Finite Group Databasebeta

← #318262

Subgroups — #318262

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28284C211yesmaximal, characteristic
2#28283C211yesmaximal, characteristic
3#21712C211yesmaximal, characteristic
4C4×#487C211yesmaximal, characteristic
5C2×#1704C212yesmaximal
6#10486C212yesmaximal
7#10482C212yesmaximal
8#8805C211yesmaximal, characteristic
9#8657C212yesmaximal
10#8587C211yesmaximal, characteristic
11#8586C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#477C2212yes—
14C2×#475C2212yes—
15C2×#446C2212yes—
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2068—22no—
20#2066—22no—
21#1990C2211yescharacteristic
22#1984C2212yes—
23#1981—21no—
24#1981—21no—
25#1956C2211yescharacteristic
26#1948—21no—
27#1947—21no—
28#1939—21no—
29#1939—21no—
30#1933—21no—
31#1933—21no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1834C2211yescharacteristic
36#1803C2212yes—
37#1803C2212yes—
38#1770—21no—
39#1770—21no—
40C4×#103—21no—
41C4×#103—21no—
42C42×C8C2211yescharacteristic
43#1704—28no—
44C2×#334C2212yes—
45C2×#334C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2312yes—
56#487—22no—
57#487D414yes—
58#477—24no—
59#477—42no—
60#475—24no—
61#475—42no—
62#446—24no—
63#446—24no—
64C2×#107—22no—
65C2×#106—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#103—21no—
77C2×#103—21no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#97—22no—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#391—21no—
93#391D412yes—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#384D411yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C43C2311yescharacteristic
110#340—21no—
111#340—21no—
112#340—21no—
113#340—21no—
114#340—22no—
115#340—22no—
116#336—22no—
117#336—22no—
118#334—24no—
119#334—42no—
120#333—24no—
121#332—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8C2×D412yes—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8C2×D412yes—
141C4⋊Q8C2×D412yes—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—82no—
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—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C2×D412yes—
167C2×C42—22no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#107—44no—
178#106—44no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8—44no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C4⋊C8—81no—
191C4⋊C8—81no—
192#103—24no—
193#103—24no—
194#103—44no—
195#103—44no—
196#103—42no—
197#103—42no—
198#103—82no—
199#103—82no—
200#97—44no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—41no—
207C4×C8C2×D412yes—
208C4×C8#4112yes—
209#95—22no—
210#95—22no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×Q8—81no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8#12112yes—
263C2×C8#12112yes—
264C2×C8#13712yes—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
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—81no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—41no—
299C42—42no—
300C42—42no—
301C42—82no—
302C42—82no—
303C42#13512yes—
304C42#13712yes—
305C42#13712yes—
306C42#13712yes—
307C42C22×D411yescharacteristic
308C42C2×#4111yescharacteristic
309C42H(5,𝔽2)11yescharacteristic
310C422−1+411yescharacteristic
311C23C2×#12111yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—42no—
316Q8—42no—
317Q8—42no—
318Q8—82no—
319Q8—82no—
320Q8—81no—
321Q8—81no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
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#46412yes—
350C2×C4#46412yes—
351C2×C4#47012yes—
352C2×C4#47012yes—
353C2×C4#47112yes—
354C2×C4C2×#12111yescharacteristic
355C2×C4#53711yescharacteristic
356C2×C4D4211yescharacteristic
357C2×C4#54711yescharacteristic
358C2×C4#54711yescharacteristic
359C2×C4#55511yescharacteristic
360C2×C4C2×#13511yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4C2×#13711yescharacteristic
363C2×C4C2×#13711yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4#57511yescharacteristic
366C2×C4#57711yescharacteristic
367C2×C4#57711yescharacteristic
368C8—22no—
369C8—22no—
370C8—22no—
371C8—41no—
372C8—82no—
373C8—82no—
374C8—82no—
375C22#253711yescharacteristic
376C22C2×#46411yescharacteristic
377C22#303711yescharacteristic
378C22C2×#47011yescharacteristic
379C22#306011yescharacteristic
380C22C2×#47111yescharacteristic
381C22#306711yescharacteristic
382C4—22no—
383C4—21no—
384C4—21no—
385C4—22no—
386C4—41no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—82no—
391C4—161no—
392C4—161no—
393C4#170414yescomplemented
394C4#303511yescharacteristic
395C4#303911yescharacteristic
396C4#324811yescharacteristic
397C4#325211yescharacteristic
398C4#327811yescharacteristic
399C4#327811yescharacteristic
400C4#335611yescharacteristic
401C4#335611yescharacteristic
402C2C2×#170411yescharacteristic
403C2#1921611yescharacteristic
404C2#1928711yescharacteristic
405C2#1934311yescharacteristic
406C2#2823211yescharacteristic
407C2#2823911yescharacteristic
408C2#2825411yescharacteristic
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
393. C4#1704semidirect

Maximal subgroups

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

subgroupindex
#282842
#282832
#217122
C4×#4872
C2×#17042
#104862
#104822
#88052
#86572
#85872
#85862