Finite Group Databasebeta

← #508968

Subgroups — #508968

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic
2#33215C211yesmaximal, characteristic
3#13507C211yesmaximal, characteristic, complemented
4#13507C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13453C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13452C211yesmaximal, characteristic, complemented
10#13451C211yesmaximal, characteristic, complemented
11#12836C211yesmaximal, characteristic
12#12836C211yesmaximal, characteristic
13#12835C211yesmaximal, characteristic
14#12834C211yesmaximal, characteristic
15#12545C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#499C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2629—21no—
22#2628—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31#1993—21no—
32#1992—21no—
33#1944—21no—
34#1943—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1846C2211yescharacteristic
40#1845C2211yescharacteristic
41#1809—21no—
42#1809—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×M16D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#125—21no—
63C2×#117D411yescharacteristic
64C2×M32—41no—
65C2×M32—41no—
66C2×M32Q814yescomplemented
67C22×C16—41no—
68C22×C16—41no—
69C22×C16Q814yescomplemented
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500—41no—
75#500—41no—
76#499—41no—
77#499—41no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90#405—41no—
91#405—41no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#399—41no—
98#398—41no—
99#398—41no—
100#396—41no—
101#396—41no—
102#395—41no—
103#393—41no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#367—41no—
115#367—41no—
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120#366—41no—
121#366—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130C22×Q8—41no—
131C23×C4C2×D411yescharacteristic
132C2×M16—41no—
133C2×M16—41no—
134C2×M16Q1614yes—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×Q812yes—
143C22×C8C2×Q812yes—
144C22×C8C2411yescharacteristic
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#125—42no—
148#125—42no—
149#117—41no—
150#117—41no—
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—21no—
158C2×C4⋊C4—21no—
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⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4D811yescharacteristic
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4D811yescharacteristic
175C2×C22⋊C4D811yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178M32—44no—
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—44no—
188C2×C16C2×Q812yes—
189C2×C16C2×Q812yes—
190C2×C16C2×Q812yes—
191C2×C16C2×Q812yes—
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—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#107—42no—
210#106—42no—
211#106—42no—
212#97—41no—
213#97—41no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228C24C2×D811yescharacteristic
229C2×Q8—82no—
230C2×Q8—161no—
231C2×Q8—161no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C4⋊Q812yes—
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×Q1612yes—
251C22×C4C22×D411yescharacteristic
252M16—44no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8C4⋊Q812yes—
264C2×C8C2×Q1612yes—
265C2×C8C2×Q1612yes—
266C2×C8C22×Q811yescharacteristic
267C2×C8C22×Q811yescharacteristic
268C2×C8H(5,𝔽2)11yescharacteristic
269C2×C8H(5,𝔽2)11yescharacteristic
270C2×C8H(5,𝔽2)11yescharacteristic
271C2×C82−1+411yescharacteristic
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
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—82no—
292C4⋊C4—82no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C16—22no—
310C16—22no—
311C16—22no—
312C16—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#50012yes—
320C23C22×D811yescharacteristic
321Q8—164no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
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—21no—
339C2×C4—21no—
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—81no—
363C2×C4—81no—
364C2×C4#50012yes—
365C2×C4C2×C4⋊Q811yescharacteristic
366C2×C4#53411yescharacteristic
367C2×C4#53711yescharacteristic
368C2×C4C22×Q1611yescharacteristic
369C2×C4#57511yescharacteristic
370C2×C4#57511yescharacteristic
371C8—22no—
372C8—22no—
373C8—22no—
374C8#54311yescharacteristic
375C8#54311yescharacteristic
376C8#54411yescharacteristic
377C8#54411yescharacteristic
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22C2×#50011yescharacteristic
386C22#318611yescharacteristic
387C22#324511yescharacteristic
388C22#341511yescharacteristic
389C22#341611yescharacteristic
390C22#341611yescharacteristic
391C22#341711yescharacteristic
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—161no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4#265911yescharacteristic
406C4#316211yescharacteristic
407C4#318911yescharacteristic
408C4#324911yescharacteristic
409C2—42no—
410C2#3029111yescharacteristic
411C2#3372911yescharacteristic
412C2#3373011yescharacteristic
413C2#3376411yescharacteristic
414C2#3376711yescharacteristic
415C2#3382311yescharacteristic
416C2#3382411yescharacteristic
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. #13507C2semidirect
4. #13507C2semidirect
5. #13506C2semidirect
6. #13505C2semidirect
7. #13453C2semidirect
8. #13453C2semidirect
9. #13452C2semidirect
10. #13451C2semidirect
66. C2×M32Q8semidirect
69. C22×C16Q8semidirect

Maximal subgroups

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

subgroupindex
#332162
#332152
#135072
#135062
#135052
#134532
#134522
#134512
#128362
#128352
#128342
#125452