Finite Group Databasebeta

← #216101

Subgroups — #216101

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3#20752C211yesmaximal, characteristic
4#9873C212yesmaximal, complemented
5#9641C212yesmaximal, complemented
6#8942C214yesmaximal
7#8570C214yesmaximal, complemented
8C2×#423C2212yes—
9#2864—21no—
10#2863—21no—
11#2863C412yes—
12#2862C411yescharacteristic
13#2859C411yescharacteristic
14#2857C411yescharacteristic
15#2857C411yescharacteristic
16#2855—21no—
17#2855C412yes—
18#2851—21no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#125—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8C2212yes—
24#2295—21no—
25#2295C2212yes—
26#2283—21no—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1831—21no—
31#1831—21no—
32#1831C2212yes—
33#1831C2212yes—
34#1817—21no—
35#1797—21no—
36C4×C4⋊C8C2212yes—
37C4×C22⋊C8C2212yes—
38#1744C2212yes—
39#1728—21no—
40#1728—21no—
41C2×Q8⋊C8C2218yes—
42#1279C2218yes—
43C22×M16—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2312yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#423—44no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2314yes—
62C2×C22⋊C8—22no—
63C2×C4×C8C2314yes—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#384C2×C411yescharacteristic
95#384C2×C411yescharacteristic
96#383—22no—
97#382—21no—
98#382—21no—
99#382—21no—
100#381—22no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2312yes—
111C4×C22⋊C4—22no—
112#376—22no—
113C43C2×C411yescharacteristic
114C43C2×C411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129Q8⋊C8—48no—
130C22×Q8—21no—
131C22×Q8C22×C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—44no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—24no—
143C22×C8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146#125—44no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C22⋊C8—44no—
212C4×C8—24no—
213C4×C8—24no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228C24C2×C22⋊C411yescharacteristic
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4#13714yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275M16—88no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—44no—
330C42#13114yes—
331C42H(5,𝔽2)12yes—
332C422−1+412yes—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#41912yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—48no—
342Q8—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4#40812yes—
394C2×C4#40812yes—
395C2×C4#41912yes—
396C2×C4#42112yes—
397C2×C4#42112yes—
398C2×C4#52012yes—
399C2×C4#53511yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#53712yes—
402C2×C4C2×#13112yes—
403C2×C4#56812yes—
404C2×C4C2×#13712yes—
405C2×C4#57512yes—
406C8—84no—
407C8—84no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22#231111yescharacteristic
416C22C2×#40811yescharacteristic
417C22#287911yescharacteristic
418C22C2×#41911yescharacteristic
419C22#289611yescharacteristic
420C22C2×#42111yescharacteristic
421C22C2×#42111yescharacteristic
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4#299314yes—
439C4#321512yes—
440C4#323612yes—
441C2—42no—
442C2#1747911yescharacteristic
443C2#1748011yescharacteristic
444C2#1749111yescharacteristic
445C2#1749911yescharacteristic
446C2#2185011yescharacteristic
447C2#2185711yescharacteristic
448C2#2186611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9873C2semidirect
5. #9641C2semidirect
7. #8570C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207772
#207522
#98732
#96412
#89422
#85702