Finite Group Databasebeta

← #6371919

Subgroups — #6371919

Order 512.

2790 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2912C212yesmaximal
2C2×#2906C211yesmaximal, characteristic
3#34183C211yesmaximal, characteristic
4C4×#520C211yesmaximal, characteristic
5#21175C216yesmaximal, complemented
6#21174C216yesmaximal, complemented
7#21171C213yesmaximal, complemented
8#21170C213yesmaximal, complemented
9#21169C213yesmaximal, complemented
10C2×#1948C213yesmaximal, complemented
11#20348C212yesmaximal
12C22×#131C2212yes—
13C2×#520C2212yes—
14C2×#520C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17#2912—216no—
18C2×#428C2216yes—
19C2×#428C2216yes—
20C2×#427C2216yes—
21C22×#106C2216yes—
22#2906C4116yescomplemented
23C2×#424C2216yes—
24C2×#423C2213yes—
25C22×C4⋊C8C2213yes—
26Q8×C4⋊C4—21no—
27Q8×C4⋊C4—21no—
28D4×C4⋊C4—23no—
29D4×C4⋊C4—23no—
30#2317C2216yes—
31#2285C2213yes—
32C2×#384C2213yes—
33#2276—21no—
34#2275—23no—
35#2274C2213yes—
36#2274C2213yes—
37C42×Q8—21no—
38C42×D4—23no—
39C4×#117C2213yes—
40C2×C4×C4⋊C4C2213yes—
41#1949C2216yes—
42#1949C2216yes—
43#1949C22112yes—
44#1948C2213yes—
45#1948C2216yes—
46#1948C22112yes—
47#1948C2211yescharacteristic
48#1948C2216yes—
49#1947C2213yes—
50#1947C2213yes—
51#1947C2216yes—
52#1736C2216yes—
53#1736C2216yes—
54#1735C2216yes—
55C2×#340C2216yes—
56C22×#41C2311yescharacteristic
57C2×#131—28no—
58C2×#131—28no—
59C2×#131C2×C4116yes—
60C22×M16C2316yes—
61C23×C8C2316yes—
62#520—24no—
63#520—24no—
64#520—28no—
65#520—28no—
66#520D418yes—
67#520Q818yes—
68C4×#41C2×C418yes—
69C4×#41Q818yes—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×D4—23no—
76C2×C4×D4—23no—
77C2×C4×D4—26no—
78C2×C4×D4D413yes—
79C2×C4×D4D413yes—
80C2×#117C2316yes—
81C2×#117C2313yes—
82C2×#117C2313yes—
83C2×#117C2313yes—
84C22×C4⋊C4—23no—
85C22×C4⋊C4—23no—
86C22×C4⋊C4C2316yes—
87C22×C4⋊C4C2313yes—
88C22×C4⋊C4C2313yes—
89C22×C42—23no—
90C22×C42C2313yes—
91#428—224no—
92#428—224no—
93#427—224no—
94C2×#107C2316yes—
95C2×#107C2316yes—
96C2×#106—224no—
97C2×#106C2316yes—
98C2×#106C23112yes—
99C2×#106C2312yes—
100#424C2×C4148yescomplemented
101#423C2×C4124yescomplemented
102C2×C4⋊C8C2×C4124yescomplemented
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2313yes—
105C2×C4⋊C8C2316yes—
106C2×C4⋊C8C2316yes—
107#389—26no—
108#389—26no—
109#385—212no—
110#384—23no—
111#384—23no—
112#384C2316yes—
113#384C2311yescharacteristic
114#384C2313yes—
115#382C2316yes—
116#380—23no—
117#380—23no—
118C4×C4⋊C4—23no—
119C4×C4⋊C4—23no—
120C4×C4⋊C4—23no—
121C4×C4⋊C4—23no—
122C4×C4⋊C4—23no—
123C4×C4⋊C4—23no—
124C4×C4⋊C4—23no—
125C4×C4⋊C4—23no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2313yes—
128C4×C4⋊C4C2313yes—
129C4×C4⋊C4C2316yes—
130C4×C4⋊C4C2313yes—
131C4×C22⋊C4—23no—
132C4×C22⋊C4—23no—
133C4×C22⋊C4—26no—
134C4×C22⋊C4—26no—
135#376—23no—
136C2×#95—23no—
137C43—23no—
138#340C23112yes—
139#340C2316yes—
140#340C2316yes—
141#340C2316yes—
142#340C2312yes—
143C2×#41C22×C414yes—
144C2×#41C2×D414yes—
145C2×#41C2×Q814yes—
146C2×#41C2×Q814yes—
147C2×#41#4114yes—
148C2×#41#4114yes—
149C22×Q8C2×D411yescharacteristic
150C22×D4C2×D413yes—
151C23×C4—23no—
152C23×C4—23no—
153C23×C4—26no—
154C23×C4C2×D413yes—
155C23×C4C2×D413yes—
156C23×C4C2413yes—
157#131—264no—
158C2×M16—212no—
159C2×M16—212no—
160C2×M16C22×C4124yes—
161C22×C8—212no—
162C22×C8—212no—
163C22×C8C22×C4124yes—
164C22×C8C2412yes—
165C22×C8C2416yes—
166C4×Q8—28no—
167C4×Q8—28no—
168C4×Q8—48no—
169C4×Q8C4⋊C418yes—
170C4×Q8C4⋊C418yes—
171C4×D4—224no—
172C4×D4—224no—
173C4×D4—424no—
174C4×D4C4⋊C4124yes—
175C4×D4C4⋊C4124yes—
176#117—26no—
177#117—26no—
178#117—212no—
179#117—212no—
180#117C22×C4112yes—
181#117C2×D4112yes—
182#117C2×Q8112yes—
183#117C2×Q8112yes—
184C2×C4⋊C4—23no—
185C2×C4⋊C4—23no—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—26no—
188C2×C4⋊C4—212no—
189C2×C4⋊C4—212no—
190C2×C4⋊C4—23no—
191C2×C4⋊C4—23no—
192C2×C4⋊C4—26no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—26no—
196C2×C4⋊C4—26no—
197C2×C4⋊C4—86no—
198C2×C4⋊C4—86no—
199C2×C4⋊C4C2×D4112yes—
200C2×C4⋊C4C2×D413yes—
201C2×C4⋊C4C2×D413yes—
202C2×C4⋊C4C2×Q8112yes—
203C2×C4⋊C4C2416yes—
204C2×C4⋊C4C2412yes—
205C2×C4⋊C4C2413yes—
206C2×C4⋊C4C2411yescharacteristic
207C2×C4⋊C4C2413yes—
208C2×C4⋊C4C2411yescharacteristic
209C2×C22⋊C4—23no—
210C2×C22⋊C4—23no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4C2×D416yes—
213C2×C42—23no—
214C2×C42—23no—
215C2×C42—26no—
216C2×C42—21no—
217C2×C42—23no—
218C2×C42—23no—
219C2×C42—23no—
220C2×C42—26no—
221C2×C42—86no—
222C2×C42C22×C4112yes—
223C2×C42C2×D413yes—
224C2×C42C2×D413yes—
225C2×C42C2×Q8112yes—
226C2×C42C2413yes—
227C2×C42C2411yescharacteristic
228#107—212no—
229#107—212no—
230#106—212no—
231#106—224no—
232#106—24no—
233C4⋊C8C22×C414yes—
234C4⋊C8C22×C4112yes—
235C4⋊C8C22×C4124yes—
236C4⋊C8C22×C4124yes—
237#95—23no—
238#95—26no—
239#95—21no—
240#95—23no—
241#95—23no—
242#95—26no—
243#95—26no—
244C24C22×D413yes—
245#41C4×D4116yes—
246#41C4×Q8116yes—
247#41#125116yes—
248#41C4⋊Q8116yes—
249C2×Q8C2×C4⋊C414yes—
250C2×Q8#12114yes—
251C2×Q8C22⋊Q814yes—
252C2×D4C2×C4⋊C4112yes—
253C2×D4#121112yes—
254C2×D4C22⋊Q8112yes—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—212no—
262C22×C4—212no—
263C22×C4—22no—
264C22×C4—26no—
265C22×C4—43no—
266C22×C4—43no—
267C22×C4—412no—
268C22×C4—83no—
269C22×C4—83no—
270C22×C4—83no—
271C22×C4—83no—
272C22×C4—86no—
273C22×C4—86no—
274C22×C4C2×C4⋊C4112yes—
275C22×C4C2×C4⋊C4112yes—
276C22×C4C23×C416yes—
277C22×C4C22×D413yes—
278C22×C4C22×D413yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D413yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D416yes—
283C22×C4C22×Q816yes—
284C22×C4C22×Q816yes—
285C22×C4C2×#4116yes—
286C22×C4C2×#4116yes—
287C22×C4C2511yescharacteristic
288M16—248no—
289C2×C8—248no—
290C2×C8—22no—
291C2×C8—26no—
292C2×C8—22no—
293C2×C8—26no—
294C2×C8C23×C414yes—
295C2×C8C23×C4112yes—
296C4⋊C4—212no—
297C4⋊C4—212no—
298C4⋊C4—23no—
299C4⋊C4—23no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—26no—
303C4⋊C4—22no—
304C4⋊C4—26no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—46no—
308C4⋊C4—412no—
309C4⋊C4—42no—
310C4⋊C4—46no—
311C4⋊C4—86no—
312C4⋊C4—86no—
313C4⋊C4C2×C4⋊C4112yes—
314C4⋊C4C2×C4⋊C4112yes—
315C4⋊C4C22×D416yes—
316C4⋊C4C22×D412yes—
317C4⋊C4C22×Q816yes—
318C4⋊C4C22×Q812yes—
319C22⋊C4—212no—
320C22⋊C4—212no—
321C22⋊C4—412no—
322C22⋊C4C2×C4⋊C4124yes—
323C42—212no—
324C42—212no—
325C42—412no—
326C42—42no—
327C42—46no—
328C42—83no—
329C42—83no—
330C42—86no—
331C42C2×C4⋊C4112yes—
332C42C2×C4⋊C4112yes—
333C42C23×C412yes—
334C42C23×C416yes—
335C42C22×Q812yes—
336C42C22×Q816yes—
337C23—23no—
338C23—23no—
339C23—23no—
340C23—23no—
341C23C22×C4⋊C416yes—
342C23C2×#12116yes—
343C23C2×C22⋊Q816yes—
344C23C23×D411yescharacteristic
345Q8#389116yes—
346D4#389148yes—
347C2×C4—23no—
348C2×C4—21no—
349C2×C4—23no—
350C2×C4—21no—
351C2×C4—23no—
352C2×C4—23no—
353C2×C4—224no—
354C2×C4—22no—
355C2×C4—26no—
356C2×C4—22no—
357C2×C4—26no—
358C2×C4—412no—
359C2×C4—41no—
360C2×C4—43no—
361C2×C4—41no—
362C2×C4—43no—
363C2×C4—43no—
364C2×C4—43no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—46no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—46no—
371C2×C4—46no—
372C2×C4—42no—
373C2×C4—812no—
374C2×C4—812no—
375C2×C4—1612no—
376C2×C4C22×C4⋊C412yes—
377C2×C4C22×C4⋊C416yes—
378C2×C4C22×C4⋊C416yes—
379C2×C4C22×C4⋊C412yes—
380C2×C4C22×C4⋊C416yes—
381C2×C4C2×C4×D4112yes—
382C2×C4C2×C4×Q8112yes—
383C2×C4C2×#12116yes—
384C2×C4C2×C22⋊Q816yes—
385C2×C4C2×#125112yes—
386C2×C4C2×C4⋊Q8112yes—
387C2×C4#57712yes—
388C2×C4#57712yes—
389C2×C4C24×C411yescharacteristic
390C2×C4C23×D411yescharacteristic
391C2×C4C23×Q811yescharacteristic
392C2×C4C23×Q811yescharacteristic
393C2×C4C22×#4111yescharacteristic
394C2×C4C22×#4111yescharacteristic
395C8—24no—
396C8—212no—
397C22—26no—
398C22—26no—
399C22—26no—
400C22—26no—
401C22—26no—
402C22—26no—
403C22C2×#389112yes—
404C22C23×C4⋊C411yescharacteristic
405C22C22×#12111yescharacteristic
406C22C22×C22⋊Q811yescharacteristic
407C22C2×#57711yescharacteristic
408C22C2×#57711yescharacteristic
409C22#358511yescharacteristic
410C22#358611yescharacteristic
411C4—26no—
412C4—22no—
413C4—26no—
414C4—46no—
415C4—42no—
416C4—46no—
417C4—42no—
418C4—82no—
419C4—86no—
420C4C2×#389112yes—
421C4#291214yescomplemented
422C4C22×C4×D411yescharacteristic
423C4C22×C4×Q811yescharacteristic
424C4C22×#12511yescharacteristic
425C4C22×C4⋊Q811yescharacteristic
426C2—212no—
427C2C22×#38911yescharacteristic
428C2C2×#291211yescharacteristic
429C2#6010911yescharacteristic
430C2#6042911yescharacteristic
431C2#6043111yescharacteristic
432C2#6049511yescharacteristic
433C2#6049711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #21175C2semidirect
6. #21174C2semidirect
7. #21171C2semidirect
8. #21170C2semidirect
9. #21169C2semidirect
10. C2×#1948C2semidirect
22. #2906C4semidirect
100. #424C2×C4semidirect
101. #423C2×C4semidirect
102. C2×C4⋊C8C2×C4semidirect
421. C4#2912semidirect

Maximal subgroups

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

subgroupindex
C2×#29122
C2×#29062
#341832
C4×#5202
#211752
#211742
#211712
#211702
#211692
C2×#19482
#203482