Finite Group Databasebeta

← #6380331

Subgroups — #6380331

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2960C211yesmaximal, characteristic
2#34213C212yesmaximal
3#22745C214yesmaximal, complemented
4#22742C214yesmaximal, complemented
5#22741C214yesmaximal, complemented
6#22739C212yesmaximal, complemented
7#22738C212yesmaximal, complemented
8#21174C214yesmaximal
9#21170C212yesmaximal
10#20763C212yesmaximal
11#20760C211yesmaximal, characteristic
12#20453C212yesmaximal
13C2×C4×#106C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15#2960C4116yescomplemented
16C2×#446C2212yes—
17C2×#446C2212yes—
18C2×C8×Q8C2212yes—
19C2×C8×Q8C2212yes—
20C2×#428C2214yes—
21C22×#106C2212yes—
22C2×#424C2212yes—
23C2×#423C2211yescharacteristic
24C2×#405C2212yes—
25C22×C4×C8C2211yescharacteristic
26#2370—24no—
27#2369—22no—
28#2365—22no—
29#2355C2214yes—
30#2355C2218yes—
31Q8×C4⋊C4C2214yes—
32#2329—24no—
33#2317C2214yes—
34C4×C22⋊Q8—24no—
35#2285C2212yes—
36#2274C2214yes—
37C2×C4×C4⋊C4C2212yes—
38#1949C2218yes—
39#1949C2218yes—
40#1949C2218yes—
41#1948C2214yes—
42#1948C2218yes—
43#1948C2218yes—
44#1947C2214yes—
45#1836C2218yes—
46#1835C2212yes—
47#1835C2212yes—
48#1835C2214yes—
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2212yes—
52#1834C2214yes—
53#1776C2218yes—
54C4×#107C2214yes—
55C4×#106C2212yes—
56C4×#106C2214yes—
57C4×#106C2212yes—
58C4×#106C2212yes—
59C4×#106C2212yes—
60#1736C2214yes—
61#1735C2212yes—
62#1735C2212yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C22×M16C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68#533C2×C418yes—
69#533Q818yes—
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8D411yescharacteristic
72C2×#125D412yes—
73C2×C22⋊Q8D412yes—
74C2×C22⋊Q8D412yes—
75C2×C4×Q8—24no—
76C2×C4×Q8C2312yes—
77C2×C4×Q8C2312yes—
78C2×#117C2314yes—
79C2×#117C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C42—22no—
83C22×C42C2311yescharacteristic
84#446C2×C4116yescomplemented
85#446C2×C4116yescomplemented
86C8×Q8C2×C4116yescomplemented
87C8×Q8C2×C4116yescomplemented
88#428—216no—
89C2×#107C2318yes—
90C2×#106—28no—
91C2×#106C2314yes—
92C2×#106C2314yes—
93C2×#106C2318yes—
94#424C2×C4116yes—
95#423C2×C418yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2314yes—
99C2×C4⋊C8C2314yes—
100#405C2×C4116yes—
101C2×#97C2314yes—
102C2×C4×C8C2×C418yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2314yes—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—28no—
109#391—24no—
110#389—24no—
111#389—24no—
112#389—28no—
113#389—24no—
114#387—28no—
115#387—28no—
116#387—28no—
117#385—28no—
118#384C2318yes—
119#384C2312yes—
120#384C2312yes—
121#382C2318yes—
122#382C2314yes—
123#380—22no—
124#380—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2314yes—
134C4×C4⋊C4C2318yes—
135C4×C4⋊C4C2318yes—
136C4×C4⋊C4C2314yes—
137C4×C22⋊C4—28no—
138C2×#95—22no—
139#340C2312yes—
140#340C2312yes—
141#340C2312yes—
142#340C2314yes—
143#340C2312yes—
144#340C2314yes—
145#340C2312yes—
146#340C2318yes—
147#340C2318yes—
148#340C2312yes—
149#340C2312yes—
150#340C2312yes—
151C22×Q8C2×D412yes—
152C22×Q8C2×D412yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—28no—
159C2×M16C22×C418yes—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8C22×C414yes—
163C22×C8C22×C414yes—
164C22×C8C2411yescharacteristic
165C22×C8C2411yescharacteristic
166C22×C8C2411yescharacteristic
167C22×C8C2411yescharacteristic
168C22×C8C2414yes—
169C4⋊Q8C4⋊C418yes—
170C4⋊Q8C4⋊C418yes—
171#125C4⋊C4116yes—
172C22⋊Q8C4⋊C4116yes—
173C22⋊Q8C4⋊C4116yes—
174C4×Q8—88no—
175C4×Q8C22×C418yes—
176C4×Q8C22×C418yes—
177C4×Q8C2×Q818yes—
178C4×Q8C2×Q818yes—
179#117—28no—
180#117—28no—
181#117C22×C418yes—
182#117C2×Q818yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4118yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4C2×D414yes—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—24no—
212C2×C42—84no—
213C2×C42C22×C414yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q814yes—
217C2×C42C2414yes—
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—216no—
221#106—28no—
222#106—28no—
223#106—216no—
224C4⋊C8C22×C418yes—
225C4⋊C8C22×C418yes—
226C4⋊C8C22×C4116yes—
227C4⋊C8C22×C4116yes—
228#97C22×C4116yes—
229C4×C8C22×C414yes—
230C4×C8C22×C414yes—
231C4×C8C22×C414yes—
232C4×C8C22×C414yes—
233C4×C8C22×C4116yes—
234#95—24no—
235#95—24no—
236#95—28no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—28no—
241#95—28no—
242C24C22×D411yescharacteristic
243C2×Q8—28no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8C2×C4⋊C418yes—
247C2×Q8C2×C4⋊C418yes—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C2×C4⋊C414yes—
262C22×C4C2×C4⋊C414yes—
263C22×C4C23×C412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D414yes—
267C22×C4C22×D414yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q812yes—
272C22×C4C2×#4114yes—
273C22×C42−1+414yes—
274C22×C4C2511yescharacteristic
275M16—216no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—28no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8C23×C412yes—
284C2×C8C23×C412yes—
285C2×C8C23×C412yes—
286C2×C8C23×C412yes—
287C2×C8C23×C418yes—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C4⋊C4C2×C4⋊C418yes—
304C4⋊C4C2×C4⋊C418yes—
305C4⋊C4C2×C4⋊C4116yes—
306C4⋊C4C2×C4⋊C4116yes—
307C4⋊C4C2×C4⋊C418yes—
308C4⋊C4C2×C4⋊C418yes—
309C4⋊C4C2×#4118yes—
310C22⋊C4—48no—
311C22⋊C4C2×C4⋊C4116yes—
312C42—48no—
313C42—48no—
314C42—88no—
315C42—84no—
316C42—88no—
317C42C2×C4⋊C414yes—
318C42C2×C4⋊C414yes—
319C42C23×C412yes—
320C42C23×C412yes—
321C42C23×C418yes—
322C42C22×Q812yes—
323C42C22×Q812yes—
324C42C22×Q818yes—
325C23—21no—
326C23—21no—
327C23—22no—
328C23C22×C4⋊C412yes—
329C23#54814yes—
330C23C23×D411yescharacteristic
331Q8—216no—
332Q8—48no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—88no—
356C2×C4—88no—
357C2×C4C22×C4⋊C412yes—
358C2×C4C22×C4⋊C412yes—
359C2×C4C22×C4⋊C412yes—
360C2×C4C22×C4⋊C418yes—
361C2×C4C22×C4⋊C418yes—
362C2×C4C22×C4⋊C412yes—
363C2×C4C22×C4⋊C412yes—
364C2×C4C4×#4114yes—
365C2×C4#51914yes—
366C2×C4#54814yes—
367C2×C4#55718yes—
368C2×C4#57714yes—
369C2×C4C24×C411yescharacteristic
370C2×C4C23×Q811yescharacteristic
371C2×C4C22×#4112yes—
372C2×C4C2×2−1+412yes—
373C8—22no—
374C8—22no—
375C8—28no—
376C8—22no—
377C8—22no—
378C22—24no—
379C22—24no—
380C22—22no—
381C22—22no—
382C22#235114yes—
383C22C23×C4⋊C411yescharacteristic
384C22C2×#54812yes—
385C22C2×#57712yes—
386C22#358411yescharacteristic
387C22#358411yescharacteristic
388C4—22no—
389C4—28no—
390C4—22no—
391C4—22no—
392C4—28no—
393C4—48no—
394C4—84no—
395C4—84no—
396C4—84no—
397C4#235114yes—
398C4#291214yes—
399C4C2×C4×#4111yescharacteristic
400C4C2×#51911yescharacteristic
401C4C2×#55712yes—
402C2—24no—
403C2C2×#235111yescharacteristic
404C2C2×#291211yescharacteristic
405C2#6010811yescharacteristic
406C2#6132612yes—
407C2#6132712yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22745C2semidirect
4. #22742C2semidirect
5. #22741C2semidirect
6. #22739C2semidirect
7. #22738C2semidirect
15. #2960C4semidirect
84. #446C2×C4semidirect
85. #446C2×C4semidirect
86. C8×Q8C2×C4semidirect
87. C8×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29602
#342132
#227452
#227422
#227412
#227392
#227382
#211742
#211702
#207632
#207602
#204532
C2×C4×#1062