Finite Group Databasebeta

← #6382719

Subgroups — #6382719

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60660C211yesmaximal, characteristic
2#60193C212yesmaximal
3#23172C218yesmaximal, complemented
4#23171C218yesmaximal, complemented
5#21307C212yesmaximal
6#21163C211yesmaximal, characteristic
7C2×#1946C211yesmaximal, characteristic
8#21042C212yesmaximal
9#21041C212yesmaximal
10#20460C211yesmaximal, characteristic
11#20446C211yesmaximal, characteristic
12C2×#1773C211yesmaximal, characteristic
13#20436C211yesmaximal, characteristic
14#3458C2212yes—
15#3164C414yescomplemented
16#3164C414yescomplemented
17#3163C414yescomplemented
18#3163C414yescomplemented
19#3162C2214yes—
20#3160C2214yes—
21#2962—28no—
22#2962C22116yes—
23#2961—24no—
24#2960—24no—
25C22×#108C2211yescharacteristic
26C22×#108C2211yescharacteristic
27#2911C2212yes—
28C2×#427C2211yescharacteristic
29C2×#427C2211yescharacteristic
30#2905C2212yes—
31C2×#424C2212yes—
32C2×#424C2211yescharacteristic
33#2873C2212yes—
34#2873C2212yes—
35C2×#405C2212yes—
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C22×#97C2211yescharacteristic
39#2296C2211yescharacteristic
40#1986C22116yes—
41#1946C2214yes—
42#1946C2214yes—
43#1946C2214yes—
44#1945C2214yes—
45#1922C22116yes—
46#1922C22116yes—
47#1778C2214yes—
48C4×#108C2214yes—
49#1773C2214yes—
50#1773C2214yes—
51#1773C2214yes—
52#1772C2214yes—
53#1771C2214yes—
54#1770C2214yes—
55#1738C2212yes—
56#1733C2212yes—
57#1733C2212yes—
58C2×#339C2212yes—
59C2×#339C2212yes—
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C22×M16C2312yes—
63C22×M16C2312yes—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68#544—24no—
69#544C2×C418yes—
70#544C2×C418yes—
71#543—24no—
72#533—24no—
73#533—24no—
74#533C2318yes—
75C2×#117C2312yes—
76C2×#117C2312yes—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#501C2×C4116yescomplemented
81#501C2×C4116yescomplemented
82#500C2×C418yescomplemented
83#499C2×C418yescomplemented
84#499C2×C418yescomplemented
85#498C2×C418yescomplemented
86#446—216no—
87#446—216no—
88#446—216no—
89C8×Q8—216no—
90C2×#108—24no—
91C2×#108Q818yescomplemented
92C2×#108C2312yes—
93C2×#108C2312yes—
94C2×#108C2312yes—
95C2×#108C2312yes—
96#428C2×C418yes—
97#428C2×C418yes—
98#427C2×C412yes—
99#427C2×C412yes—
100#427C2×C412yes—
101#427C2×C412yes—
102#427C2312yes—
103#427C2312yes—
104#427C2318yes—
105#427C2312yes—
106#427C2312yes—
107C2×#107C2×C412yes—
108C2×#107C2×C412yes—
109C2×#106C2×C412yes—
110C2×#106C2×C412yes—
111#424—24no—
112#424—22no—
113#424—22no—
114#424D414yes—
115#424D414yes—
116#424C2318yes—
117#424C2318yes—
118#423—24no—
119C2×C4⋊C8—24no—
120C2×C4⋊C8C2314yes—
121#405—24no—
122#405—24no—
123#405—24no—
124#405—24no—
125#405—24no—
126#405C2×C412yes—
127#405C2×C412yes—
128#405C2×C412yes—
129#405C2×C412yes—
130#405C2318yes—
131#405C2318yes—
132C4×M16—24no—
133C4×M16C2318yes—
134C2×#97—22no—
135C2×#97—24no—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×C4×C8—22no—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#389—22no—
144#385—22no—
145C4×C4⋊C4C2314yes—
146#341C2318yes—
147#339C2318yes—
148#339C2318yes—
149#339C2318yes—
150#339C2318yes—
151#336C2314yes—
152#336C2314yes—
153C23×C4—21no—
154C23×C4C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—24no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16C22×C412yes—
165C2×M16C22×C412yes—
166C2×M16C2×D412yes—
167C2×M16C2×D412yes—
168C2×M16C2×Q818yes—
169C2×M16C2414yes—
170C2×M16C2414yes—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×Q814yes—
182C22×C8#4114yes—
183C22×C8C2412yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C22×C8C2411yescharacteristic
187C22×C8C2411yescharacteristic
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8C22×C4116yes—
191#125—28no—
192#125—28no—
193#125C22×C4116yes—
194C22⋊Q8—216no—
195C22⋊Q8—216no—
196C22⋊Q8—216no—
197C4×Q8—216no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117C22×C414yes—
205#117C22×C414yes—
206#117C22×C412yes—
207#117C22×C412yes—
208#117C2×D412yes—
209#117C2×D412yes—
210#117C2414yes—
211#117C2414yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4#4112yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4#4112yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42C2412yes—
227C2×C42C2412yes—
228C2×C42C2412yes—
229#108—24no—
230#108—24no—
231#108C2×Q818yes—
232#108C2×Q818yes—
233#107C22×C414yes—
234#107C22×C414yes—
235#107C22×C414yes—
236#107C22×C414yes—
237#107C22×C4116yes—
238#106C22×C414yes—
239#106C22×C414yes—
240#106C22×C414yes—
241#106C22×C414yes—
242#106C22×C4116yes—
243C4⋊C8—28no—
244C4⋊C8—28no—
245C4⋊C8—216no—
246C4⋊C8C2×D4116yes—
247#97—28no—
248#97—28no—
249#97—24no—
250#97—24no—
251#97—24no—
252#97—28no—
253#97C22×C414yes—
254#97C22×C414yes—
255C4×C8—28no—
256C4×C8—28no—
257C4×C8—24no—
258C4×C8—28no—
259C4×C8C22×C414yes—
260C4×C8C22×C414yes—
261#95—24no—
262C24C2×#4111yescharacteristic
263C2×Q8—216no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4C23×C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×Q812yes—
275C22×C4C2×#4114yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4112yes—
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C22×C4C2511yescharacteristic
281M16—24no—
282M16—24no—
283M16—24no—
284M16—24no—
285M16—28no—
286M16—24no—
287M16—24no—
288M16—24no—
289M16—44no—
290M16C22×Q818yes—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—28no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—44no—
303C2×C8C4×Q814yes—
304C2×C8C4×Q814yes—
305C2×C8C4×Q814yes—
306C2×C8C4×Q814yes—
307C2×C8C23×C414yes—
308C2×C8C23×C411yescharacteristic
309C2×C8C23×C411yescharacteristic
310C2×C8C23×C411yescharacteristic
311C2×C8C23×C411yescharacteristic
312C2×C8C22×D414yes—
313C2×C8C22×D414yes—
314C2×C8C22×Q812yes—
315C2×C8C22×Q812yes—
316C2×C8C2×#4112yes—
317C2×C8C2×#4112yes—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—216no—
321C4⋊C4—216no—
322C4⋊C4—216no—
323C4⋊C4—42no—
324C4⋊C4—84no—
325C4⋊C4C4×D414yes—
326C4⋊C4C4×D414yes—
327C4⋊C4C23×C418yes—
328C4⋊C4C23×C418yes—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—42no—
334C22⋊C4C4×D414yes—
335C22⋊C4C4×D414yes—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42C23×C414yes—
346C42C22×D414yes—
347C23—42no—
348C23C2×C4×D411yescharacteristic
349C23C2×C4×D411yescharacteristic
350C23#55612yes—
351C23#55712yes—
352C23C22×#4111yescharacteristic
353Q8—816no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4C2×C4×D418yes—
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C4×Q812yes—
369C2×C4C2×C4×Q812yes—
370C2×C4#51912yes—
371C2×C4#51912yes—
372C2×C4#54712yes—
373C2×C4#54712yes—
374C2×C4D4×Q812yes—
375C2×C4D4×Q812yes—
376C2×C4#55612yes—
377C2×C4#55712yes—
378C2×C4C24×C411yescharacteristic
379C2×C4C23×D411yescharacteristic
380C2×C4C23×Q811yescharacteristic
381C2×C4C22×#4111yescharacteristic
382C2×C4C2×2−1+411yescharacteristic
383C2×C4C2×2−1+411yescharacteristic
384C8—24no—
385C8—24no—
386C8—24no—
387C8—28no—
388C8—44no—
389C8C2×C4×Q812yes—
390C8C2×C4×Q812yes—
391C8C2×C4×Q812yes—
392C8C2×C4×Q812yes—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22#236512yes—
399C22#236512yes—
400C22C22×C4×D411yescharacteristic
401C22C2×#55611yescharacteristic
402C22C2×#55711yescharacteristic
403C4—28no—
404C4—48no—
405C4—48no—
406C4—82no—
407C4—82no—
408C4#236512yes—
409C4#236512yes—
410C4C22×C4×Q811yescharacteristic
411C4C2×#51911yescharacteristic
412C4C2×#54711yescharacteristic
413C4C2×D4×Q811yescharacteristic
414C2—21no—
415C2—22no—
416C2—22no—
417C2—21no—
418C2C2×#236511yescharacteristic
419C2#6018412yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC24C2×#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#424 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23172C2semidirect
4. #23171C2semidirect
15. #3164C4semidirect
16. #3164C4semidirect
17. #3163C4semidirect
18. #3163C4semidirect
80. #501C2×C4semidirect
81. #501C2×C4semidirect
82. #500C2×C4semidirect
83. #499C2×C4semidirect
84. #499C2×C4semidirect
85. #498C2×C4semidirect
91. C2×#108Q8semidirect

Maximal subgroups

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

subgroupindex
#606602
#601932
#231722
#231712
#213072
#211632
C2×#19462
#210422
#210412
#204602
#204462
C2×#17732
#204362