Finite Group Databasebeta

← #184205

Subgroups — #184205

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21114C211yesmaximal, characteristic
2#20574C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic
4C2×#1590C211yesmaximal, characteristic
5C2×#1588C211yesmaximal, characteristic
6#8288C211yesmaximal, characteristic, complemented
7#8282C211yesmaximal, characteristic, complemented
8#8279C211yesmaximal, characteristic, complemented
9#8210C211yesmaximal, characteristic
10#8209C211yesmaximal, characteristic, complemented
11#8208C211yesmaximal, characteristic
12#8203C211yesmaximal, characteristic, complemented
13#8182C211yesmaximal, characteristic, complemented
14#8174C211yesmaximal, characteristic
15#7845C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21#2295C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1928C412yes—
24#1919C412yes—
25#1877C412yes—
26#1862C412yes—
27C4×#107C2211yescharacteristic
28#1771—21no—
29#1770—21no—
30#1770C2211yescharacteristic
31#1770C2211yescharacteristic
32#1770C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1768—21no—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37C4×#103C2211yescharacteristic
38C4×#102C2211yescharacteristic
39#1753—21no—
40#1753—21no—
41#1749—21no—
42#1746—21no—
43#1744C2211yescharacteristic
44#1744C2211yescharacteristic
45#1744C2211yescharacteristic
46#1728—21no—
47#1590C418yescomplemented
48#1588C418yescomplemented
49C2×#335C2211yescharacteristic
50C2×#334C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65C22×SD16C2×C411yescharacteristic
66C22×M16—21no—
67C22×M16—21no—
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2311yescharacteristic
71C22×C42—21no—
72#446—22no—
73#446C2×C414yescomplemented
74C8×Q8—22no—
75C8×Q8C2×C414yescomplemented
76C4×SD16C2×C414yes—
77C4×SD16C2×C414yes—
78#436—22no—
79#436—22no—
80#436C2×C414yes—
81#436C2×C414yes—
82C2×#107C2×C412yes—
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103C2×C412yes—
91C2×#103C2×C411yescharacteristic
92C2×#102C2×C412yes—
93C2×#102C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391C2×C412yes—
111#389C2×C412yes—
112#387—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2311yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119#376—21no—
120C43C2311yescharacteristic
121#340C2×C412yes—
122#340C2×C412yes—
123#340C2×C412yes—
124#336—21no—
125#336—21no—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130#336—22no—
131#336C2×C412yes—
132#336C2×C412yes—
133#336D411yescharacteristic
134#336D411yescharacteristic
135#336D411yescharacteristic
136#336D411yescharacteristic
137#335C2×C414yescomplemented
138#334C2×C414yescomplemented
139Q8⋊C8C2×C414yescomplemented
140Q8⋊C8C2×C414yescomplemented
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yes—
143#322—22no—
144#322—22no—
145#322C2×C414yescomplemented
146C82C2×C414yescomplemented
147C22×Q8C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—22no—
151C2×SD16—42no—
152C2×SD16C4214yes—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—81no—
156C2×M16—81no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C22×C412yes—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8C22×C412yes—
175C4×Q8C22×C412yes—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4C22×C412yes—
181C4×D4C22×C412yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4112yes—
200C2×C42C2411yescharacteristic
201#107—42no—
202#106C4214yes—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8C22×C412yes—
210C4⋊C8C22×C412yes—
211C4⋊C8C22×C412yes—
212C4⋊C8C22×C412yes—
213#103—42no—
214#103C4214yes—
215#102—42no—
216#102C4214yes—
217C22⋊C8—24no—
218C22⋊C8—24no—
219#97—24no—
220#97—81no—
221#97—81no—
222#97—81no—
223#97—81no—
224C4×C8—24no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—81no—
230C4×C8—81no—
231C4×C8C22×C412yes—
232C4×C8C22×C412yes—
233C4×C8C22×C412yes—
234C4×C8C22×C412yes—
235C4×C8C22×C412yes—
236C4×C8C22×C412yes—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245C24—21no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8C2×C4212yes—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4C2×C4212yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273SD16—44no—
274M16—84no—
275M16—84no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8C2×C4212yes—
306C2×C8C2×C4212yes—
307C2×C8C4×D412yes—
308C2×C8C4×D412yes—
309C2×C8C4×D412yes—
310C2×C8C4×D412yes—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—82no—
320C4⋊C4C2×C4212yes—
321C4⋊C4C2×C4212yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—44no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—42no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—82no—
339C42C23×C411yescharacteristic
340C42C23×C411yescharacteristic
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C23—22no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23C2×C4×D411yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—42no—
352D4—22no—
353D4—22no—
354D4—42no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—88no—
380C2×C4C22×C4211yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C4×#4111yescharacteristic
384C2×C4C4×#4111yescharacteristic
385C2×C4C4×#4111yescharacteristic
386C2×C4#56812yes—
387C2×C4#56812yes—
388C2×C4#57611yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—22no—
393C8—22no—
394C8—22no—
395C8—22no—
396C8—22no—
397C8—22no—
398C8—42no—
399C8—42no—
400C8—42no—
401C8—42no—
402C8—42no—
403C8—42no—
404C8—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22C42×D411yescharacteristic
411C22#293012yes—
412C22#294611yescharacteristic
413C22#294711yescharacteristic
414C22#295712yes—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—82no—
423C4—82no—
424C4#287311yescharacteristic
425C4#287511yescharacteristic
426C4#294611yescharacteristic
427C4#294711yescharacteristic
428C4#294911yescharacteristic
429C4#295111yescharacteristic
430C4#296712yes—
431C2—82no—
432C2#1794612yes—
433C2#1794912yes—
434C2#2048711yescharacteristic
435C2#2050411yescharacteristic
436C2#2051411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20574C2semidirect
6. #8288C2semidirect
7. #8282C2semidirect
8. #8279C2semidirect
10. #8209C2semidirect
12. #8203C2semidirect
13. #8182C2semidirect
15. #7845C2semidirect
47. #1590C4semidirect
48. #1588C4semidirect
73. #446C2×C4semidirect
75. C8×Q8C2×C4semidirect
137. #335C2×C4semidirect
138. #334C2×C4semidirect
139. Q8⋊C8C2×C4semidirect
140. Q8⋊C8C2×C4semidirect
145. #322C2×C4semidirect
146. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#211142
#205742
#205682
C2×#15902
C2×#15882
#82882
#82822
#82792
#82102
#82092
#82082
#82032
#81822
#81742
#78452