Finite Group Databasebeta

← #195253

Subgroups — #195253

Order 512.

1234 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
1C2×#3209C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#9040C218yesmaximal
7#8026C211yesmaximal, characteristic
8#7906C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—24no—
12#3209C418yescomplemented
13C2×#494—21no—
14C2×#487—21no—
15C2×#462—22no—
16C2×#451C2214yes—
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#1981—22no—
32#1956—22no—
33#1787C2214yes—
34#1518—24no—
35C2×#328C2214yes—
36#1446C2214yes—
37#1445—22no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8C2312yes—
43C22×Q16—24no—
44#533—22no—
45#533—22no—
46#533C2×C414yes—
47#533D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D411yescharacteristic
57C2×C4⋊Q8D411yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2312yes—
61C2×C4×Q8—21no—
62C2×C4×Q8D412yes—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71#494—28no—
72#487—28no—
73#462—216no—
74#451—28no—
75#451C2×C4116yes—
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#103—24no—
83C2×#103—28no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#391—24no—
89#391—24no—
90#389—24no—
91#389—22no—
92#389—22no—
93#389—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#385—24no—
99#384D412yes—
100#382—21no—
101#382—22no—
102#382—22no—
103#380—21no—
104#380C2312yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4—24no—
112#376—22no—
113#376—21no—
114#376—21no—
115#376—21no—
116#376—21no—
117C2×#95—21no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95D412yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C43—22no—
124#328—28no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—84no—
132C22×Q8C22×C414yes—
133C22×Q8C2×D412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×Q16—216no—
142C2×Q16—88no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8C22⋊C414yes—
156C4⋊Q8C22⋊C414yes—
157#125—24no—
158#125—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8C22×C414yes—
164C22⋊Q8C2×D414yes—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8C22⋊C418yes—
169#117—24no—
170#117—24no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
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—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42Q1614yescomplemented
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C4⋊C8—24no—
218C4⋊C8—42no—
219#103—216no—
220#103—88no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8C22×C414yes—
225C4×C8—24no—
226C4×C8—42no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—42no—
240#95—84no—
241#95—84no—
242C24C22×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—44no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—82no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8C2×C22⋊C414yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4C2×Q1614yes—
291C22×C4C2×Q1612yes—
292C22×C4#13512yes—
293C22×C4#13712yes—
294C22×C4#13712yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301Q16—816no—
302C2×C8—24no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4C2×C22⋊C414yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C42—28no—
329C42—24no—
330C42—24no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—84no—
335C42—84no—
336C42C2×C22⋊C412yes—
337C42C2×C22⋊C412yes—
338C42C4×D412yes—
339C42C4×D412yes—
340C42#12112yes—
341C42#12112yes—
342C42#12312yes—
343C42#12312yes—
344C23—22no—
345C23—22no—
346C23C2×C4×D411yescharacteristic
347C23C2×#12011yescharacteristic
348C23C2×#12111yescharacteristic
349C23C22×Q1611yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×#13711yescharacteristic
352C23C2×#13711yescharacteristic
353Q8—44no—
354Q8—48no—
355Q8—48no—
356Q8—82no—
357Q8—84no—
358Q8—84no—
359Q8—84no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#38612yes—
400C2×C4#38612yes—
401C2×C4C4×Q1614yes—
402C2×C4#44114yes—
403C2×C4#45112yes—
404C2×C4#45112yes—
405C2×C4#45212yes—
406C2×C4#45212yes—
407C2×C4#46214yes—
408C2×C4#46414yes—
409C2×C4C22×C22⋊C411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×#12111yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C8—44no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22C2×#38611yescharacteristic
424C22C2×C4×Q1611yescharacteristic
425C22C2×#44111yescharacteristic
426C22C2×#45111yescharacteristic
427C22C2×#45211yescharacteristic
428C22#302212yes—
429C22#302712yes—
430C22C2×#46211yescharacteristic
431C22C2×#46411yescharacteristic
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—44no—
440C4—44no—
441C4—84no—
442C4—82no—
443C4—84no—
444C4—162no—
445C4#187814yes—
446C4#187814yes—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2#1830312yes—
452C2#1830512yes—
453C2C2×#187811yescharacteristic
454C2#2094411yescharacteristic
455C2#2096111yescharacteristic
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. #3209C4semidirect
208. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#340742
#218042
C2×#15182
#142822
#90402
#80262
#79062