Finite Group Databasebeta

← #482199

Subgroups — #482199

Order 512.

933 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35701C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11853C214yesmaximal, complemented
5#11847C214yesmaximal, complemented
6#11723C212yesmaximal, complemented
7#11697C212yesmaximal, complemented
8C2×#554C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2982—21no—
11#2879—21no—
12#2633C412yes—
13#2628C2211yescharacteristic
14#2611C2212yes—
15#2598C2212yes—
16#2542C411yescharacteristic
17#2535C411yescharacteristic
18#2463—22no—
19#2449—22no—
20#2423—21no—
21#2407—21no—
22#2397C412yes—
23#2392C2211yescharacteristic
24#2361—21no—
25#2356C411yescharacteristic
26D4×C4⋊C4—21no—
27#1822C2214yes—
28#1821—22no—
29#1742C2212yes—
30C2×#343—21no—
31#1460—21no—
32#1460—22no—
33#1458C2214yes—
34#1458C2212yes—
35#1353C2218yes—
36#1346C2214yes—
37#1345C2214yes—
38C22×#41D411yescharacteristic
39C22×M16—22no—
40#554—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×#121—21no—
47#520—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D412yes—
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C8×Q8—44no—
65#431—22no—
66#431—24no—
67#408—44no—
68#407—24no—
69#400C2312yes—
70#399—22no—
71#399C2×C412yes—
72#399C2×C412yes—
73#399C2×C412yes—
74#398—22no—
75#398C2311yescharacteristic
76#397C2×C412yes—
77#397C2×C411yescharacteristic
78#396—21no—
79#395—21no—
80#394—22no—
81#394—21no—
82#394C2311yescharacteristic
83#393—21no—
84#392—22no—
85#392C2×C412yes—
86#392C2×C411yescharacteristic
87#391—21no—
88#390—21no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384—21no—
103#383C2×C412yes—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110#376C2×C412yes—
111#343—22no—
112#343—24no—
113#343C2314yes—
114#324C2314yes—
115#323—24no—
116#323—22no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—21no—
121C22×Q8C22⋊C411yescharacteristic
122C22×D4—21no—
123C22×D4—22no—
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22⋊C412yes—
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C23×C4—22no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C2×M16—24no—
135C2×M16—42no—
136C2×M16—44no—
137C2×M16—44no—
138C22×C8—42no—
139C4⋊Q8—42no—
140#125—44no—
141C22⋊Q8—48no—
142C22⋊Q8—44no—
143C22⋊Q8—84no—
144#121—44no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×D4—44no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—44no—
151C4×D4—42no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—84no—
155C4×D4—84no—
156#117—44no—
157#117—42no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2411yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22×C412yes—
191C2×C22⋊C4C22×C411yescharacteristic
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C42—22no—
195C2×C42—21no—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C4⋊C8—44no—
201C4⋊C8—42no—
202C22⋊C8—24no—
203C22⋊C8—48no—
204C4×C8—44no—
205C4×C8—42no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95C22×C411yescharacteristic
216#95C22×C412yes—
217#95C22×C411yescharacteristic
218C24—21no—
219C24C2×C22⋊C412yes—
220C24C2×C22⋊C411yescharacteristic
221#41—48no—
222#41—48no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—82no—
227C2×Q8#9914yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—82no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#13514yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—44no—
275M16—44no—
276M16—84no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—84no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—24no—
314C42—22no—
315C42—24no—
316C42—22no—
317C42—44no—
318C42—42no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#41612yes—
336C23C22×C22⋊C411yescharacteristic
337Q8—28no—
338Q8—162no—
339D4—48no—
340D4—42no—
341D4—48no—
342D4—42no—
343D4—44no—
344D4—88no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—48no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—84no—
380C2×C4—162no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4—164no—
384C2×C4C2×#9914yes—
385C2×C4C2×#9912yes—
386C2×C4#41612yes—
387C2×C4#42112yes—
388C2×C4C2×#13512yes—
389C8—84no—
390C8—84no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C22×#9911yescharacteristic
411C22C2×#41611yescharacteristic
412C22C2×#42111yescharacteristic
413C4—22no—
414C4—24no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—81no—
423C4—82no—
424C4—82no—
425C4—81no—
426C4#151614yes—
427C2—21no—
428C2—21no—
429C2—42no—
430C2—82no—
431C2—82no—
432C2C2×#151611yescharacteristic
433C2#3334911yescharacteristic
434C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#35701C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20695C2semidirect
3. #17353C2semidirect
4. #11853C2semidirect
5. #11847C2semidirect
6. #11723C2semidirect
7. #11697C2semidirect

Maximal subgroups

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

subgroupindex
#357012
#206952
#173532
#118532
#118472
#117232
#116972