Finite Group Databasebeta

← #497659

Subgroups — #497659

Order 512.

915 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39174C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13226C212yesmaximal, complemented
4#13226C214yesmaximal, complemented
5#12803C212yesmaximal, complemented
6#12801C211yesmaximal, characteristic, complemented
7#12632C212yesmaximal, complemented
8C4×#355C211yesmaximal, characteristic, complemented
9C2×#557C2211yescharacteristic
10C2×#550C411yescharacteristic
11Q8×M16—22no—
12#2858C411yescharacteristic
13#2711—24no—
14#2641C412yes—
15#2566C412yes—
16#2548C2212yes—
17#2548C2211yescharacteristic
18#2542C411yescharacteristic
19#2432—22no—
20#2426—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#355C2212yes—
25C2×#355C2214yes—
26C2×#355C2212yes—
27C2×#355C2214yes—
28#1998—24no—
29#1997—22no—
30#1801C2212yes—
31#1801C2214yes—
32C4×#101C2212yes—
33C4×#101C2214yes—
34#1394C2212yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37C22×#41—21no—
38#557—42no—
39#557—42no—
40#550—42no—
41#550—42no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8D412yes—
44C2×#127C2×C411yescharacteristic
45C2×#127C2×C412yes—
46C2×#125C2×C412yes—
47C2×#125D414yes—
48C2×#124—22no—
49C2×#124C2312yes—
50C2×#124C2311yescharacteristic
51C2×C22⋊Q8—22no—
52C2×#121—22no—
53C2×#121—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8C2312yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60#446—44no—
61#446—42no—
62C8×Q8—42no—
63#423—22no—
64#423—24no—
65C2×#101C2314yes—
66C2×#101C2312yes—
67C2×#101C2312yes—
68C4×M16—22no—
69C4×M16—24no—
70#399—22no—
71#396—24no—
72#394—24no—
73#394—24no—
74#394—24no—
75#394—22no—
76#391—21no—
77#391C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390—24no—
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C411yescharacteristic
85#390C2×C412yes—
86#389—21no—
87#388—22no—
88#388—24no—
89#387—22no—
90#385—24no—
91#385—24no—
92#384C2×C412yes—
93#382C2×C411yescharacteristic
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C22⋊C4—22no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#355—24no—
100#355—28no—
101#355—42no—
102#355—44no—
103#343—24no—
104#343—22no—
105#343—24no—
106#343—22no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—84no—
110C22×Q8C22⋊C411yescharacteristic
111C22×Q8C22⋊C412yes—
112C22×Q8C2411yescharacteristic
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4C22×C412yes—
118C22×D4C22×C411yescharacteristic
119C23×C4—21no—
120C23×C4—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—24no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128#127—44no—
129#127—41no—
130#127—41no—
131#125—44no—
132#125—44no—
133#125—44no—
134#124—24no—
135#124—22no—
136#124—22no—
137#124—21no—
138#124—22no—
139#124—21no—
140#124—84no—
141C22⋊Q8—84no—
142#121—48no—
143#121—42no—
144#121—42no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—44no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D414yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188C4⋊C8—42no—
189C4⋊C8—44no—
190C4⋊C8—42no—
191#101—22no—
192#101—22no—
193#101—24no—
194#101—24no—
195#101—22no—
196#101#4114yescomplemented
197#97—44no—
198#97—42no—
199C4×C8—44no—
200C4×C8—42no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—22no—
207C24—21no—
208C24—22no—
209#41—88no—
210#41—88no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—44no—
219C2×Q8C2×#4112yes—
220C2×Q8H(5,𝔽2)11yescharacteristic
221C2×Q82−1+411yescharacteristic
222C2×D4—22no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—44no—
236C2×D4—84no—
237C2×D4—84no—
238C2×D4—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—84no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C414yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D412yes—
256M16—24no—
257M16—42no—
258M16—42no—
259M16—44no—
260M16—42no—
261M16—44no—
262C2×C8—42no—
263C2×C8—44no—
264C2×C8—42no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—84no—
278C4⋊C4#9912yes—
279C4⋊C4#9912yes—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—82no—
287C22⋊C4—84no—
288C22⋊C4—84no—
289C22⋊C4—84no—
290C22⋊C4—84no—
291C42—24no—
292C42—22no—
293C42—24no—
294C42—22no—
295C42—24no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—21no—
301C42—22no—
302C42—41no—
303C42—42no—
304C42—81no—
305C23—42no—
306C23—44no—
307C23—42no—
308C23—44no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23—41no—
314C23—42no—
315C23C22×C22⋊C411yescharacteristic
316Q8—42no—
317Q8—41no—
318Q8—41no—
319Q8—42no—
320Q8—44no—
321Q8—42no—
322Q8—82no—
323D4—81no—
324D4—82no—
325D4—81no—
326D4—82no—
327D4—82no—
328D4—88no—
329D4—82no—
330D4—164no—
331D4—162no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4C2×#9914yes—
361C2×C4C2×#9911yescharacteristic
362C2×C4C2×#9911yescharacteristic
363C2×C4C4×#4111yescharacteristic
364C2×C4#52011yescharacteristic
365C2×C4#54812yes—
366C2×C4#55012yes—
367C8—44no—
368C8—84no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—82no—
373C22—84no—
374C22—82no—
375C22—84no—
376C22—81no—
377C22—82no—
378C22—81no—
379C22—82no—
380C22—81no—
381C22—82no—
382C22#234111yescharacteristic
383C22C22×#9911yescharacteristic
384C22#288411yescharacteristic
385C4—22no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—24no—
391C4—42no—
392C4—44no—
393C4—42no—
394C4—41no—
395C4—81no—
396C4—82no—
397C4—81no—
398C4—81no—
399C4#213412yes—
400C2—21no—
401C2—21no—
402C2—161no—
403C2—162no—
404C2#2253411yescharacteristic
405C2C2×#213411yescharacteristic
406C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22517C2semidirect
3. #13226C2semidirect
4. #13226C2semidirect
5. #12803C2semidirect
6. #12801C2semidirect
7. #12632C2semidirect
8. C4×#355C2semidirect
196. #101#41semidirect

Maximal subgroups

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

subgroupindex
#391742
#225172
#132262
#128032
#128012
#126322
C4×#3552