Finite Group Databasebeta

← #1409136

Subgroups — #1409136

Order 512.

2791 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C213yesmaximal, complemented
2C2×#2519C213yesmaximal, complemented
3#17071C213yesmaximal, complemented
4#17062C213yesmaximal, complemented
5#16984C213yesmaximal, complemented
6#16692C213yesmaximal, complemented
7#15211C211yesmaximal, characteristic, complemented
8#15084C213yesmaximal, complemented
9#15079C213yesmaximal, complemented
10#14900C211yesmaximal, characteristic
11#14900C211yesmaximal, characteristic
12#14354C211yesmaximal, characteristic, complemented
13#14321C211yesmaximal, characteristic
14#13856C211yesmaximal, characteristic, complemented
15#13810C211yesmaximal, characteristic, complemented
16C22×#126C2213yes—
17C22×#125C2211yescharacteristic
18#2519—224no—
19#2512—224no—
20C2×#399C2213yes—
21C2×#399C2216yes—
22C2×#398C2213yes—
23C2×#398C2213yes—
24C2×#397C2216yes—
25C2×#396C2216yes—
26C2×#396C2213yes—
27C2×#392C2211yescharacteristic
28C2×#390C2213yes—
29C2×#390C2213yes—
30C2×#390C2213yes—
31C2×#389C2213yes—
32C2×#389C2213yes—
33C2×#388C2213yes—
34C2×#388C2213yes—
35C2×#388C2213yes—
36C2×#388C2213yes—
37C2×#388C2216yes—
38C2×#387C2213yes—
39C2×#384C2213yes—
40C2×#382C2213yes—
41C2×C4×C4⋊C4C2213yes—
42C2×C4×C4⋊C4C2213yes—
43C2×C4×C4⋊C4C2213yes—
44C2×#376C2211yescharacteristic
45C2×#376C2211yescharacteristic
46#1445C2216yes—
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2213yes—
50#1444C2213yes—
51#1444C2211yescharacteristic
52#1444C2213yes—
53#1444C2213yes—
54#1444C2213yes—
55#1444C2213yes—
56#1439C2213yes—
57#1439C2213yes—
58#1439C2216yes—
59#1439C2216yes—
60#1439C2213yes—
61#1438C2213yes—
62#1438C2213yes—
63#1438C2213yes—
64C4⋊#95C2213yes—
65#1434C2213yes—
66#1434C2213yes—
67C4×#95C2211yescharacteristic
68C23×D4C2313yes—
69C2×#126—224no—
70C2×#126—212no—
71C2×#125—212no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2313yes—
80C22×C4⋊C4C2313yes—
81C22×C4⋊C4C2316yes—
82C22×C4⋊C4C2313yes—
83C22×C4⋊C4C2313yes—
84C22×C4⋊C4C2313yes—
85C22×C4⋊C4C2313yes—
86C22×C4⋊C4C2313yes—
87C22×C4⋊C4C2313yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4C2313yes—
90C22×C22⋊C4C2313yes—
91C22×C22⋊C4C2313yes—
92C22×C22⋊C4C2313yes—
93C22×C22⋊C4C2313yes—
94C22×C22⋊C4C2316yes—
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2316yes—
97C22×C22⋊C4C2316yes—
98C22×C22⋊C4C2313yes—
99C22×C42C2311yescharacteristic
100C22×C42C2313yes—
101C22×C42C2313yes—
102C22×C42C2313yes—
103C22×C42C2313yes—
104#399—212no—
105#399—412no—
106#398—212no—
107#398—212no—
108#397—224no—
109#396—224no—
110#396—212no—
111#392—24no—
112#390—212no—
113#390—212no—
114#390—212no—
115#389—212no—
116#389—212no—
117#388—212no—
118#388—212no—
119#388—212no—
120#388—224no—
121#388—212no—
122#387—212no—
123#384—212no—
124#382—212no—
125C4×C4⋊C4—212no—
126C4×C4⋊C4—212no—
127C4×C4⋊C4D4124yescomplemented
128#376—24no—
129#376—24no—
130C2×#95C2316yes—
131C2×#95C2313yes—
132C2×#95C2316yes—
133C2×#95C2313yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2316yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2311yescharacteristic
147C25C2413yes—
148C22×D4—26no—
149C22×D4—26no—
150C22×D4—26no—
151C22×D4—46no—
152C22×D4—46no—
153C23×C4C2413yes—
154C23×C4C2411yescharacteristic
155C23×C4C2413yes—
156C23×C4C2413yes—
157C23×C4C2411yescharacteristic
158C23×C4C2411yescharacteristic
159C23×C4C2413yes—
160C23×C4C2413yes—
161C23×C4C2413yes—
162C23×C4C2411yescharacteristic
163C23×C4C2411yescharacteristic
164C23×C4C2411yescharacteristic
165C23×C4C2411yescharacteristic
166C23×C4C2413yes—
167#126—448no—
168#125—416no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—26no—
173C2×C4⋊C4—26no—
174C2×C4⋊C4—26no—
175C2×C4⋊C4—212no—
176C2×C4⋊C4—212no—
177C2×C4⋊C4—26no—
178C2×C4⋊C4—26no—
179C2×C4⋊C4—26no—
180C2×C4⋊C4—26no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—212no—
188C2×C4⋊C4—43no—
189C2×C4⋊C4—43no—
190C2×C4⋊C4—43no—
191C2×C4⋊C4—43no—
192C2×C4⋊C4—46no—
193C2×C4⋊C4—46no—
194C2×C4⋊C4—46no—
195C2×C4⋊C4—46no—
196C2×C4⋊C4—43no—
197C2×C4⋊C4—43no—
198C2×C4⋊C4—46no—
199C2×C4⋊C4—46no—
200C2×C4⋊C4C2×D4112yes—
201C2×C4⋊C4C2×D4112yes—
202C2×C22⋊C4—26no—
203C2×C22⋊C4—212no—
204C2×C22⋊C4—26no—
205C2×C22⋊C4—26no—
206C2×C22⋊C4—212no—
207C2×C22⋊C4—212no—
208C2×C22⋊C4—26no—
209C2×C22⋊C4—26no—
210C2×C22⋊C4—26no—
211C2×C22⋊C4—212no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—26no—
215C2×C22⋊C4—26no—
216C2×C22⋊C4—212no—
217C2×C22⋊C4—212no—
218C2×C22⋊C4—43no—
219C2×C22⋊C4—46no—
220C2×C22⋊C4—46no—
221C2×C22⋊C4—46no—
222C2×C22⋊C4—43no—
223C2×C22⋊C4—43no—
224C2×C22⋊C4—46no—
225C2×C22⋊C4—46no—
226C2×C22⋊C4—46no—
227C2×C22⋊C4—46no—
228C2×C42—26no—
229C2×C42—26no—
230C2×C42—26no—
231C2×C42—26no—
232C2×C42—26no—
233C2×C42—26no—
234C2×C42—212no—
235C2×C42—46no—
236C2×C42C2×D4112yes—
237C2×C42C2×D4112yes—
238C2×C42C2×D4112yes—
239#95—26no—
240#95—26no—
241#95—212no—
242#95—26no—
243#95—26no—
244#95—26no—
245#95—26no—
246#95—26no—
247#95—212no—
248#95—26no—
249#95—26no—
250#95—26no—
251#95—26no—
252#95—46no—
253#95C2×D4112yes—
254#95C2×D4112yes—
255#95#4114yes—
256C24—23no—
257C24—26no—
258C24—23no—
259C24—23no—
260C24—23no—
261C24—26no—
262C24—23no—
263C24—26no—
264C24—26no—
265C24—23no—
266C24—23no—
267C24C2511yescharacteristic
268C2×D4—412no—
269C2×D4—412no—
270C2×D4—412no—
271C2×D4—812no—
272C2×D4—812no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—23no—
276C22×C4—23no—
277C22×C4—23no—
278C22×C4—26no—
279C22×C4—21no—
280C22×C4—26no—
281C22×C4—23no—
282C22×C4—23no—
283C22×C4—23no—
284C22×C4—23no—
285C22×C4—23no—
286C22×C4—23no—
287C22×C4—26no—
288C22×C4—23no—
289C22×C4—23no—
290C22×C4—26no—
291C22×C4—23no—
292C22×C4—23no—
293C22×C4—23no—
294C22×C4—23no—
295C22×C4—23no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—23no—
299C22×C4—23no—
300C22×C4—23no—
301C22×C4—23no—
302C22×C4—26no—
303C22×C4—23no—
304C22×C4—23no—
305C22×C4—23no—
306C22×C4—26no—
307C22×C4—26no—
308C22×C4—21no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—21no—
313C22×C4—23no—
314C22×C4—23no—
315C22×C4—23no—
316C22×C4—26no—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—21no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4C22×D416yes—
324C22×C4C22×D416yes—
325C22×C4C22×D416yes—
326C22×C4C22×D416yes—
327C22×C4C22×D416yes—
328C22×C4C22×D416yes—
329C22×C4C22×D416yes—
330C22×C4C2×#4116yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+416yes—
333C4⋊C4—412no—
334C4⋊C4—412no—
335C4⋊C4—412no—
336C4⋊C4—412no—
337C4⋊C4—412no—
338C4⋊C4—412no—
339C4⋊C4—412no—
340C4⋊C4—412no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—412no—
347C4⋊C4—412no—
348C4⋊C4—812no—
349C4⋊C4#127116yes—
350C22⋊C4—412no—
351C22⋊C4—412no—
352C22⋊C4—424no—
353C22⋊C4—412no—
354C22⋊C4—412no—
355C22⋊C4—412no—
356C22⋊C4—412no—
357C22⋊C4—412no—
358C22⋊C4—812no—
359C22⋊C4—812no—
360C42—224no—
361C42—224no—
362C42—412no—
363C42—44no—
364C42—412no—
365C23—43no—
366C23—43no—
367C23—46no—
368C23—43no—
369C23—46no—
370C23—43no—
371C23—46no—
372C23—46no—
373C23—43no—
374C23—43no—
375C23—46no—
376C23—43no—
377C23—46no—
378C23—43no—
379C23—43no—
380C23—46no—
381C23—43no—
382C23—46no—
383C23—46no—
384C23—46no—
385C23—46no—
386C23—43no—
387C23—46no—
388C23C23×D413yes—
389C23C22×#4111yescharacteristic
390C23C2×H(5,𝔽2)11yescharacteristic
391C23C2×2−1+413yes—
392C23#58511yescharacteristic
393C23#58513yes—
394C23#58513yes—
395D4—824no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—26no—
400C2×C4—26no—
401C2×C4—26no—
402C2×C4—26no—
403C2×C4—26no—
404C2×C4—212no—
405C2×C4—212no—
406C2×C4—26no—
407C2×C4—26no—
408C2×C4—26no—
409C2×C4—26no—
410C2×C4—26no—
411C2×C4—26no—
412C2×C4—26no—
413C2×C4—26no—
414C2×C4—26no—
415C2×C4—26no—
416C2×C4—41no—
417C2×C4—43no—
418C2×C4—43no—
419C2×C4—43no—
420C2×C4—43no—
421C2×C4—43no—
422C2×C4—46no—
423C2×C4—43no—
424C2×C4—43no—
425C2×C4—46no—
426C2×C4—43no—
427C2×C4—43no—
428C2×C4—43no—
429C2×C4—43no—
430C2×C4—41no—
431C2×C4—43no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—46no—
435C2×C4—43no—
436C2×C4—43no—
437C2×C4—43no—
438C2×C4—43no—
439C2×C4—46no—
440C2×C4—43no—
441C2×C4—43no—
442C2×C4—43no—
443C2×C4—46no—
444C2×C4—46no—
445C2×C4—43no—
446C2×C4—43no—
447C2×C4—46no—
448C2×C4—46no—
449C2×C4—43no—
450C2×C4—43no—
451C2×C4—43no—
452C2×C4C2×#12714yes—
453C2×C4C2×#12714yes—
454C2×C4C2×#12714yes—
455C2×C4#547112yes—
456C22—86no—
457C22—83no—
458C22—83no—
459C22—86no—
460C22—83no—
461C22—83no—
462C22—86no—
463C22—83no—
464C22—83no—
465C22—86no—
466C22—83no—
467C22C22×#12711yescharacteristic
468C22#344913yes—
469C22#344913yes—
470C22C2×#54713yes—
471C22#348713yes—
472C22#348813yes—
473C22#349913yes—
474C22#351711yescharacteristic
475C22#352113yes—
476C22#354713yes—
477C22#356416yes—
478C22#356513yes—
479C4—24no—
480C4—24no—
481C4—24no—
482C4—212no—
483C4—46no—
484C4—46no—
485C4—46no—
486C4—42no—
487C4—83no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—83no—
492C4—83no—
493C2—163no—
494C2#3464111yescharacteristic
495C2#3606611yescharacteristic
496C2#3617911yescharacteristic
497C2#3804713yes—
498C2#3804913yes—
499C2#4231213yes—
500C2#4231413yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2512C2semidirect
2. C2×#2519C2semidirect
3. #17071C2semidirect
4. #17062C2semidirect
5. #16984C2semidirect
6. #16692C2semidirect
7. #15211C2semidirect
8. #15084C2semidirect
9. #15079C2semidirect
12. #14354C2semidirect
14. #13856C2semidirect
15. #13810C2semidirect
127. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#25192
#170712
#170622
#169842
#166922
#152112
#150842
#150792
#149002
#143542
#143212
#138562
#138102