Finite Group Databasebeta

← #119929

Subgroups — #119929

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2C2×#1561C211yesmaximal, characteristic
3C2×#1485C211yesmaximal, characteristic
4#10857C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7880C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1990—21no—
16#1956—21no—
17#1872—22no—
18#1803—21no—
19#1803—21no—
20C2×#343—22no—
21#1561—22no—
22#1561C414yescomplemented
23C2×#332—21no—
24C2×#332—21no—
25#1485—22no—
26#1485—22no—
27C2×Q8⋊C8—22no—
28#1299—22no—
29#1296—22no—
30#1277—21no—
31#1277—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533D412yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D412yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16D412yes—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#343—44no—
95#340—22no—
96#340—22no—
97#332—24no—
98#332—24no—
99Q8⋊C8—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—81no—
119C2×M16—81no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8C22⋊C412yes—
133C4⋊Q8C22⋊C412yes—
134#125—22no—
135#125—22no—
136#125—41no—
137#125—41no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154#117—22no—
155#117—22no—
156#117—41no—
157#117—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—81no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203#103—82no—
204#103—82no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—22no—
210#97—41no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C24#12311yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4#11711yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264M16—24no—
265M16—42no—
266M16—42no—
267M16—42no—
268M16—44no—
269M16—82no—
270M16—82no—
271M16—82no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—81no—
289C2×C8—81no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—81no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C42—42no—
313C42—42no—
314C42C2×C22⋊C411yescharacteristic
315C42#11711yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42#12011yescharacteristic
319C42#12111yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42#12311yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12411yescharacteristic
329C42#12611yescharacteristic
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#38811yescharacteristic
337C23#39411yescharacteristic
338C23#39711yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
346Q8—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
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—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—164no—
383C2×C4#38111yescharacteristic
384C2×C4#38611yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#39411yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39711yescharacteristic
393C2×C4#39711yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#45311yescharacteristic
396C2×C4#45411yescharacteristic
397C2×C4#45511yescharacteristic
398C2×C4#45611yescharacteristic
399C2×C4#47112yes—
400C2×C4#47111yescharacteristic
401C2×C4#47111yescharacteristic
402C2×C4#47211yescharacteristic
403C2×C4#47312yes—
404C2×C4#47311yescharacteristic
405C8—44no—
406C8—82no—
407C8—82no—
408C8—82no—
409C8—81no—
410C8—81no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22#144011yescharacteristic
422C22#203711yescharacteristic
423C22#203811yescharacteristic
424C22#204611yescharacteristic
425C22#204711yescharacteristic
426C22#205111yescharacteristic
427C22#205211yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—82no—
435C4—82no—
436C4—162no—
437C4#188211yescharacteristic
438C4#188311yescharacteristic
439C4#190711yescharacteristic
440C4#190811yescharacteristic
441C4#193012yes—
442C4#203911yescharacteristic
443C4#204011yescharacteristic
444C4#205212yes—
445C4#207811yescharacteristic
446C4#207911yescharacteristic
447C2—41no—
448C2—41no—
449C2#1049211yescharacteristic
450C2#1049311yescharacteristic
451C2#1049411yescharacteristic
452C2#1049511yescharacteristic
453C2#1116411yescharacteristic
454C2#1119211yescharacteristic
455C2#1119511yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1561C4semidirect

Maximal subgroups

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

subgroupindex
#218602
C2×#15612
C2×#14852
#108572
C2×#12992
#80432
#78802