Finite Group Databasebeta

← #165533

Subgroups — #165533

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36457C211yesmaximal, characteristic
2#22020C212yesmaximal, complemented
3#7719C212yesmaximal, complemented
4#7699C212yesmaximal, complemented
5#7659C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7618C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7584C212yesmaximal, complemented
10C2×Q82C411yescharacteristic
11C2×#550C411yescharacteristic
12#2858C2212yes—
13#2858C2211yescharacteristic
14#2563C411yescharacteristic
15#2563C412yes—
16#2548C412yes—
17#2548C411yescharacteristic
18#2491—22no—
19#2491—22no—
20#2491—22no—
21#2341—22no—
22C4×#124C2212yes—
23C4×#124C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1940—22no—
27#1916—22no—
28#1916—22no—
29#1839C2212yes—
30#1839C2212yes—
31#1831C2212yes—
32#1831C2212yes—
33C4×C4⋊C8C2212yes—
34#1745C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1270C2211yescharacteristic
39#1270C2212yes—
40#1270C2212yes—
41#1270C2212yes—
42#1270C2212yes—
43#1270C2212yes—
44#1270C2211yescharacteristic
45C22×#41—21no—
46Q82—44no—
47#550—44no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127C2×C412yes—
53C2×#127C2×C411yescharacteristic
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124C2×C412yes—
58C2×#124C2×C411yescharacteristic
59C2×#121—22no—
60C2×#121—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C412yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×#97—22no—
72C2×C4×C8—22no—
73#394—21no—
74#394—21no—
75#394—22no—
76#394—22no—
77#393—21no—
78#393—21no—
79#393—22no—
80#393—22no—
81#391C2311yescharacteristic
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2311yescharacteristic
92#390C2312yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#382C2×C412yes—
97#382C2×C411yescharacteristic
98#380—22no—
99#380—22no—
100#380—22no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C43C2311yescharacteristic
110C43C2312yes—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C2×#41—44no—
123C2×#41—48no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C411yescharacteristic
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4C22×C411yescharacteristic
133C22×D4C22×C412yes—
134C23×C4—22no—
135C23×C4—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4⋊Q8—44no—
141C4⋊Q8—42no—
142C4⋊Q8—48no—
143C4⋊Q8—44no—
144#127—44no—
145#127—42no—
146#124—24no—
147#124—22no—
148#124—42no—
149#124—41no—
150#124—82no—
151#124—82no—
152#124—82no—
153#124—82no—
154#124—82no—
155#124—82no—
156#121—48no—
157#121—44no—
158C4×Q8—24no—
159C4×Q8—44no—
160C4×Q8—44no—
161C4×Q8—42no—
162C4×Q8—44no—
163C4×D4—42no—
164C4×D4—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8—82no—
207#97—44no—
208C4×C8—82no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95C22⋊C411yescharacteristic
214#95C22⋊C411yescharacteristic
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217C24—21no—
218C24—22no—
219#41—816no—
220C2×Q8—21no—
221C2×Q8—22no—
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—44no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8#13112yes—
231C2×D4—21no—
232C2×D4—22no—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D412yes—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4#10014yes—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—21no—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—24no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—41no—
324C42—44no—
325C42—42no—
326C23—41no—
327C23—42no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23C22×C22⋊C411yescharacteristic
339Q8—28no—
340Q8—42no—
341Q8—42no—
342Q8—164no—
343Q8—162no—
344D4—81no—
345D4—82no—
346D4—81no—
347D4—82no—
348D4—84no—
349D4—88no—
350D4—164no—
351D4—162no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—84no—
383C2×C4—88no—
384C2×C4C2×#10012yes—
385C2×C4C2×#10014yes—
386C2×C4#42114yes—
387C2×C4C2×#13111yescharacteristic
388C2×C4#56811yescharacteristic
389C2×C4#57612yes—
390C2×C4#57712yes—
391C8—82no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—81no—
396C22—82no—
397C22—81no—
398C22—82no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22#288011yescharacteristic
408C22#288411yescharacteristic
409C22C22×#10011yescharacteristic
410C22#289712yes—
411C22C2×#42112yes—
412C4—22no—
413C4—24no—
414C4—21no—
415C4—21no—
416C4—24no—
417C4—22no—
418C4—22no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—81no—
425C4—81no—
426C4—82no—
427C2—161no—
428C2—162no—
429C2#1742511yescharacteristic
430C2#1752612yes—
431C2#1760212yes—
432C2#1766611yescharacteristic
433C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22020C2semidirect
3. #7719C2semidirect
4. #7699C2semidirect
5. #7659C2semidirect
6. #7649C2semidirect
7. #7618C2semidirect
8. #7601C2semidirect
9. #7584C2semidirect

Maximal subgroups

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

subgroupindex
#364572
#220202
#77192
#76992
#76592
#76492
#76182
#76012
#75842