Finite Group Databasebeta

← #497599

Subgroups — #497599

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36562C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13303C214yesmaximal, complemented
4#13302C212yesmaximal, complemented
5#12801C211yesmaximal, characteristic
6#12801C212yesmaximal
7C4×#355C212yesmaximal
8C4×#355C211yesmaximal, characteristic
9C2×#557C2211yescharacteristic
10C2×#554C411yescharacteristic
11Q8×M16—22no—
12#2862C411yescharacteristic
13#2859C412yes—
14#2567C2211yescharacteristic
15#2566C412yes—
16#2565C2212yes—
17#2556—24no—
18#2542C411yescharacteristic
19#2429—22no—
20#2360—22no—
21C4×#124C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C411yescharacteristic
24C2×#355C2212yes—
25C2×#355C2211yescharacteristic
26C2×#355C2211yescharacteristic
27#1998—24no—
28#1997—22no—
29#1801C2212yes—
30#1801C2214yes—
31C4×#101C2212yes—
32C4×#101C2214yes—
33#1394C2214yes—
34#1394C2212yes—
35#1394C2214yes—
36#1394C2212yes—
37#557—41no—
38#557—42no—
39#557—41no—
40#554—42no—
41#554—41no—
42#554—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8D412yes—
46C2×#125C2×C412yes—
47C2×#125C2×C412yes—
48C2×#125D414yes—
49C2×#124C2311yescharacteristic
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8C2312yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C22×C4⋊C4—21no—
61#446—44no—
62#446—42no—
63C8×Q8—42no—
64#423—22no—
65#423—24no—
66C2×#101C2314yes—
67C2×#101C2312yes—
68C2×#101C2312yes—
69C4×M16—22no—
70C4×M16—24no—
71#399—22no—
72#398—24no—
73#396—24no—
74#395—22no—
75#391—22no—
76#391—22no—
77#391C2311yescharacteristic
78#391C2312yes—
79#390C2×C411yescharacteristic
80#390C2×C412yes—
81#389—24no—
82#389—24no—
83#389—22no—
84#388—24no—
85#388—22no—
86#387—24no—
87#385—24no—
88#384C2×C412yes—
89#384C2×C412yes—
90#382C2×C412yes—
91#382C2×C411yescharacteristic
92#382C2×C412yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#355—24no—
102#355—22no—
103#355Q814yescomplemented
104#343—24no—
105#343—22no—
106#343—24no—
107#343—22no—
108C22×Q8—21no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C2411yescharacteristic
112C22×D4C22×C411yescharacteristic
113C23×C4—21no—
114C23×C4—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—24no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—42no—
121C4⋊Q8—42no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124#125—42no—
125#125—42no—
126#125—44no—
127#125—44no—
128#125—42no—
129#125—42no—
130#124—22no—
131#124C2×Q812yes—
132#124#4112yes—
133C22⋊Q8—48no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—22no—
142C4×Q8—24no—
143C4×Q8—22no—
144C4×Q8—42no—
145C4×Q8—41no—
146C4×Q8—42no—
147C4×Q8—41no—
148C4×Q8—44no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D414yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C2×D414yes—
189C2×C42C2×D412yes—
190C4⋊C8—44no—
191C4⋊C8—42no—
192C4⋊C8—44no—
193C4⋊C8—42no—
194#101—22no—
195#101—24no—
196#101—22no—
197#101—24no—
198#101—22no—
199#101C2×Q814yes—
200#97—44no—
201#97—42no—
202C4×C8—44no—
203C4×C8—42no—
204#95—22no—
205#95—24no—
206#95—22no—
207#95—21no—
208#95—24no—
209#95—22no—
210#95—24no—
211C24—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—42no—
221C2×Q8—82no—
222C2×Q8C22×Q811yescharacteristic
223C2×Q8C2×#4111yescharacteristic
224C2×Q82−1+412yes—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4C4×Q812yes—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—84no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—24no—
249M16—42no—
250M16—42no—
251M16—44no—
252M16—42no—
253M16—44no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—42no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—84no—
278C4⋊C4#9912yes—
279C4⋊C4#9911yescharacteristic
280C4⋊C4#9911yescharacteristic
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—48no—
286C22⋊C4—82no—
287C42—21no—
288C42—24no—
289C42—22no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—21no—
295C42—21no—
296C42—42no—
297C42—41no—
298C42—42no—
299C42—44no—
300C42—82no—
301C42—81no—
302C42C4×Q812yes—
303C23—22no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23C22×C22⋊C411yescharacteristic
309Q8—41no—
310Q8—41no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314Q8—44no—
315Q8—82no—
316Q8—162no—
317D4—42no—
318D4—42no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—162no—
349C2×C4—162no—
350C2×C4C2×#9912yes—
351C2×C4C2×#9912yes—
352C2×C4C2×#9911yescharacteristic
353C2×C4C2×#9911yescharacteristic
354C2×C4C2×C4×Q811yescharacteristic
355C2×C4#51911yescharacteristic
356C2×C4#54812yes—
357C2×C4D4×Q812yes—
358C8—44no—
359C8—84no—
360C8—82no—
361C22—21no—
362C22—21no—
363C22—44no—
364C22—42no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22Q8×C22⋊C411yescharacteristic
369C22C22×#9911yescharacteristic
370C22#288411yescharacteristic
371C4—21no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—21no—
377C4—21no—
378C4—44no—
379C4—42no—
380C4—42no—
381C4—44no—
382C4—42no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4#213212yes—
388C2—21no—
389C2—21no—
390C2—82no—
391C2Q8×#9911yescharacteristic
392C2C2×#213211yescharacteristic
393C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22×C4
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. #13303C2semidirect
4. #13302C2semidirect
103. #355Q8semidirect

Maximal subgroups

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

subgroupindex
#365622
#225172
#133032
#133022
#128012
C4×#3552