Finite Group Databasebeta

← #223146

Subgroups — #223146

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36174C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic, complemented
3#21270C211yesmaximal, characteristic, complemented
4#9891C211yesmaximal, characteristic, complemented
5#9891C211yesmaximal, characteristic, complemented
6#9124C212yesmaximal, complemented
7#9123C212yesmaximal, complemented
8#8720C212yesmaximal, complemented
9#8579C211yesmaximal, characteristic, complemented
10C4×#329C211yesmaximal, characteristic, complemented
11#8032C212yesmaximal, complemented
12C2×#432C2212yes—
13C2×#432C2212yes—
14#2859C2211yescharacteristic
15#2850C2212yes—
16#2538C411yescharacteristic
17#2537C411yescharacteristic
18#2526—22no—
19#2518—22no—
20#2495—21no—
21#2490C412yes—
22#2489C412yes—
23#2420—21no—
24#2363—21no—
25#2355C411yescharacteristic
26#2354C411yescharacteristic
27#2346—21no—
28C4×#124C2211yescharacteristic
29#2283C2212yes—
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1924—22no—
36#1924—22no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39#1749C2212yes—
40#1749C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#329C2211yescharacteristic
44C2×#329C2211yescharacteristic
45C2×#329C2212yes—
46#1295C2211yescharacteristic
47#1295C2212yes—
48#1295C2212yes—
49#1295C2212yes—
50#1295C2211yescharacteristic
51#1295C2212yes—
52#1295C2212yes—
53C2×#125C2×C412yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8C2×C412yes—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117—21no—
64C22×C42C2312yes—
65#432—24no—
66#432—24no—
67#432—42no—
68#432—42no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97C2312yes—
80C2×#97C2312yes—
81C2×#97C2311yescharacteristic
82C2×#97C2311yescharacteristic
83C2×#97C2311yescharacteristic
84C2×#97C2311yescharacteristic
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#400—22no—
88#397—22no—
89#397—22no—
90#396—22no—
91#394—22no—
92#394—22no—
93#391C2×C412yes—
94#391C2×C411yescharacteristic
95#390—21no—
96#390—22no—
97#390C2×C412yes—
98#390C2×C411yescharacteristic
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—21no—
108#388—21no—
109#388—22no—
110#388—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#383—22no—
117#382—21no—
118#382—22no—
119#382C2312yes—
120#382C2311yescharacteristic
121#381—22no—
122#380—21no—
123#380—21no—
124#380C2×C412yes—
125#380C2×C412yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4C2×C412yes—
136C4×C22⋊C4D412yes—
137C4×C22⋊C4D412yes—
138#376—22no—
139C43C2311yescharacteristic
140#336—21no—
141#336—21no—
142#336—22no—
143#336—22no—
144#329—22no—
145#329—22no—
146#329—42no—
147C22×Q8C22×C411yescharacteristic
148C22×D4—22no—
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161#125—82no—
162#124—21no—
163#124—21no—
164#124—21no—
165#124—21no—
166C22⋊Q8—82no—
167#121—44no—
168#121—82no—
169C4×Q8—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—44no—
222C4⋊C8—44no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229#97—21no—
230#97—21no—
231#97—22no—
232#97—22no—
233#97—21no—
234#97—21no—
235#97—22no—
236#97—22no—
237#97—21no—
238#97—21no—
239#97—42no—
240#97—42no—
241#97#4112yes—
242#97#4112yes—
243C4×C8—42no—
244C4×C8—42no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24—21no—
261C24C2×C22⋊C412yes—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—82no—
275C2×D4—82no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—44no—
368C42—42no—
369C42—82no—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C42H(5,𝔽2)11yescharacteristic
373C422−1+411yescharacteristic
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C22×C22⋊C411yescharacteristic
389Q8—81no—
390Q8—81no—
391D4—44no—
392D4—81no—
393D4—81no—
394D4—164no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4#41312yes—
442C2×C4#42112yes—
443C2×C4#42112yes—
444C2×C4C4×#4111yescharacteristic
445C2×C4#52011yescharacteristic
446C2×C4#54611yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4#54711yescharacteristic
449C2×C4#54711yescharacteristic
450C8—42no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—82no—
455C8—82no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22#233911yescharacteristic
471C22C2×#41311yescharacteristic
472C22#288811yescharacteristic
473C22C2×#42111yescharacteristic
474C22C2×#42111yescharacteristic
475C22#290111yescharacteristic
476C22#290111yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4#153514yes—
495C2—82no—
496C2—161no—
497C2C2×#153511yescharacteristic
498C2#1768111yescharacteristic
499C2#1768411yescharacteristic
500C2#1768711yescharacteristic
501C2#2248311yescharacteristic
502C2#2250711yescharacteristic
503C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21270C2semidirect
3. #21270C2semidirect
4. #9891C2semidirect
5. #9891C2semidirect
6. #9124C2semidirect
7. #9123C2semidirect
8. #8720C2semidirect
9. #8579C2semidirect
10. C4×#329C2semidirect
11. #8032C2semidirect

Maximal subgroups

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

subgroupindex
#361742
#212702
#98912
#91242
#91232
#87202
#85792
C4×#3292
#80322