Finite Group Databasebeta

← #508592

Subgroups — #508592

Order 512.

489 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33215C211yesmaximal, characteristic
2#21158C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13539C211yesmaximal, characteristic, complemented
5#13516C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13515C211yesmaximal, characteristic, complemented
8#13514C211yesmaximal, characteristic, complemented
9#13452C211yesmaximal, characteristic, complemented
10#13451C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic
12#12921C211yesmaximal, characteristic
13#12867C211yesmaximal, characteristic
14#12866C211yesmaximal, characteristic
15#12784C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2628—21no—
21#2316—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#1993—21no—
31#1944—21no—
32#1943—21no—
33#1943—21no—
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1846C2211yescharacteristic
41#1809—21no—
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1736—21no—
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×M16D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#500—41no—
70#500—41no—
71#427—41no—
72#427—41no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—21no—
76C2×#107Q814yescomplemented
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87#405—41no—
88#405—41no—
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#400—41no—
94#399—41no—
95#395—41no—
96#393—41no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—41no—
104#389—41no—
105#382—21no—
106#380—21no—
107#380—21no—
108#380—41no—
109#380—41no—
110#367—41no—
111#367—41no—
112#366—41no—
113#366—41no—
114C4⋊C16—41no—
115C4⋊C16—41no—
116C22⋊C16—22no—
117C22⋊C16—22no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×Q8—41no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—41no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×Q812yes—
140C22×C8#4112yes—
141C22×C8C2411yescharacteristic
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144#117—41no—
145#117—41no—
146#117—42no—
147#117—42no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4D811yescharacteristic
167C2×C4⋊C4D811yescharacteristic
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4D811yescharacteristic
172C2×C22⋊C4D811yescharacteristic
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188#107—24no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—42no—
204#107C2×Q812yes—
205#107C2×Q812yes—
206#106—42no—
207#106—42no—
208#97—41no—
209#97—41no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224C24C2×D811yescharacteristic
225C2×Q8—82no—
226C2×Q8—161no—
227C2×Q8—161no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—161no—
242C22×C4—161no—
243C22×C4C22⋊Q812yes—
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4#13712yes—
250C22×C4C22×D411yescharacteristic
251M16—44no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8C22⋊Q812yes—
264C2×C8#13712yes—
265C2×C8C22×Q811yescharacteristic
266C2×C8C2×#4111yescharacteristic
267C2×C8H(5,𝔽2)11yescharacteristic
268C2×C8H(5,𝔽2)11yescharacteristic
269C2×C82−1+411yescharacteristic
270C2×C82−1+411yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—82no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—82no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—82no—
307C16—41no—
308C16—41no—
309C16—41no—
310C16—41no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23#47412yes—
318C23C22×D811yescharacteristic
319Q8—164no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—164no—
363C2×C4#47412yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4#53411yescharacteristic
366C2×C4#53611yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4#57511yescharacteristic
370C8—22no—
371C8—22no—
372C8—41no—
373C8#53911yescharacteristic
374C8#53911yescharacteristic
375C8#54411yescharacteristic
376C8#54411yescharacteristic
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22C2×#47411yescharacteristic
385C22#318611yescharacteristic
386C22#322611yescharacteristic
387C22#341511yescharacteristic
388C22#341611yescharacteristic
389C22#341611yescharacteristic
390C22#341711yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—82no—
397C4—82no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4#262711yescharacteristic
405C4#307811yescharacteristic
406C4#319511yescharacteristic
407C4#323611yescharacteristic
408C2—42no—
409C2#2996611yescharacteristic
410C2#3363111yescharacteristic
411C2#3363211yescharacteristic
412C2#3376411yescharacteristic
413C2#3376711yescharacteristic
414C2#3380911yescharacteristic
415C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13539C2semidirect
4. #13539C2semidirect
5. #13516C2semidirect
6. #13516C2semidirect
7. #13515C2semidirect
8. #13514C2semidirect
9. #13452C2semidirect
10. #13451C2semidirect
76. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
#332152
#211582
#135392
#135162
#135152
#135142
#134522
#134512
#129212
#128672
#128662
#127842