Finite Group Databasebeta

← #117470

Subgroups — #117470

Order 512.

754 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3C2×#1556C211yesmaximal, characteristic
4#14274C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8033C211yesmaximal, characteristic
7#8022C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#391—21no—
13C2×#391—21no—
14#2285C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#376C2211yescharacteristic
17#1993—22no—
18#1990—21no—
19#1990—21no—
20#1949—22no—
21#1853—21no—
22#1852—21no—
23#1847—21no—
24#1836—21no—
25#1736—22no—
26#1556—22no—
27#1556C414yescomplemented
28C2×#332—22no—
29#1445—22no—
30#1445—22no—
31#1300—22no—
32#1295—22no—
33#1294—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#391—44no—
78#391—44no—
79#384D411yescharacteristic
80#384D411yescharacteristic
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90#376—21no—
91#376—21no—
92#376—21no—
93#376—21no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#332—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—41no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—42no—
124C2×M16—41no—
125C2×M16—41no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145#117—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C4⋊C414yes—
170C2×C22⋊C4—41no—
171C2×C42—24no—
172C2×C42—42no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188#107—82no—
189#106—82no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8C22⋊C414yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—41no—
204C4×C8—41no—
205C4×C8—41no—
206#95—22no—
207#95—22no—
208#95—48no—
209#95—48no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218C24#12011yescharacteristic
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12411yescharacteristic
282C22×C4#12711yescharacteristic
283M16—44no—
284M16—44no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—81no—
293C2×C8—81no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—82no—
313C42—44no—
314C42—44no—
315C42—82no—
316C42C2×C22⋊C411yescharacteristic
317C42#11711yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12411yescharacteristic
329C42#12411yescharacteristic
330C42#12611yescharacteristic
331C42#12611yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#38511yescharacteristic
337C23#39211yescharacteristic
338C23#39411yescharacteristic
339C23#45412yes—
340C23#45612yes—
341Q8—84no—
342Q8—84no—
343Q8—84no—
344Q8—84no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#38011yescharacteristic
396C2×C4#38511yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38811yescharacteristic
399C2×C4#38811yescharacteristic
400C2×C4#39011yescharacteristic
401C2×C4#39311yescharacteristic
402C2×C4#39411yescharacteristic
403C2×C4#39611yescharacteristic
404C2×C4#39611yescharacteristic
405C2×C4#39711yescharacteristic
406C2×C4#39911yescharacteristic
407C2×C4#45312yes—
408C2×C4#45412yes—
409C8—44no—
410C8—44no—
411C8—82no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22#143911yescharacteristic
421C22#185712yes—
422C22#200211yescharacteristic
423C22#200311yescharacteristic
424C22#200311yescharacteristic
425C22#200511yescharacteristic
426C22#200911yescharacteristic
427C22#200911yescharacteristic
428C22#203812yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—84no—
440C4—84no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4#188312yes—
446C4#202712yes—
447C2—22no—
448C2—22no—
449C2#1009411yescharacteristic
450C2#1009511yescharacteristic
451C2#1010211yescharacteristic
452C2#1010311yescharacteristic
453C2#1106311yescharacteristic
454C2#1113011yescharacteristic
455C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#213242
C2×#15562
#142742
C2×#13002
#80332
#80222