Finite Group Databasebeta

← #109879

Subgroups — #109879

Order 512.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20544C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#10853C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8087C211yesmaximal, characteristic
6#7988C211yesmaximal, characteristic
7#7973C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1771—21no—
16#1771—21no—
17#1768—21no—
18#1768—21no—
19#1752—21no—
20C8×C22⋊C4—21no—
21C2×#341—21no—
22C2×#341—21no—
23#1573—22no—
24#1571—21no—
25#1570—21no—
26#1569C414yescomplemented
27#1303—21no—
28#1303—21no—
29#1302—22no—
30#1290C414yescomplemented
31#1288C414yes—
32C22×M16—21no—
33C22×M16—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C22×C42C2311yescharacteristic
37#432—21no—
38#432—21no—
39#432—21no—
40#432C2×C412yes—
41#432C2×C412yes—
42#432C2×C412yes—
43#432C2×C412yes—
44#432D412yes—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423C2×C412yes—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8Q812yes—
65#407—21no—
66#407—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C4×M16C2×C412yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#383C2×C412yes—
82#382—21no—
83#381Q812yes—
84#380—21no—
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4D411yescharacteristic
87C4×C22⋊C4—21no—
88#376—21no—
89#341—42no—
90#341—42no—
91#336—21no—
92#336—21no—
93#336—21no—
94#336—21no—
95#335—24no—
96#334—24no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8C4⋊C412yes—
127C22×C8C4⋊C412yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42C22×C411yescharacteristic
134C2×C42C22×C411yescharacteristic
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8C22⋊C412yes—
160C4⋊C8C22⋊C412yes—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—41no—
167#97—22no—
168#97—22no—
169#97—22no—
170#97—22no—
171#97—41no—
172#97—81no—
173#97—81no—
174#97C4214yes—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—41no—
178C4×C8—41no—
179C4×C8—81no—
180C4×C8—81no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95C4⋊C412yes—
185#95C4⋊C412yes—
186C24C22⋊Q811yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4C2×C4⋊C411yescharacteristic
196C22×C4C2×C4⋊C411yescharacteristic
197C22×C4#11711yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×Q811yescharacteristic
203C22×C4C4×Q811yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4C22⋊Q811yescharacteristic
207C22×C4#12311yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4#12511yescharacteristic
210C22×C4#12511yescharacteristic
211C22×C4#12611yescharacteristic
212C22×C4C4⋊Q811yescharacteristic
213M16—24no—
214M16—42no—
215M16—42no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—82no—
220M16—82no—
221M16—81no—
222M16—81no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C22⋊C4—44no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42C2×C4211yescharacteristic
255C42C2×C22⋊C411yescharacteristic
256C42#11711yescharacteristic
257C42#11711yescharacteristic
258C42C4×D411yescharacteristic
259C42C4×D411yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×Q811yescharacteristic
263C42C4×Q811yescharacteristic
264C42#12111yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42C22⋊Q811yescharacteristic
267C42#12311yescharacteristic
268C42#12411yescharacteristic
269C42#12611yescharacteristic
270C23—21no—
271C23—21no—
272C23—22no—
273C23—41no—
274C23#38511yescharacteristic
275C23#38711yescharacteristic
276C23#38911yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4C4×C22⋊C411yescharacteristic
306C2×C4C4×C4⋊C411yescharacteristic
307C2×C4C4×C4⋊C411yescharacteristic
308C2×C4#38511yescharacteristic
309C2×C4#38611yescharacteristic
310C2×C4#38711yescharacteristic
311C2×C4#38711yescharacteristic
312C2×C4#38711yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38911yescharacteristic
316C2×C4#39111yescharacteristic
317C2×C4#45311yescharacteristic
318C2×C4#45411yescharacteristic
319C2×C4#45511yescharacteristic
320C2×C4#45611yescharacteristic
321C8—44no—
322C8—44no—
323C8—42no—
324C8—42no—
325C8—42no—
326C8—42no—
327C8—82no—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22#143611yescharacteristic
339C22#185911yescharacteristic
340C22#186011yescharacteristic
341C22#195312yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—82no—
354C4#185611yescharacteristic
355C4#185711yescharacteristic
356C4#188211yescharacteristic
357C4#188311yescharacteristic
358C2—21no—
359C2—21no—
360C2—22no—
361C2—41no—
362C2#1089011yescharacteristic
363C2#1089811yescharacteristic
364C2#1089911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1436
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1569C4semidirect
30. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#205442
#178222
#108532
#80932
#80872
#79882
#79732