Finite Group Databasebeta

← #277161

Subgroups — #277161

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#28748C211yesmaximal, characteristic
3#22722C211yesmaximal, characteristic
4C4×#477C211yesmaximal, characteristic
5C2×#1704C212yesmaximal
6#10588C212yesmaximal
7#10541C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9228C212yesmaximal
11#8587C211yesmaximal, characteristic
12C2×#487C2212yes—
13C2×#477C2211yescharacteristic
14C2×#475C2212yes—
15C2×#446C2212yes—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2043—22no—
20#1988C2212yes—
21#1987C2211yescharacteristic
22#1981—21no—
23#1981—21no—
24#1956C2212yes—
25#1948—21no—
26#1948—21no—
27#1947C2211yescharacteristic
28#1940—22no—
29#1939C2212yes—
30#1921—21no—
31#1863—21no—
32#1842—21no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1803—21no—
36#1803—21no—
37#1803C2211yescharacteristic
38C4×#107—21no—
39#1767—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43#1704—28no—
44C2×#334C2212yes—
45C2×#333C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1277C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#487—24no—
59#487—42no—
60#477—22no—
61#477D414yes—
62#475—24no—
63#475—24no—
64#446—24no—
65#446—42no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#106—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#103—21no—
82C2×#103—22no—
83C2×#103—21no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#391—21no—
92#389—21no—
93#389—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2311yescharacteristic
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C43—21no—
108C43C2311yescharacteristic
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#336—22no—
116#336—21no—
117#336D412yes—
118#334—24no—
119#333—24no—
120#332—24no—
121#332—42no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8C2×D411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—81no—
141C4⋊Q8—81no—
142C4⋊Q8C2×D412yes—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—41no—
149C4×Q8—82no—
150C4×Q8C2×D412yes—
151C4×Q8#4112yes—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42C2411yescharacteristic
184#107—24no—
185#107—42no—
186#106—44no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—44no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—81no—
201C4⋊C8—82no—
202C4⋊C8C2×D412yes—
203#103—24no—
204#103—44no—
205#103—44no—
206#103—42no—
207#103—44no—
208#103—82no—
209#103—82no—
210#97—44no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—42no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—41no—
218#95—41no—
219C2×Q8—21no—
220C2×Q8—22no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—81no—
233C2×Q8—81no—
234C2×Q8#12112yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4#12112yes—
294C4⋊C4#13712yes—
295C42—22no—
296C42—22no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—41no—
303C42—42no—
304C42—41no—
305C42—41no—
306C42—42no—
307C42—42no—
308C42—81no—
309C42—82no—
310C42#13512yes—
311C42#13712yes—
312C42#13712yes—
313C42#13712yes—
314C42C22×D411yescharacteristic
315C42C2×#4111yescharacteristic
316C42H(5,𝔽2)11yescharacteristic
317C422−1+411yescharacteristic
318C23C2×#12111yescharacteristic
319Q8—22no—
320Q8—22no—
321Q8—41no—
322Q8—41no—
323Q8—82no—
324Q8—82no—
325Q8—82no—
326Q8—82no—
327Q8—82no—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#46412yes—
360C2×C4#46412yes—
361C2×C4#47012yes—
362C2×C4#47012yes—
363C2×C4#47112yes—
364C2×C4C2×#12111yescharacteristic
365C2×C4#53711yescharacteristic
366C2×C4D4211yescharacteristic
367C2×C4#54711yescharacteristic
368C2×C4#54811yescharacteristic
369C2×C4#55311yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4#57511yescharacteristic
376C2×C4#57611yescharacteristic
377C2×C4#57711yescharacteristic
378C8—42no—
379C8—42no—
380C8—42no—
381C8—81no—
382C8—82no—
383C8—82no—
384C22#241611yescharacteristic
385C22C2×#46411yescharacteristic
386C22#304511yescharacteristic
387C22C2×#47011yescharacteristic
388C22#305611yescharacteristic
389C22C2×#47111yescharacteristic
390C22#306911yescharacteristic
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—161no—
402C4—162no—
403C4#170414yescomplemented
404C4#303711yescharacteristic
405C4#303911yescharacteristic
406C4#325211yescharacteristic
407C4#325511yescharacteristic
408C4#327811yescharacteristic
409C4#328711yescharacteristic
410C4#336511yescharacteristic
411C4#337511yescharacteristic
412C2C2×#170411yescharacteristic
413C2#1917511yescharacteristic
414C2#1944711yescharacteristic
415C2#1948211yescharacteristic
416C2#2473311yescharacteristic
417C2#2476011yescharacteristic
418C2#2477111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
403. C4#1704semidirect

Maximal subgroups

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

subgroupindex
#331962
#287482
#227222
C4×#4772
C2×#17042
#105882
#105412
#96922
#96022
#92282
#85872