164no—
487C4#182812yes—
488C2—21no—
489C2—22no—
490C2—21no—
491C2—22no—
492C2—22no—
493C2—22no—
494C2—24no—
495C2—44no—
496C2C2×#182311yescharacteristic
497C2#2070811yescharacteristic
498C2C2×#182811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8760C2semidirect
5. #8023C2semidirect
6. C2×#1289C2semidirect
11. #2349C4semidirect
18. #2285C4semidirect
19. C2×#380C4semidirect
30. #1294C22semidirect
31. #1289C22semidirect
76. #382C2×C4semidirect
90. #380C2×C4semidirect
94. C4×C4⋊C4C2×C4semidirect
97. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#17992
C2×#14592
#87602
#80232
C2×#12892