Finite Group Databasebeta

← #195259

Subgroups — #195259

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#9046C218yesmaximal
7#8026C211yesmaximal, characteristic
8#7906C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—24no—
12#3230C418yescomplemented
13C2×#500—21no—
14C2×#487—21no—
15C2×#484C2214yes—
16C2×#477—22no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2331—21no—
20#2331—21no—
21#2330—22no—
22#2329—22no—
23#2329C2214yes—
24#2296—21no—
25#2295—22no—
26#2295—22no—
27#2285C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C4×#117—21no—
30C2×#376C2211yescharacteristic
31#1990—22no—
32#1987—22no—
33#1824C2214yes—
34#1518—24no—
35C2×#328C2214yes—
36#1446—22no—
37#1445C2214yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8—21no—
43#533—22no—
44#533—22no—
45#533C2×C414yes—
46#533D414yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#500—28no—
71#487—28no—
72#484—28no—
73#484C2×C4116yes—
74#477—216no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431C2×C414yes—
79C2×#107—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#103—24no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#391—24no—
88#391—24no—
89#389—24no—
90#389—22no—
91#389—22no—
92#389—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#385—24no—
98#384D412yes—
99#382—21no—
100#382—22no—
101#382—22no—
102#380—21no—
103#380C2312yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—24no—
111#376—22no—
112#376—21no—
113#376—21no—
114#376—21no—
115#376—21no—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95D412yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122C43—22no—
123#340—28no—
124#328—28no—
125C22×Q8—24no—
126C22×Q8—21no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—84no—
131C22×Q8C2×D412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8C22⋊C414yes—
152C4⋊Q8C22⋊C414yes—
153#125—24no—
154#125—24no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8C22×C414yes—
160C22⋊Q8C2×D414yes—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8C22⋊C418yes—
165#117—24no—
166#117—24no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42Q1614yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215#107—216no—
216C4⋊C8—24no—
217C4⋊C8—42no—
218#103—216no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8C22×C414yes—
223C4×C8—24no—
224C4×C8—42no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—42no—
238#95—84no—
239#95—84no—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—82no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8—88no—
250C2×Q8—88no—
251C2×Q8C2×C22⋊C414yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4C2×Q1614yes—
285C22×C4C2×Q1612yes—
286C22×C4#13512yes—
287C22×C4#13712yes—
288C22×C4#13712yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—24no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—42no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4C2×C22⋊C414yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—28no—
324C42—24no—
325C42—24no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—84no—
330C42—84no—
331C42C2×C22⋊C412yes—
332C42C2×C22⋊C412yes—
333C42C4×D412yes—
334C42C4×D412yes—
335C42#12112yes—
336C42#12112yes—
337C42#12312yes—
338C42#12312yes—
339C23—22no—
340C23—22no—
341C23C2×C4×D411yescharacteristic
342C23C2×#12011yescharacteristic
343C23C2×#12111yescharacteristic
344C23C22×Q1611yescharacteristic
345C23C2×#13511yescharacteristic
346C23C2×#13711yescharacteristic
347C23C2×#13711yescharacteristic
348Q8—44no—
349Q8—82no—
350Q8—84no—
351Q8—88no—
352Q8—88no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#38612yes—
391C2×C4#38612yes—
392C2×C4#38612yes—
393C2×C4#38612yes—
394C2×C4C4×Q1614yes—
395C2×C4#44114yes—
396C2×C4#45112yes—
397C2×C4#45112yes—
398C2×C4#45212yes—
399C2×C4#45212yes—
400C2×C4#46214yes—
401C2×C4#46414yes—
402C2×C4C22×C22⋊C411yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×#12111yescharacteristic
405C2×C4C2×#12311yescharacteristic
406C8—44no—
407C8—82no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22C2×#38611yescharacteristic
417C22C2×C4×Q1611yescharacteristic
418C22C2×#44111yescharacteristic
419C22C2×#45111yescharacteristic
420C22C2×#45211yescharacteristic
421C22#302112yes—
422C22#302812yes—
423C22C2×#46211yescharacteristic
424C22C2×#46411yescharacteristic
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—44no—
433C4—44no—
434C4—84no—
435C4—82no—
436C4—84no—
437C4—162no—
438C4#187814yes—
439C4#187814yes—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2#1831112yes—
445C2#1831312yes—
446C2C2×#187811yescharacteristic
447C2#2094511yescharacteristic
448C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#340742
#218342
C2×#15182
#142822
#90462
#80262
#79062