Finite Group Databasebeta

← #119075

Subgroups — #119075

Order 512.

703 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#21768C211yesmaximal, characteristic
2#21753C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4#8104C211yesmaximal, characteristic
5#8009C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7929C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1901—22no—
16#1901C414yescomplemented
17#1894—22no—
18#1873—22no—
19#1824—21no—
20#1821—21no—
21#1787—21no—
22#1784—21no—
23#1508—22no—
24#1508—22no—
25C2×#328—22no—
26#1305—22no—
27#1293—22no—
28#1284—22no—
29#1282—22no—
30C23×Q8D411yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16C2×C412yes—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C42—21no—
54C2×#103—24no—
55C2×#101—24no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407—21no—
68#407—21no—
69#407—21no—
70#407—21no—
71#407C2×C412yes—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C2×C22⋊C8D412yes—
80#381—21no—
81#380—21no—
82#380D412yes—
83#379—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4D412yes—
90C2×#95—21no—
91#343—24no—
92#340—24no—
93#328—28no—
94C25#4111yescharacteristic
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—82no—
101C22×Q8C22⋊C412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—41no—
105C23×C4—82no—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—82no—
129C2×M16—82no—
130C2×M16—82no—
131C2×M16—82no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152#117—82no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—82no—
182#103—84no—
183#101—84no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24—42no—
205C24—42no—
206C24—81no—
207C24#11711yescharacteristic
208C24#12111yescharacteristic
209C24#12311yescharacteristic
210C24#12311yescharacteristic
211C24#12311yescharacteristic
212C24#12411yescharacteristic
213C24#12611yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12411yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12411yescharacteristic
276C22×C4#12611yescharacteristic
277C22×C4#12611yescharacteristic
278C22×C4#12611yescharacteristic
279M16—44no—
280M16—44no—
281M16—82no—
282M16—82no—
283M16—82no—
284M16—82no—
285M16—82no—
286M16—82no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—82no—
315C42—82no—
316C42—82no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—88no—
339C23#38111yescharacteristic
340C23#38811yescharacteristic
341C23#38811yescharacteristic
342C23#39411yescharacteristic
343C23#39411yescharacteristic
344C23#39711yescharacteristic
345C23#39711yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—88no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38811yescharacteristic
396C2×C4#38811yescharacteristic
397C2×C4#39411yescharacteristic
398C2×C4#39411yescharacteristic
399C2×C4#39711yescharacteristic
400C2×C4#39711yescharacteristic
401C8—82no—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—44no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22#144011yescharacteristic
432C22#205012yes—
433C22#208212yes—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—84no—
442C4—82no—
443C4—82no—
444C4—162no—
445C4#191312yes—
446C2—22no—
447C2—22no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2#1115111yescharacteristic
454C2#1115811yescharacteristic
455C2#1117211yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1901C4semidirect

Maximal subgroups

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

subgroupindex
#217682
#217532
C2×#15082
#81042
#80092
#79422
#79292