Finite Group Databasebeta

← #101378

Subgroups — #101378

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24212C211yesmaximal, characteristic
2#22287C211yesmaximal, characteristic
3#11384C211yesmaximal, characteristic
4#8572C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7020C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#462—21no—
10C2×#451—21no—
11C4×C22⋊Q8C2211yescharacteristic
12#2043C412yescomplemented
13#2016C412yescomplemented
14#1984—21no—
15#1981—21no—
16#1939—21no—
17#1916—21no—
18#1916—21no—
19#1850—21no—
20#1797C2211yescharacteristic
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1787—21no—
24C8×C22⋊C4—21no—
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#332—21no—
29C2×#332—21no—
30C2×#328—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33C23×Q8—21no—
34C22×Q16—22no—
35C23×C8D411yescharacteristic
36C23×C8D411yescharacteristic
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8—42no—
42C2×C4⋊Q8C2×C412yes—
43C2×C22⋊Q8D411yescharacteristic
44C2×C4×Q8D411yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C42C2311yescharacteristic
49#462—42no—
50#462—42no—
51#451—42no—
52#451—42no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67#391C2×C411yescharacteristic
68#389C2×C411yescharacteristic
69#387—21no—
70#385—21no—
71#380D411yescharacteristic
72C4×C4⋊C4—21no—
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C2×#95—21no—
76C2×#95—21no—
77#340—42no—
78#340—42no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336C2×C412yes—
88#332—42no—
89#332—42no—
90#328—42no—
91C22×Q8—22no—
92C22×Q8—44no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—82no—
99C22×Q8C22⋊C411yescharacteristic
100C23×C4—41no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×Q16—44no—
106C2×Q16—44no—
107C2×Q16—82no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8Q1612yescomplemented
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C22⋊Q8—21no—
133C22⋊Q8—21no—
134C22⋊Q8D812yes—
135C22⋊Q8SD1612yes—
136C4×Q8—41no—
137C4×Q8—41no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—81no—
176C4⋊C8—81no—
177#103—44no—
178#103—44no—
179#103—82no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C4×C8—81no—
187C4×C8—81no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—41no—
194#95—41no—
195#95—81no—
196C24#12111yescharacteristic
197C2×Q8—21no—
198C2×Q8—24no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—81no—
209C2×Q8—81no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—81no—
215C2×Q8—81no—
216C2×Q8#10212yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×SD1611yescharacteristic
253C22×C4C2×Q1611yescharacteristic
254C22×C4#13511yescharacteristic
255C22×C4#13511yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13711yescharacteristic
258C22×C4#13711yescharacteristic
259Q16—88no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—81no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—81no—
281C2×C8—81no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4#10212yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#38611yescharacteristic
321C23#46211yescharacteristic
322C23#46411yescharacteristic
323C23#46511yescharacteristic
324C23#46611yescharacteristic
325C23#47111yescharacteristic
326C23#47311yescharacteristic
327Q8—42no—
328Q8—44no—
329Q8—44no—
330Q8—81no—
331Q8—82no—
332Q8—82no—
333Q8—82no—
334Q8—82no—
335Q8—82no—
336Q8—82no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—161no—
386C2×C4—161no—
387C2×C4—161no—
388C2×C4—161no—
389C2×C4C2×#10211yescharacteristic
390C2×C4#41611yescharacteristic
391C2×C4C4×Q1611yescharacteristic
392C2×C4#44111yescharacteristic
393C2×C4#45111yescharacteristic
394C2×C4#45211yescharacteristic
395C2×C4#45511yescharacteristic
396C2×C4#45611yescharacteristic
397C2×C4B(4,𝔽2)11yescharacteristic
398C2×C4B(4,𝔽2)11yescharacteristic
399C2×C4#46511yescharacteristic
400C2×C4#46611yescharacteristic
401C8—42no—
402C8—82no—
403C8—81no—
404C8—81no—
405C8—81no—
406C8—161no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22#167211yescharacteristic
420C22#167411yescharacteristic
421C22#168011yescharacteristic
422C22#168211yescharacteristic
423C22#187811yescharacteristic
424C22#188211yescharacteristic
425C22#188911yescharacteristic
426C22#189311yescharacteristic
427C22#190211yescharacteristic
428C22#190811yescharacteristic
429C22#191011yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—84no—
443C4—162no—
444C4#160211yescharacteristic
445C4#160311yescharacteristic
446C4#160511yescharacteristic
447C4#160611yescharacteristic
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2#904011yescharacteristic
453C2#921811yescharacteristic
454C2#929011yescharacteristic
455C2#931611yescharacteristic
456C2#937611yescharacteristic
457C2#946811yescharacteristic
458C2#951911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2043C4semidirect
13. #2016C4semidirect
121. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#242122
#222872
#113842
#85722
#83402
#70392
#70202