Finite Group Databasebeta

← #115158

Subgroups — #115158

Order 512.

1106 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
1C2×#2396C211yesmaximal, characteristic
2#20627C211yesmaximal, characteristic
3C2×#1487C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5C2×#1303C212yesmaximal, complemented
6C22×C4×Q8C2211yescharacteristic
7C22×#117C2211yescharacteristic
8C2×#431C2212yes—
9C2×#423C2212yes—
10C2×#407—21no—
11#2396—24no—
12#2396—24no—
13C2×#400C411yescharacteristic
14C2×#399C411yescharacteristic
15C2×#398—21no—
16C2×#396—21no—
17C2×#395C411yescharacteristic
18C2×#393C411yescharacteristic
19C2×C4×C4⋊C4—21no—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4—21no—
22#1798—24no—
23#1728—21no—
24#1487—24no—
25#1487—24no—
26C2×Q8⋊C8—24no—
27#1458—24no—
28C2×#324—21no—
29C2×#323—21no—
30#1303—24no—
31C23×Q8C2×C411yescharacteristic
32C24×C4D411yescharacteristic
33C22×M16—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8D414yes—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#431—22no—
64#431D414yes—
65#423—22no—
66#423—22no—
67#423—22no—
68#423—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8D414yes—
72#407—42no—
73#407—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C4×C8—22no—
77#400—22no—
78#400—22no—
79#399—22no—
80#399—22no—
81#398—28no—
82#396—28no—
83#395—22no—
84#395—22no—
85#393—22no—
86#393—22no—
87C4×C4⋊C4—28no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C4×C22⋊C4—28no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97#336—24no—
98Q8⋊C8—216no—
99#324—24no—
100#323—24no—
101C25C22⋊C411yescharacteristic
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—44no—
106C22×Q8C22⋊C412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4SD1612yes—
123C23×C4Q1612yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—24no—
134C22×C8—42no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148#117—42no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—41no—
192C2×C42—42no—
193C2×C42—41no—
194C2×C42—42no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C22⋊C8—24no—
210C22⋊C8—48no—
211C22⋊C8—42no—
212C4×C8—24no—
213C4×C8—42no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—21no—
219#95—21no—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—42no—
229C24—42no—
230C24#10312yes—
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—28no—
233C2×Q8—24no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—48no—
238C2×Q8—42no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#10312yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4#12011yescharacteristic
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4C2×SD1611yescharacteristic
305C22×C4C2×Q1611yescharacteristic
306C22×C4#13512yes—
307C22×C4AGL(1,ℤ/8)11yescharacteristic
308C22×C4#13712yes—
309C22×C4#13711yescharacteristic
310M16—82no—
311M16—82no—
312M16—82no—
313M16—82no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—84no—
319C4⋊C4—24no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—42no—
337C42—44no—
338C42—42no—
339C42—44no—
340C42—44no—
341C42—44no—
342C42#12012yes—
343C42#12112yes—
344C42#12112yes—
345C42#12112yes—
346C42#12112yes—
347C42#12312yes—
348C42#12412yes—
349C42#12712yes—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—44no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—82no—
366C23—82no—
367C23—81no—
368C23—81no—
369C23#37911yescharacteristic
370C23#38611yescharacteristic
371C23#38611yescharacteristic
372C23C2×#10311yescharacteristic
373C23#41611yescharacteristic
374C23#41711yescharacteristic
375C23#41911yescharacteristic
376Q8—28no—
377Q8—44no—
378Q8—48no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4#38612yes—
427C2×C4#39012yes—
428C2×C4#39212yes—
429C2×C4#39212yes—
430C2×C4#39412yes—
431C2×C4#39412yes—
432C2×C4#44812yes—
433C2×C4#44912yes—
434C2×C4#45011yescharacteristic
435C2×C4#45111yescharacteristic
436C2×C4#45112yes—
437C2×C4#45212yes—
438C2×C4#45212yes—
439C2×C4#46512yes—
440C2×C4#46712yes—
441C2×C4#46812yes—
442C2×C4#47012yes—
443C2×C4#48311yescharacteristic
444C2×C4#48411yescharacteristic
445C2×C4#48512yes—
446C8—84no—
447C8—84no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22—81no—
465C22#143811yescharacteristic
466C22#178711yescharacteristic
467C22#178811yescharacteristic
468C22#187211yescharacteristic
469C22#187311yescharacteristic
470C22#188412yes—
471C22#189112yes—
472C22#189211yescharacteristic
473C22#189411yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—44no—
479C4—42no—
480C4—44no—
481C4—82no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4#200012yes—
486C4#200112yes—
487C4#203512yes—
488C4#203612yes—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—82no—
494C2#903312yes—
495C2#903412yes—
496C2#1097011yescharacteristic
497C2#1097911yescharacteristic
498C2#1099311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1487C2semidirect
5. C2×#1303C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
#206272
C2×#14872
C2×#14582
C2×#13032