Finite Group Databasebeta

← #188523

Subgroups — #188523

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#13966C211yesmaximal, characteristic
6#8576C214yesmaximal, complemented
7#8574C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9C2×#1294C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2368—22no—
14#2360C414yescomplemented
15#2345—22no—
16#2345C412yescomplemented
17Q8×C22⋊C4C412yescomplemented
18#2329—22no—
19#2315—21no—
20C4×C22⋊Q8C2214yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2273—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1850—22no—
27#1850—22no—
28#1797C2214yes—
29#1520—24no—
30C2×#328—24no—
31C2×#328C2214yes—
32C2×#328C2212yes—
33C2×#328C2212yes—
34#1442—22no—
35#1434C2212yes—
36#1434C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C23×Q8C2×C411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8—21no—
46C23×C8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D414yes—
53C2×C22⋊Q8Q814yes—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8C2×C414yescomplemented
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—24no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75#391—24no—
76#391C2×C414yescomplemented
77#389—24no—
78#389C2×C414yescomplemented
79#387—24no—
80#387—24no—
81#387—24no—
82#385—24no—
83#385—24no—
84#380—21no—
85#380—21no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380C2×C412yes—
91#380C2312yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4C2×C414yescomplemented
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101C4×C22⋊C4D412yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111C2×#95C2312yes—
112C2×#95C2311yescharacteristic
113C2×#95C2311yescharacteristic
114#336—24no—
115#328—28no—
116#328—24no—
117#328—24no—
118C25C2×D411yescharacteristic
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—82no—
122C22×Q8C4⋊C414yes—
123C22×Q8C22×C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×Q812yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—82no—
153C22×C8—82no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8C4⋊C418yes—
162C22⋊Q8C2×D414yes—
163C22⋊Q8C2×Q814yes—
164C4×Q8—84no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C4⋊C414yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C414yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—82no—
202C2×C42C22×C414yes—
203C2×C42C2×D414yes—
204C4⋊C8—84no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211#95—22no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—82no—
227#95—82no—
228#95C22⋊C414yes—
229#95C22⋊C414yes—
230C24—42no—
231C24—42no—
232C24C2×C22⋊C411yescharacteristic
233C24C2×C22⋊C411yescharacteristic
234C24#12112yes—
235C24C22⋊Q812yes—
236C24#13512yes—
237C24#13512yes—
238C24#13712yes—
239C24#13712yes—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8C2×C4⋊C414yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4C2×C22⋊C414yes—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C4⋊C412yes—
303C22×C4C2×C4⋊C412yes—
304C22×C4#11712yes—
305C22×C4#11712yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—84no—
323C2×C8—84no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—44no—
330C4⋊C4—84no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4C2×C4⋊C414yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—84no—
346C42—44no—
347C42—44no—
348C42—84no—
349C42—84no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23#38012yes—
365C23#38012yes—
366C23#41612yes—
367C23#41612yes—
368C23#41912yes—
369C23#41912yes—
370C23#46414yes—
371C23#46414yes—
372C23#47914yes—
373C23#47914yes—
374C23C22×C22⋊C411yescharacteristic
375C23C2×#12111yescharacteristic
376C23C2×C22⋊Q811yescharacteristic
377C23C2×#13511yescharacteristic
378C23C2×#13511yescharacteristic
379C23C2×#13711yescharacteristic
380C23C2×#13711yescharacteristic
381Q8—44no—
382Q8—44no—
383Q8—84no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—44no—
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—82no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—816no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—164no—
440C2×C4#38012yes—
441C2×C4#38012yes—
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×#11711yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C8—82no—
447C8—82no—
448C22—21no—
449C22—22no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—22no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—24no—
459C22—24no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22#180514yes—
469C22#180514yes—
470C22C2×#38011yescharacteristic
471C22#288312yes—
472C22C2×#41611yescharacteristic
473C22C2×#41911yescharacteristic
474C22#290112yes—
475C22C2×#46411yescharacteristic
476C22C2×#46411yescharacteristic
477C22C2×#47911yescharacteristic
478C22C2×#47911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—22no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—84no—
490C4—84no—
491C4—161no—
492C4—161no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—24no—
498C2—24no—
499C2#1761512yes—
500C2#1762112yes—
501C2#2063311yescharacteristic
502C2C2×#180511yescharacteristic
503C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8576C2semidirect
7. #8574C2semidirect
14. #2360C4semidirect
16. #2345C4semidirect
17. Q8×C22⋊C4C4semidirect
55. C2×C4×Q8C2×C4semidirect
76. #391C2×C4semidirect
78. #389C2×C4semidirect
93. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342122
#219302
C2×#15202
C22×#3282
#139662
#85762
#85742
#80232
C2×#12942