Finite Group Databasebeta

← #118856

Subgroups — #118856

Order 512.

662 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#21753C211yesmaximal, characteristic
2#17578C211yesmaximal, characteristic
3C2×#1511C211yesmaximal, characteristic
4C2×#1306C211yesmaximal, characteristic
5#8104C211yesmaximal, characteristic
6#8009C211yesmaximal, characteristic
7#7943C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#381C2211yescharacteristic
15#1901—22no—
16#1901—22no—
17#1821C412yescomplemented
18#1784C412yescomplemented
19#1520—22no—
20#1511—22no—
21#1511—22no—
22#1511—22no—
23#1508—22no—
24C2×#328—22no—
25#1306—22no—
26#1305—22no—
27#1293—22no—
28#1284—22no—
29C23×Q8C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C2×C22⋊Q8—21no—
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⋊Q8D412yes—
46C2×C22⋊Q8D412yes—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C2×#101—24no—
53#407—21no—
54#407—21no—
55#407—21no—
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#407C2×C412yes—
67#407C2×C412yes—
68#407D412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8D412yes—
77#381—21no—
78#381—21no—
79#381—21no—
80#381—21no—
81#381D412yes—
82#380—21no—
83#380—21no—
84#379—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4D412yes—
87C2×#95—21no—
88C2×#95—21no—
89#343—24no—
90#328—28no—
91#328—28no—
92C25#4111yescharacteristic
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—82no—
99C22×Q8C22⋊C412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—41no—
103C23×C4—82no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—82no—
127C2×M16—82no—
128C2×M16—82no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150#117—82no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—82no—
176C2×C42—22no—
177#101—84no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—41no—
198#95—41no—
199#95—41no—
200C24—42no—
201C24—42no—
202C24—81no—
203C24#11711yescharacteristic
204C24#12111yescharacteristic
205C24#12311yescharacteristic
206C24#12311yescharacteristic
207C24#12311yescharacteristic
208C24#12411yescharacteristic
209C24#12611yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4#11711yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12411yescharacteristic
272C22×C4#12611yescharacteristic
273C22×C4#12611yescharacteristic
274C22×C4#12611yescharacteristic
275M16—44no—
276M16—44no—
277M16—82no—
278M16—82no—
279M16—82no—
280M16—82no—
281M16—82no—
282M16—82no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—82no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—88no—
332C23#38111yescharacteristic
333C23#38811yescharacteristic
334C23#38811yescharacteristic
335C23#39411yescharacteristic
336C23#39411yescharacteristic
337C23#39711yescharacteristic
338C23#39711yescharacteristic
339Q8—82no—
340Q8—82no—
341Q8—82no—
342Q8—82no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
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—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—164no—
386C2×C4#38611yescharacteristic
387C2×C4#38611yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39711yescharacteristic
393C2×C4#39711yescharacteristic
394C8—82no—
395C8—82no—
396C8—82no—
397C8—82no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
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—44no—
418C22—44no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22#144011yescharacteristic
425C22#182812yes—
426C22#208212yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—82no—
436C4—82no—
437C4—162no—
438C4—162no—
439C2—22no—
440C2—22no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—41no—
445C2—41no—
446C2#1115811yescharacteristic
447C2#1115911yescharacteristic
448C2#1116611yescharacteristic
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
17. #1821C4semidirect
18. #1784C4semidirect

Maximal subgroups

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

subgroupindex
#217532
#175782
C2×#15112
C2×#13062
#81042
#80092
#79432