Finite Group Databasebeta

← #120235

Subgroups — #120235

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2#21944C211yesmaximal, characteristic
3#8107C211yesmaximal, characteristic
4C2×#1305C211yesmaximal, characteristic
5#8023C211yesmaximal, characteristic
6#7983C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1855—22no—
16#1855C414yescomplemented
17#1854—22no—
18#1853—21no—
19#1852—21no—
20#1810—22no—
21#1809—22no—
22#1305—22no—
23#1305—22no—
24#1294—22no—
25#1289—21no—
26#1289—21no—
27#1282—22no—
28C24×C4C2311yescharacteristic
29C22×M16—21no—
30C22×M16C2×C412yes—
31C22×M16D412yes—
32C23×C8—21no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C2×#108—24no—
39C2×#107—22no—
40C2×#106—22no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407C2×C412yes—
52#407D412yes—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8C2×C412yes—
62C2×C22⋊C8D412yes—
63C2×C22⋊C8Q812yes—
64C2×C22⋊C8Q812yes—
65#381—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380D411yescharacteristic
74#380D411yescharacteristic
75#380D412yes—
76#380Q812yes—
77#379—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80C2×#95—21no—
81C2×#95—21no—
82#341—24no—
83#340—22no—
84#340—22no—
85C25#4111yescharacteristic
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—82no—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×Q811yescharacteristic
96C23×C4C2×Q811yescharacteristic
97C23×C4C2×Q811yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—82no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16C22⋊C414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—82no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—81no—
137C2×C22⋊C4—81no—
138C2×C22⋊C4—81no—
139C2×C22⋊C4—81no—
140C2×C42—41no—
141#108—84no—
142#107—82no—
143#106—82no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8C4⋊C414yes—
160#95—22no—
161#95—22no—
162#95—21no—
163#95—21no—
164#95—41no—
165#95—41no—
166#95—41no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—81no—
172C24—81no—
173C24C4×Q811yescharacteristic
174C24#12111yescharacteristic
175C24C22⋊Q811yescharacteristic
176C24#12311yescharacteristic
177C24#12411yescharacteristic
178C24#12511yescharacteristic
179C24#12611yescharacteristic
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
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—81no—
204C22×C4—81no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4#11711yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×Q811yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12511yescharacteristic
227C22×C4#12511yescharacteristic
228C22×C4#12511yescharacteristic
229C22×C4#12611yescharacteristic
230C22×C4C4⋊Q811yescharacteristic
231C22×C4C4⋊Q811yescharacteristic
232C22×C4C4⋊Q811yescharacteristic
233M16—44no—
234M16—44no—
235M16—82no—
236M16—82no—
237M16—82no—
238M16—82no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C42—82no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23#38711yescharacteristic
300C23#38911yescharacteristic
301C23#39111yescharacteristic
302C23#39411yescharacteristic
303C23#39611yescharacteristic
304C23#39811yescharacteristic
305C23#40011yescharacteristic
306C23#47111yescharacteristic
307C23#47111yescharacteristic
308C23#47211yescharacteristic
309C23#47311yescharacteristic
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
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—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—162no—
351C2×C4—162no—
352C2×C4—164no—
353C2×C4#38011yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38711yescharacteristic
356C2×C4#38911yescharacteristic
357C2×C4#39511yescharacteristic
358C2×C4#39611yescharacteristic
359C2×C4#39811yescharacteristic
360C2×C4#39811yescharacteristic
361C8—44no—
362C8—44no—
363C8—82no—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22#144111yescharacteristic
400C22#195111yescharacteristic
401C22#195211yescharacteristic
402C22#205111yescharacteristic
403C22#205211yescharacteristic
404C22#206311yescharacteristic
405C22#206411yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—161no—
414C4—161no—
415C4—162no—
416C4—162no—
417C4#191312yes—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—41no—
425C2—41no—
426C2—41no—
427C2—41no—
428C2#1120511yescharacteristic
429C2#1124911yescharacteristic
430C2#1125111yescharacteristic
subgroupG/H
center, socleC22#1441
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. #1855C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#219442
#81072
C2×#13052
#80232
#79832
#79252