Finite Group Databasebeta

← #215356

Subgroups — #215356

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9677C212yesmaximal, complemented
5#9673C212yesmaximal, complemented
6#9647C212yesmaximal, complemented
7#9641C212yesmaximal, complemented
8#8314C212yesmaximal
9#8311C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2584—22no—
13#2583—22no—
14#2567C412yes—
15#2546C412yes—
16#2502C412yes—
17#2434C412yes—
18#2399—21no—
19#2399—21no—
20C4×C4⋊Q8C2212yes—
21C4×C4⋊Q8C2212yes—
22#2296—21no—
23#2296—21no—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1840—21no—
28#1831C2212yes—
29#1818—21no—
30#1797C2212yes—
31#1767C2211yescharacteristic
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2212yes—
34#1750—21no—
35C4×C22⋊C8C2212yes—
36#1746—21no—
37#1744C2211yescharacteristic
38C4×#97C2211yescharacteristic
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C23×Q8C2×C411yescharacteristic
48#533—42no—
49#533—42no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2312yes—
64C2×C4×Q8C2312yes—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81C2×C4×C8C2312yes—
82#398—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#395—22no—
87#395—22no—
88#395—22no—
89#395—22no—
90#391C2×C412yes—
91#391C2×C412yes—
92#391C2×C412yes—
93#391C2×C412yes—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384C2×C412yes—
107#384C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D412yes—
116#376D411yescharacteristic
117#376D411yescharacteristic
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128Q8⋊C8—42no—
129Q8⋊C8—42no—
130Q8⋊C8—42no—
131Q8⋊C8—42no—
132Q8⋊C8—42no—
133Q8⋊C8—42no—
134Q8⋊C8—42no—
135Q8⋊C8—42no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—24no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—42no—
142C22×Q8C22×C412yes—
143C22×Q8C22×C412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161#125—44no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—44no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—44no—
176C4×Q8—44no—
177C4×Q8—44no—
178#117—24no—
179#117—22no—
180#117—22no—
181#117—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—44no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233#97—44no—
234#97—42no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—42no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24C2×C22⋊C411yescharacteristic
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—44no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—44no—
268C2×Q8—48no—
269C2×Q8—44no—
270C2×Q8—82no—
271C2×Q8#13114yes—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4#13714yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—44no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—21no—
350C42—21no—
351C42—42no—
352C42—42no—
353C42—44no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42#13712yes—
359C42#13712yes—
360C42H(5,𝔽2)12yes—
361C42H(5,𝔽2)12yes—
362C23—21no—
363C23—22no—
364C23—21no—
365C23—22no—
366C23#41912yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—24no—
369Q8—24no—
370Q8—48no—
371Q8—42no—
372Q8—42no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—84no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4#41914yes—
417C2×C4#41912yes—
418C2×C4#51811yescharacteristic
419C2×C4#51811yescharacteristic
420C2×C4#53512yes—
421C2×C4#53512yes—
422C2×C4C2×#13112yes—
423C2×C4#56812yes—
424C2×C4C2×#13712yes—
425C2×C4C2×#13711yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C8—44no—
428C8—82no—
429C8—82no—
430C8—82no—
431C22—22no—
432C22—24no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22#230911yescharacteristic
437C22#288012yes—
438C22C2×#41911yescharacteristic
439C22C2×#41911yescharacteristic
440C22#290112yes—
441C4—22no—
442C4—22no—
443C4—24no—
444C4—41no—
445C4—44no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4#297612yes—
454C4#297711yescharacteristic
455C4#297811yescharacteristic
456C4#321812yes—
457C4#321812yes—
458C2—42no—
459C2#1751412yes—
460C2#1751412yes—
461C2#2179211yescharacteristic
462C2#2181111yescharacteristic
463C2#2181711yescharacteristic
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 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9677C2semidirect
5. #9673C2semidirect
6. #9647C2semidirect
7. #9641C2semidirect

Maximal subgroups

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

subgroupindex
#349532
#214312
#207862
#96772
#96732
#96472
#96412
#83142
#83112