Finite Group Databasebeta

← #496979

Subgroups — #496979

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33584C211yesmaximal, characteristic
2#21694C211yesmaximal, characteristic, complemented
3#21295C211yesmaximal, characteristic, complemented
4#13258C211yesmaximal, characteristic, complemented
5#13250C211yesmaximal, characteristic, complemented
6#13249C211yesmaximal, characteristic
7#13183C211yesmaximal, characteristic, complemented
8#13155C211yesmaximal, characteristic, complemented
9#13154C211yesmaximal, characteristic, complemented
10#13038C211yesmaximal, characteristic, complemented
11#13037C211yesmaximal, characteristic
12#12957C211yesmaximal, characteristic, complemented
13#12956C211yesmaximal, characteristic, complemented
14#12816C211yesmaximal, characteristic, complemented
15#12657C211yesmaximal, characteristic, complemented
16C2×#510C2211yescharacteristic
17C2×#509C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×#442C2211yescharacteristic
20C2×#441C2211yescharacteristic
21C4×#135C2211yescharacteristic
22C2×C4×SD16—21no—
23#2923—21no—
24#2326—21no—
25#2211C414yescomplemented
26#2210C414yescomplemented
27C2×#365C2211yescharacteristic
28C2×#365C2211yescharacteristic
29#2150C2211yescharacteristic
30#2150C2211yescharacteristic
31C2×#362C2211yescharacteristic
32C2×#362C2211yescharacteristic
33C2×#361C2211yescharacteristic
34C2×#361C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C2×#360C2211yescharacteristic
37#2141C2211yescharacteristic
38#2140C2211yescharacteristic
39C2×#347C2211yescharacteristic
40C2×#347C2211yescharacteristic
41#1975C2211yescharacteristic
42#1975C2211yescharacteristic
43#1972C2211yescharacteristic
44#1971C2211yescharacteristic
45#1968—21no—
46#1959—21no—
47#1946C2211yescharacteristic
48#1946C2211yescharacteristic
49#1937—21no—
50#1937—21no—
51#1844C2211yescharacteristic
52C4×#108C2211yescharacteristic
53#1759—21no—
54#1726C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1375C2211yescharacteristic
60#1375C2211yescharacteristic
61C2×#135C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C2×#124—21no—
66C4×#41—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69#510—41no—
70#510—41no—
71#509—41no—
72#509—41no—
73C2×Q32C2×C412yescomplemented
74C2×SD32C2×C412yes—
75C2×SD32C2×C412yescomplemented
76C2×D16C2×C412yes—
77C2×M32C2311yescharacteristic
78C2×M32C2311yescharacteristic
79C2×M32C2311yescharacteristic
80C2×M32C2311yescharacteristic
81#495—22no—
82#495C2×C414yes—
83#442—41no—
84#442—41no—
85#441—41no—
86#441—41no—
87C4×Q16C2×C412yescomplemented
88C4×SD16—22no—
89C4×SD16—44no—
90C4×D8C2×C412yescomplemented
91#433—42no—
92#432—42no—
93C2×#108C2311yescharacteristic
94C2×#108C2311yescharacteristic
95#427C2311yescharacteristic
96C2×#106—21no—
97C2×#106C2311yescharacteristic
98C2×#106C2311yescharacteristic
99#424—21no—
100#424—21no—
101C2×#104—21no—
102C2×#104—21no—
103#416—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108#408—21no—
109#408—21no—
110C4×M16—21no—
111C2×#97D411yescharacteristic
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#390—21no—
119#388—42no—
120#385—42no—
121#384—21no—
122#376D411yescharacteristic
123#365—41no—
124#365—41no—
125C4⋊C16—22no—
126C4⋊C16C2×C414yescomplemented
127#362—41no—
128#362—41no—
129#361—41no—
130#361—41no—
131#360—41no—
132#360—41no—
133#358C2×C414yescomplemented
134#357C2×C414yes—
135#347—41no—
136#347—41no—
137#341—21no—
138#339—21no—
139#339—21no—
140C2×#41—21no—
141C22×Q8—21no—
142C22×D4—21no—
143C23×C4—41no—
144#135—21no—
145#135—21no—
146#135—21no—
147#135—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16C22×C412yes—
154C2×Q16C22×C411yescharacteristic
155C2×SD16—24no—
156C2×SD16—21no—
157C2×SD16—42no—
158C2×SD16—42no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8C22×C412yes—
165C2×D8C22×C411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2411yescharacteristic
172#124—24no—
173#124—41no—
174#124—41no—
175C4×Q8—22no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—22no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182#117—21no—
183#117—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—42no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196Q32—42no—
197SD32—42no—
198SD32—42no—
199D16—42no—
200M32—21no—
201M32—21no—
202M32—21no—
203M32—21no—
204M32—21no—
205M32—21no—
206M32—21no—
207M32—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16C22×C412yes—
211C2×C16C22×C412yes—
212#108—21no—
213#108—21no—
214#108—21no—
215#108—21no—
216#107C22×C412yes—
217#106—21no—
218#106—21no—
219#106—21no—
220#106—21no—
221#106—21no—
222#106—42no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#104—42no—
226#104—42no—
227#103—22no—
228#103—42no—
229#103—42no—
230#102—22no—
231#102—42no—
232#102—42no—
233C22⋊C8—42no—
234C22⋊C8—42no—
235#97—21no—
236#97—21no—
237#97—21no—
238#97—21no—
239#97—21no—
240#97—21no—
241#97—41no—
242C4×C8—22no—
243C4×C8—41no—
244C4×C8—41no—
245C4×C8C22×C412yes—
246C4×C8#4112yes—
247#95—22no—
248#95—42no—
249C24—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—162no—
278C22×C4AGL(1,ℤ/8)12yes—
279C22×C4C22×D411yescharacteristic
280Q16—21no—
281Q16—22no—
282Q16—22no—
283Q16—21no—
284Q16—41no—
285Q16—41no—
286SD16—48no—
287SD16—41no—
288SD16—41no—
289D8—21no—
290D8—22no—
291D8—22no—
292D8—21no—
293D8—41no—
294D8—41no—
295M16—41no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8C4×D412yes—
310C2×C8C23×C411yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C82−1+411yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—82no—
323C4⋊C4—162no—
324C22⋊C4—41no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—82no—
329C22⋊C4—162no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—41no—
334C42—42no—
335C42—82no—
336C42C4×D412yes—
337C42AGL(1,ℤ/8)12yes—
338C42AGL(1,ℤ/8)11yescharacteristic
339C42AGL(1,ℤ/8)11yescharacteristic
340C16—22no—
341C16—22no—
342C16—41no—
343C16—41no—
344C23—42no—
345C23—41no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23C2×AGL(1,ℤ/8)11yescharacteristic
351Q8—41no—
352Q8—42no—
353Q8—42no—
354Q8—41no—
355Q8—81no—
356Q8—81no—
357D4—42no—
358D4—41no—
359D4—42no—
360D4—41no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—162no—
382C2×C4—164no—
383C2×C4#44212yes—
384C2×C4#44212yes—
385C2×C4C2×C4×D411yescharacteristic
386C2×C4#53511yescharacteristic
387C2×C4#53711yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4#57511yescharacteristic
391C8—22no—
392C8—41no—
393C8—41no—
394C8—81no—
395C8—81no—
396C8#51811yescharacteristic
397C8#52011yescharacteristic
398C8#54011yescharacteristic
399C8#54211yescharacteristic
400C22—22no—
401C22—81no—
402C22—82no—
403C22—82no—
404C22—84no—
405C22—161no—
406C22—161no—
407C22—162no—
408C22C2×#44211yescharacteristic
409C22#321711yescharacteristic
410C22#325111yescharacteristic
411C4—22no—
412C4—22no—
413C4—42no—
414C4—81no—
415C4—81no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4#232411yescharacteristic
421C4C4×AGL(1,ℤ/8)11yescharacteristic
422C4#321411yescharacteristic
423C4#325311yescharacteristic
424C2—21no—
425C2—21no—
426C2—161no—
427C2—162no—
428C2#2215111yescharacteristic
429C2#3346912yes—
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#509C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21694C2semidirect
3. #21295C2semidirect
4. #13258C2semidirect
5. #13250C2semidirect
7. #13183C2semidirect
8. #13155C2semidirect
9. #13154C2semidirect
10. #13038C2semidirect
12. #12957C2semidirect
13. #12956C2semidirect
14. #12816C2semidirect
15. #12657C2semidirect
25. #2211C4semidirect
26. #2210C4semidirect
73. C2×Q32C2×C4semidirect
75. C2×SD32C2×C4semidirect
87. C4×Q16C2×C4semidirect
90. C4×D8C2×C4semidirect
126. C4⋊C16C2×C4semidirect
133. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335842
#216942
#212952
#132582
#132502
#132492
#131832
#131552
#131542
#130382
#130372
#129572
#129562
#128162
#126572