Finite Group Databasebeta

← #498233

Subgroups — #498233

Order 512.

553 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33648C211yesmaximal, characteristic
2#23031C211yesmaximal, characteristic
3#22607C211yesmaximal, characteristic, complemented
4#13174C211yesmaximal, characteristic, complemented
5#13155C211yesmaximal, characteristic, complemented
6#13142C211yesmaximal, characteristic, complemented
7#13119C211yesmaximal, characteristic, complemented
8#13118C211yesmaximal, characteristic, complemented
9#13067C211yesmaximal, characteristic
10#12956C211yesmaximal, characteristic, complemented
11#12930C211yesmaximal, characteristic, complemented
12#12677C211yesmaximal, characteristic, complemented
13#12647C211yesmaximal, characteristic, complemented
14#12616C211yesmaximal, characteristic, complemented
15#12567C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#442C2211yescharacteristic
19#2924—21no—
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2363—21no—
23#2234C412yescomplemented
24#2234C412yescomplemented
25#2233C412yescomplemented
26#2232C412yescomplemented
27#2159C2211yescharacteristic
28#2159C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#365C2211yescharacteristic
31C2×#361C2211yescharacteristic
32C2×#361C2211yescharacteristic
33#2140C2211yescharacteristic
34#2140C2211yescharacteristic
35#2112C2211yescharacteristic
36C2×#347C2211yescharacteristic
37C2×#347C2211yescharacteristic
38#1972C2211yescharacteristic
39#1946C2211yescharacteristic
40#1946C2211yescharacteristic
41#1943C2211yescharacteristic
42#1925—21no—
43#1922—21no—
44#1922—21no—
45#1875—21no—
46#1862—21no—
47#1826—21no—
48#1810C2211yescharacteristic
49#1776C2211yescharacteristic
50#1773C2211yescharacteristic
51#1773C2211yescharacteristic
52#1752C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#121—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×M32—21no—
70C2×M32—22no—
71C2×M32—41no—
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76C22×C16—21no—
77#466—21no—
78#466—21no—
79#466C2×C412yes—
80#466C2×C412yes—
81#442—22no—
82#442—41no—
83#433—41no—
84#433—41no—
85#432—41no—
86#431—41no—
87C2×#108C2311yescharacteristic
88C2×#108C2311yescharacteristic
89#427—21no—
90#427—21no—
91#427—22no—
92#427—41no—
93#427C2×C411yescharacteristic, complemented
94#427C2×C411yescharacteristic, complemented
95#427C2311yescharacteristic
96#427C2311yescharacteristic
97C2×#107—21no—
98C2×#107C2×C411yescharacteristic, complemented
99C2×#107C2×C411yescharacteristic, complemented
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×#106C2311yescharacteristic
103C2×#106C2311yescharacteristic
104#424—21no—
105#424—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109#407—21no—
110C2×C22⋊C8D411yescharacteristic
111#405—21no—
112#405—21no—
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115#390—21no—
116#390—41no—
117#389—21no—
118#388—41no—
119#386—41no—
120C4×C4⋊C4—21no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C412yescomplemented
124#367C2×C412yescomplemented
125#366C2×C412yescomplemented
126#366C2×C412yescomplemented
127#365—22no—
128#365—41no—
129#361—22no—
130#361—41no—
131#357C2×C412yes—
132#357C2×C412yes—
133#357C2×C412yes—
134#357C2×C412yes—
135C22⋊C16C2×C412yescomplemented
136C22⋊C16C2×C412yescomplemented
137#347—22no—
138#347—41no—
139#341—21no—
140#340—21no—
141#340—21no—
142#339—21no—
143#339—21no—
144#339—21no—
145#339—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148C22×D4—21no—
149C22×D4—41no—
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8C22×C412yes—
158C2×M16—21no—
159C2×M16—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8#4111yescharacteristic
170C22×C8#4111yescharacteristic
171C22×C8C2411yescharacteristic
172#121—22no—
173#121—22no—
174#121—41no—
175#121—41no—
176C4×D4—42no—
177C4×D4—81no—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—41no—
183#117—41no—
184#117—42no—
185#117—81no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204M32—21no—
205M32—21no—
206M32—21no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—42no—
212M32#4112yes—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—42no—
216C2×C16C22×C412yes—
217C2×C16C22×C412yes—
218#108—21no—
219#108—21no—
220#108—21no—
221#108—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—42no—
226#107C22×C411yescharacteristic
227#107C22×C411yescharacteristic
228#107C22×C411yescharacteristic
229#107C22×C411yescharacteristic
230#107C22×C412yes—
231#106—21no—
232#106—21no—
233#106—21no—
234#106—21no—
235#106—21no—
236#106—21no—
237#106—21no—
238#106#4112yes—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243#102—22no—
244#102—22no—
245#102—42no—
246#102—81no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251#97—21no—
252#97—21no—
253#97—21no—
254#97—21no—
255#97—41no—
256#97—41no—
257C4×C8—41no—
258C4×C8—41no—
259#95—21no—
260#95—41no—
261#95—41no—
262C24—41no—
263C24AGL(1,ℤ/8)11yescharacteristic
264C2×D4—22no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—161no—
273C2×D4—161no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4AGL(1,ℤ/8)11yescharacteristic
291C22×C4AGL(1,ℤ/8)11yescharacteristic
292C22×C4AGL(1,ℤ/8)11yescharacteristic
293C22×C4C22×D411yescharacteristic
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8C4×D411yescharacteristic
321C2×C8C4×D411yescharacteristic
322C2×C8C23×C411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C2×C82−1+411yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—161no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—81no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—82no—
361C42—161no—
362C16—22no—
363C16—22no—
364C16—22no—
365C16—41no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23#44211yescharacteristic
378C23#44211yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380D4—42no—
381D4—42no—
382D4—81no—
383D4—81no—
384D4—162no—
385D4—162no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—161no—
417C2×C4—164no—
418C2×C4#44211yescharacteristic
419C2×C4#44211yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57811yescharacteristic
426C8—21no—
427C8—21no—
428C8—41no—
429C8—41no—
430C8—81no—
431C8—81no—
432C8C4×#4111yescharacteristic
433C8#52011yescharacteristic
434C8#55111yescharacteristic
435C8#55211yescharacteristic
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22C2×#44211yescharacteristic
454C22#331011yescharacteristic
455C22#334311yescharacteristic
456C4—21no—
457C4—21no—
458C4—81no—
459C4—81no—
460C4—82no—
461C4—81no—
462C4—81no—
463C4—82no—
464C4—81no—
465C4—81no—
466C4—161no—
467C4#235911yescharacteristic
468C4#294711yescharacteristic
469C4#330811yescharacteristic
470C4#334511yescharacteristic
471C2—21no—
472C2—21no—
473C2—41no—
474C2—41no—
475C2—162no—
476C2#2293811yescharacteristic
477C2#3346811yescharacteristic
478C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22607C2semidirect
4. #13174C2semidirect
5. #13155C2semidirect
6. #13142C2semidirect
7. #13119C2semidirect
8. #13118C2semidirect
10. #12956C2semidirect
11. #12930C2semidirect
12. #12677C2semidirect
13. #12647C2semidirect
14. #12616C2semidirect
15. #12567C2semidirect
23. #2234C4semidirect
24. #2234C4semidirect
25. #2233C4semidirect
26. #2232C4semidirect
93. #427C2×C4semidirect
94. #427C2×C4semidirect
98. C2×#107C2×C4semidirect
99. C2×#107C2×C4semidirect
123. #367C2×C4semidirect
124. #367C2×C4semidirect
125. #366C2×C4semidirect
126. #366C2×C4semidirect
135. C22⋊C16C2×C4semidirect
136. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#336482
#230312
#226072
#131742
#131552
#131422
#131192
#131182
#130672
#129562
#129302
#126772
#126472
#126162
#125672