Finite Group Databasebeta

← #121906

Subgroups — #121906

Order 512.

695 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21938C211yesmaximal, characteristic
2#10854C211yesmaximal, characteristic
3#10846C211yesmaximal, characteristic
4#8023C211yesmaximal, characteristic
5C2×#1289C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1853—22no—
16#1853—22no—
17#1807C414yescomplemented
18C2×#342—22no—
19C2×#342—22no—
20#1294—22no—
21#1289—22no—
22#1289—22no—
23#1282—22no—
24C24×C4C2311yescharacteristic
25C23×C8—21no—
26C23×C8C2×C412yes—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4D412yes—
36C22×C42—21no—
37C2×#106—24no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8D412yes—
48C2×C22⋊C8D412yes—
49C2×C22⋊C8D412yes—
50#380—21no—
51#380—21no—
52#380—21no—
53#380—21no—
54#380—21no—
55#380—21no—
56#380—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380C2×C412yes—
64#380C2×C412yes—
65#380Q812yes—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95Q812yes—
75#342—44no—
76#342—44no—
77#340—24no—
78C25C2×D411yescharacteristic
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—82no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—82no—
105C22×C8—82no—
106C22×C8C22⋊C414yes—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4SD1614yes—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—82no—
144#106—84no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C24—81no—
164C24—81no—
165C24C4×D411yescharacteristic
166C24#12111yescharacteristic
167C24C22⋊Q811yescharacteristic
168C24C22⋊Q811yescharacteristic
169C24#12311yescharacteristic
170C24#12311yescharacteristic
171C24#12711yescharacteristic
172C24C2×SD1612yes—
173C24#13512yes—
174C24AGL(1,ℤ/8)12yes—
175C24#13712yes—
176C22×C4—24no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4#11711yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×Q811yescharacteristic
217C22×C4C4×Q811yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12511yescharacteristic
232C22×C4#12511yescharacteristic
233C22×C4#12511yescharacteristic
234C22×C4#12511yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4C4⋊Q811yescharacteristic
240C2×C8—24no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C42—82no—
272C42—82no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—82no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23#38711yescharacteristic
292C23#38711yescharacteristic
293C23#39011yescharacteristic
294C23#39711yescharacteristic
295C23#39811yescharacteristic
296C23#39911yescharacteristic
297C23#39911yescharacteristic
298C23C4×SD1612yes—
299C23#44012yes—
300C23#46512yes—
301C23#46812yes—
302C23#47512yes—
303C23#47612yes—
304C23#47812yes—
305C23#47912yes—
306C23#48112yes—
307C23#48212yes—
308C23#48312yes—
309C23#48512yes—
310C23#49211yescharacteristic
311C23#49511yescharacteristic
312C23#49611yescharacteristic
313C23#49711yescharacteristic
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—164no—
360C2×C4—164no—
361C2×C4—162no—
362C2×C4—162no—
363C2×C4—164no—
364C2×C4#38111yescharacteristic
365C2×C4#38711yescharacteristic
366C2×C4#38711yescharacteristic
367C2×C4#39111yescharacteristic
368C2×C4#39711yescharacteristic
369C2×C4#39811yescharacteristic
370C2×C4#40011yescharacteristic
371C2×C4#40011yescharacteristic
372C8—44no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22#144511yescharacteristic
405C22#191912yes—
406C22#192012yes—
407C22#196811yescharacteristic
408C22#197311yescharacteristic
409C22#207712yes—
410C22#208812yes—
411C22#209211yescharacteristic
412C22#209411yescharacteristic
413C22#209611yescharacteristic
414C22#209711yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—84no—
423C4—162no—
424C4—162no—
425C4—162no—
426C4—162no—
427C4#182812yes—
428C2—22no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—22no—
434C2—21no—
435C2—21no—
436C2—22no—
437C2—22no—
438C2#1135111yescharacteristic
439C2#1136111yescharacteristic
440C2#1137811yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
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. #1807C4semidirect

Maximal subgroups

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

subgroupindex
#219382
#108542
#108462
#80232
C2×#12892
C2×#12822