Finite Group Databasebeta

← #498269

Subgroups — #498269

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic
2#23160C211yesmaximal, characteristic, complemented
3C4⋊C4×D8C211yesmaximal, characteristic
4#13284C211yesmaximal, characteristic, complemented
5#13177C211yesmaximal, characteristic, complemented
6#13168C211yesmaximal, characteristic, complemented
7#13166C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13109C211yesmaximal, characteristic, complemented
10#12937C211yesmaximal, characteristic
11#12937C211yesmaximal, characteristic
12#12663C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic
14#12637C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2370—21no—
21D4×C4⋊C4—21no—
22#2228C418yescomplemented
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1989—21no—
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1920—21no—
41#1919—21no—
42#1862—21no—
43#1862—21no—
44#1845C2211yescharacteristic
45#1802—21no—
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#125—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C22×C4⋊C4—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#499—22no—
71#499C2×C414yescomplemented
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8—41no—
76C4×D8—41no—
77C4×D8C2×C414yes—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—41no—
99#387—41no—
100#387—41no—
101#385—41no—
102#385—41no—
103#384—21no—
104#380—41no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—41no—
109C4×C4⋊C4D411yescharacteristic
110#367—41no—
111#367—41no—
112#367C2×C414yescomplemented
113#366C2×C414yescomplemented
114C4⋊C16C2×C414yescomplemented
115#357—22no—
116#357C2×C414yes—
117#357C2×C414yes—
118#357Q814yes—
119#346—22no—
120#346—41no—
121C4×C16—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#336D411yescharacteristic
130#336D411yescharacteristic
131C22×D4—21no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8C22×C412yes—
139C2×D8C2×Q812yes—
140C2×D8#4112yes—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2411yescharacteristic
145#125—24no—
146#125—42no—
147C4×D4—24no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—81no—
152C4×D4—81no—
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—21no—
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⋊C4—84no—
169C2×C4⋊C4D811yescharacteristic
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16C22×C412yes—
190C2×C16C22×C412yes—
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—81no—
203#107C22×C412yes—
204#107C22×C412yes—
205#107C22×C412yes—
206#107C2×Q812yes—
207#106—24no—
208C4⋊C8—22no—
209#102—24no—
210#102—42no—
211#102—42no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8C22×C412yes—
217C4×C8#4112yes—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—44no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—161no—
249C22×C4—161no—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C22×D411yescharacteristic
257D8—22no—
258D8—22no—
259D8—22no—
260D8C4×Q814yes—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8C4×D412yes—
270C2×C8C23×C411yescharacteristic
271C2×C8C22×Q811yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8H(5,𝔽2)11yescharacteristic
275C2×C82−1+411yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—82no—
308C42—161no—
309C42—161no—
310C42C4×D412yes—
311C42#13512yes—
312C16—22no—
313C16—22no—
314C16—22no—
315C16—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23C22×D811yescharacteristic
324D4—24no—
325D4—42no—
326D4—42no—
327D4—42no—
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—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—88no—
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—81no—
365C2×C4—164no—
366C2×C4—164no—
367C2×C4C4×D812yes—
368C2×C4C4×D812yes—
369C2×C4#50912yes—
370C2×C4C2×C4×D411yescharacteristic
371C2×C4#54611yescharacteristic
372C2×C4D4×Q811yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C2×C4#57711yescharacteristic
375C2×C4#57811yescharacteristic
376C8—22no—
377C8—41no—
378C8—41no—
379C8C2×C4×Q811yescharacteristic
380C8#52011yescharacteristic
381C8#55211yescharacteristic
382C8#55411yescharacteristic
383C22—44no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22C2×C4×D811yescharacteristic
391C22#329411yescharacteristic
392C22Q8×D811yescharacteristic
393C22C2×#50911yescharacteristic
394C22#341411yescharacteristic
395C22#341611yescharacteristic
396C22#341611yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4#236011yescharacteristic
411C4#294711yescharacteristic
412C4#330211yescharacteristic
413C4#338111yescharacteristic
414C2—84no—
415C2#2296211yescharacteristic
416C2C4×#50911yescharacteristic
417C2#3346011yescharacteristic
418C2#3387411yescharacteristic
419C2#3387711yescharacteristic
420C2#3395111yescharacteristic
421C2#3395511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23160C2semidirect
4. #13284C2semidirect
5. #13177C2semidirect
6. #13168C2semidirect
7. #13166C2semidirect
8. #13111C2semidirect
9. #13109C2semidirect
12. #12663C2semidirect
22. #2228C4semidirect
71. #499C2×C4semidirect
112. #367C2×C4semidirect
113. #366C2×C4semidirect
114. C4⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
#231602
C4⋊C4×D82
#132842
#131772
#131682
#131662
#131112
#131092
#129372
#126632
#126372
C4×#3572