Finite Group Databasebeta

← #121484

Subgroups — #121484

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#10857C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8028C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1948—22no—
16#1948—21no—
17#1948—21no—
18#1947—21no—
19#1947—21no—
20#1853—22no—
21#1844—21no—
22#1836—22no—
23#1834—21no—
24#1834—21no—
25#1834—21no—
26#1809—22no—
27C2×#343—22no—
28#1736—22no—
29#1299C414yescomplemented
30#1295—22no—
31#1279—22no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C2×#117—21no—
36C2×#117—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D412yes—
40C22×C42C2311yescharacteristic
41#431—21no—
42#431—21no—
43#431C2×C412yes—
44#431Q812yes—
45C2×#107—22no—
46C2×#106—22no—
47C2×#106—24no—
48#423—21no—
49#423—21no—
50#423C2×C412yes—
51#423Q812yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D412yes—
55C2×C22⋊C8—21no—
56C4×M16—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16C2×C412yes—
62C4×M16Q812yes—
63C4×M16Q812yes—
64C2×#97—21no—
65C2×#97D412yes—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384D411yescharacteristic
74#384D411yescharacteristic
75#384D411yescharacteristic
76#384D411yescharacteristic
77#382—21no—
78#382—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87#343—44no—
88#340—22no—
89#340—22no—
90#340—24no—
91#340—24no—
92#340—24no—
93C23×C4—22no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16C4⋊C414yes—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115#117—44no—
116#117—82no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—82no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42C22×C411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×Q811yescharacteristic
149C2×C42C2×Q811yescharacteristic
150C2×C42C2×Q811yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155#107—82no—
156#106—82no—
157#106—84no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8C4⋊C414yes—
164C22⋊C8—22no—
165C22⋊C8—22no—
166#97—22no—
167#97—22no—
168#97—41no—
169#97C4⋊C414yes—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180C24#12111yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4C2×C4⋊C411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×Q811yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22⋊Q811yescharacteristic
206C22×C4C22⋊Q811yescharacteristic
207C22×C4C22⋊Q811yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4#12511yescharacteristic
210C22×C4#12511yescharacteristic
211C22×C4#12511yescharacteristic
212C22×C4#12711yescharacteristic
213C22×C4C4⋊Q811yescharacteristic
214C22×C4C4⋊Q811yescharacteristic
215M16—24no—
216M16—44no—
217M16—44no—
218M16—42no—
219M16—42no—
220M16—42no—
221M16—44no—
222M16—82no—
223C2×C8—24no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C42—82no—
261C42—82no—
262C42C2×C4⋊C411yescharacteristic
263C42C2×C4⋊C411yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42C22⋊Q811yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42C4⋊Q811yescharacteristic
275C42C4⋊Q811yescharacteristic
276C42C4⋊Q811yescharacteristic
277C42C4⋊Q811yescharacteristic
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#38911yescharacteristic
285C23#39211yescharacteristic
286C23#39811yescharacteristic
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—164no—
325C2×C4#38411yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38911yescharacteristic
330C2×C4#38911yescharacteristic
331C2×C4#39511yescharacteristic
332C2×C4#39511yescharacteristic
333C2×C4#39511yescharacteristic
334C2×C4#39811yescharacteristic
335C2×C4#39811yescharacteristic
336C2×C4#39911yescharacteristic
337C2×C4#45311yescharacteristic
338C2×C4#45411yescharacteristic
339C2×C4#45412yes—
340C2×C4#45511yescharacteristic
341C2×C4#45512yes—
342C2×C4#45611yescharacteristic
343C2×C4#47112yes—
344C2×C4#47112yes—
345C8—44no—
346C8—44no—
347C8—44no—
348C8—44no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#144411yescharacteristic
360C22#200311yescharacteristic
361C22#200411yescharacteristic
362C22#200611yescharacteristic
363C22#200711yescharacteristic
364C22#201911yescharacteristic
365C22#201911yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4—162no—
376C4—161no—
377C4#185611yescharacteristic
378C4#185711yescharacteristic
379C4#186012yes—
380C4#195112yes—
381C4#206011yescharacteristic
382C4#206112yes—
383C4#206111yescharacteristic
384C4#209012yes—
385C2—41no—
386C2—41no—
387C2#1010811yescharacteristic
388C2#1011111yescharacteristic
389C2#1011311yescharacteristic
390C2#1011411yescharacteristic
391C2#1133011yescharacteristic
392C2#1133911yescharacteristic
393C2#1134211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
29. #1299C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#219642
#207642
#108572
C2×#12992
#80282
#78962