Finite Group Databasebeta

← #121040

Subgroups — #121040

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14707C211yesmaximal, characteristic
2#10839C211yesmaximal, characteristic
3#8026C211yesmaximal, characteristic
4#8026C211yesmaximal, characteristic
5C2×#1286C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#389—21no—
12#2285C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C43C2211yescharacteristic
16C2×#339—21no—
17C2×#339—21no—
18#1436—22no—
19#1436—22no—
20C4⋊#95—21no—
21#1294—22no—
22#1294—22no—
23#1286—22no—
24#1286C414yescomplemented
25#1276—22no—
26C22×M16—21no—
27C2×#117—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—21no—
31C22×C42—21no—
32C22×C42—21no—
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431C2×C412yes—
43#431D412yes—
44#423—21no—
45#423—21no—
46#423—21no—
47#423C2×C412yes—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#389—42no—
57#389—42no—
58#384Q811yescharacteristic
59#384Q811yescharacteristic
60#382—21no—
61#382—21no—
62#382—21no—
63#382—21no—
64#382—21no—
65#382—21no—
66#382—21no—
67#382C2×C412yes—
68#382Q812yes—
69#380—21no—
70#380—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95D412yes—
79C2×#95D412yes—
80C43—21no—
81C43—21no—
82C43—21no—
83C43—21no—
84#339—42no—
85#339—42no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102#117—41no—
103#117—41no—
104#117—41no—
105#117—41no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42D812yes—
148C2×C42SD1612yes—
149C2×C42C22×C411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42C2×Q811yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—41no—
176C4×C8—41no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—44no—
183#95—44no—
184#95—44no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95C4⋊C414yes—
190C24#12011yescharacteristic
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×SD1611yescharacteristic
245C22×C4#13511yescharacteristic
246C22×C4#13511yescharacteristic
247C22×C4AGL(1,ℤ/8)11yescharacteristic
248C22×C4#13711yescharacteristic
249C22×C4#13711yescharacteristic
250C22×C4#13711yescharacteristic
251M16—42no—
252M16—42no—
253M16—81no—
254M16—81no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—81no—
259C2×C8—81no—
260C2×C8—81no—
261C2×C8—81no—
262C2×C8—81no—
263C2×C8—81no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—88no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—44no—
288C42—44no—
289C42—44no—
290C42—44no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—81no—
296C42—81no—
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×Q811yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C42#12511yescharacteristic
304C42#12511yescharacteristic
305C42#12511yescharacteristic
306C42#12511yescharacteristic
307C42#12511yescharacteristic
308C42#12611yescharacteristic
309C42#12611yescharacteristic
310C42#12611yescharacteristic
311C42#12611yescharacteristic
312C42C4⋊Q811yescharacteristic
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23#38511yescharacteristic
318C23#39311yescharacteristic
319C23#39411yescharacteristic
320C23#44711yescharacteristic
321C23#44811yescharacteristic
322C23#45211yescharacteristic
323C23#45211yescharacteristic
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4#38211yescharacteristic
375C2×C4#38711yescharacteristic
376C2×C4#38711yescharacteristic
377C2×C4#38811yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38911yescharacteristic
380C2×C4#39711yescharacteristic
381C2×C4#39811yescharacteristic
382C2×C4#39811yescharacteristic
383C2×C4#40011yescharacteristic
384C2×C4#40011yescharacteristic
385C2×C4#40011yescharacteristic
386C2×C4C4×D811yescharacteristic
387C2×C4C4×SD1611yescharacteristic
388C2×C4#44011yescharacteristic
389C2×C4#44111yescharacteristic
390C2×C4#46511yescharacteristic
391C2×C4#46611yescharacteristic
392C2×C4#46811yescharacteristic
393C2×C4#47011yescharacteristic
394C2×C4#47411yescharacteristic
395C2×C4#47411yescharacteristic
396C2×C4#47511yescharacteristic
397C2×C4#47611yescharacteristic
398C2×C4#47811yescharacteristic
399C2×C4#47911yescharacteristic
400C2×C4#47911yescharacteristic
401C2×C4#47911yescharacteristic
402C2×C4#48011yescharacteristic
403C2×C4#48211yescharacteristic
404C2×C4#48311yescharacteristic
405C2×C4#48511yescharacteristic
406C2×C4#48611yescharacteristic
407C2×C4#48811yescharacteristic
408C2×C4#48911yescharacteristic
409C2×C4#49111yescharacteristic
410C8—82no—
411C8—82no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22#144211yescharacteristic
426C22#186211yescharacteristic
427C22#186511yescharacteristic
428C22#202311yescharacteristic
429C22#202611yescharacteristic
430C22#203111yescharacteristic
431C22#203111yescharacteristic
432C22#203111yescharacteristic
433C22#203111yescharacteristic
434C22#203311yescharacteristic
435C22#203511yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—44no—
446C4—84no—
447C4—162no—
448C4—161no—
449C4—161no—
450C4#191811yescharacteristic
451C4#192011yescharacteristic
452C4#193411yescharacteristic
453C4#193511yescharacteristic
454C4#208611yescharacteristic
455C4#208811yescharacteristic
456C4#209911yescharacteristic
457C4#210111yescharacteristic
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2#1036211yescharacteristic
463C2#1036411yescharacteristic
464C2#1036711yescharacteristic
465C2#1036911yescharacteristic
466C2#1126311yescharacteristic
467C2#1128211yescharacteristic
468C2#1129211yescharacteristic
subgroupG/H
center, socleC23#393
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
24. #1286C4semidirect

Maximal subgroups

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

subgroupindex
#147072
#108392
#80262
C2×#12862
C2×#12762