Finite Group Databasebeta

← #224170

Subgroups — #224170

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36556C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic
3#20520C211yesmaximal, characteristic
4#9893C212yesmaximal
5#9892C212yesmaximal
6#9890C211yesmaximal, characteristic
7#9890C211yesmaximal, characteristic
8#8580C211yesmaximal, characteristic
9#8580C212yesmaximal
10#8227C212yesmaximal
11C4×#330C211yesmaximal, characteristic
12C2×#446—21no—
13#2864C2212yes—
14#2862C2211yescharacteristic
15#2588—22no—
16#2567C411yescharacteristic
17#2546C411yescharacteristic
18#2520—22no—
19#2511—21no—
20#2509C412yes—
21#2508C412yes—
22#2446—21no—
23#2369C411yescharacteristic
24#2365—21no—
25#2365C411yescharacteristic
26#2355—21no—
27C4×#125C2211yescharacteristic
28C4×#125C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1771—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#330C2211yescharacteristic
44C2×#330C2212yes—
45C2×#330C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2311yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61#446—28no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×#97—22no—
67C2×#97—21no—
68C2×#97—21no—
69C2×#97C2311yescharacteristic
70C2×#97C2311yescharacteristic
71C2×#97C2312yes—
72C2×#97C2312yes—
73C2×#97C2311yescharacteristic
74C2×#97C2311yescharacteristic
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#395—22no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391—22no—
92#389—22no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C411yescharacteristic
96#389C2×C411yescharacteristic
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#387—22no—
103#387—21no—
104#387—22no—
105#387—22no—
106#387—21no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#384—21no—
114#382—22no—
115#382C2×C412yes—
116#382C2312yes—
117#382C2311yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#330—22no—
142#330—24no—
143#330Q814yescomplemented
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—44no—
156#125—21no—
157#125—21no—
158#125C2×Q812yes—
159#125#4112yes—
160C4×Q8—24no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205C4⋊C8—24no—
206C4⋊C8—28no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209#97—28no—
210#97—21no—
211#97—21no—
212#97—22no—
213#97—22no—
214#97—21no—
215#97—21no—
216#97—21no—
217#97—21no—
218#97—22no—
219#97—22no—
220#97—42no—
221#97—42no—
222#97C2×Q812yes—
223#97C2×Q812yes—
224C4×C8—24no—
225C4×C8—42no—
226C4×C8—44no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—81no—
256C2×Q8—81no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4#10414yes—
318C4⋊C4C4×Q812yes—
319C4⋊C4C4×Q812yes—
320C42—22no—
321C42—22no—
322C42—24no—
323C42—22no—
324C42—42no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—42no—
335C42—82no—
336C42C22×Q811yescharacteristic
337C42C2×#4111yescharacteristic
338C422−1+411yescharacteristic
339C422−1+411yescharacteristic
340C23C22×C22⋊C411yescharacteristic
341Q8—44no—
342Q8—84no—
343Q8—162no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4#41312yes—
389C2×C4C2×#10414yes—
390C2×C4C2×#10412yes—
391C2×C4#42112yes—
392C2×C4C2×C4×Q811yescharacteristic
393C2×C4#51911yescharacteristic
394C2×C4#54811yescharacteristic
395C2×C4#54811yescharacteristic
396C2×C4D4×Q811yescharacteristic
397C2×C4D4×Q811yescharacteristic
398C8—24no—
399C8—42no—
400C8—44no—
401C8—42no—
402C8—42no—
403C8—82no—
404C22Q8×C22⋊C411yescharacteristic
405C22C2×#41311yescharacteristic
406C22#288911yescharacteristic
407C22C22×#10411yescharacteristic
408C22C2×#42111yescharacteristic
409C22#290111yescharacteristic
410C22#290111yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—24no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—81no—
427C4—82no—
428C4—82no—
429C4—81no—
430C4#152914yes—
431C2C2×#152911yescharacteristic
432C2#1765311yescharacteristic
433C2#1768311yescharacteristic
434C2#1769111yescharacteristic
435C2#2251711yescharacteristic
436C2Q8×#10411yescharacteristic
437C2#2252911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
143. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365562
#213452
#205202
#98932
#98922
#98902
#85802
#82272
C4×#3302