Finite Group Databasebeta

← #100353

Subgroups — #100353

Order 512.

447 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic
3#9898C211yesmaximal, characteristic
4#9692C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#7050C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#475—21no—
9C2×#446—21no—
10#2868C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1988—21no—
13#1948—21no—
14#1948C412yes—
15#1940—22no—
16#1939—21no—
17#1842—21no—
18#1842C2211yescharacteristic
19#1842C2211yescharacteristic
20#1842C2211yescharacteristic
21#1803—21no—
22#1803C412yes—
23#1767—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#332—21no—
27C2×#332—21no—
28C2×Q8⋊C8—21no—
29#1296—22no—
30#1279—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37#475—42no—
38#475—42no—
39#446—42no—
40#446—42no—
41C2×#106—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yes—
53C2×C4⋊C8D411yescharacteristic
54C2×C4⋊C8D411yescharacteristic
55C2×C4⋊C8D412yes—
56C2×#103—22no—
57C2×#97—22no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384C2×C412yes—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4C2×C411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C43—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—22no—
74#336—22no—
75#336—21no—
76#336—21no—
77#332—42no—
78#332—42no—
79Q8⋊C8—42no—
80C22×Q8—41no—
81C22×Q8C22⋊C411yescharacteristic
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—81no—
97C4⋊Q8—81no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—22no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—21no—
105C4×Q8—41no—
106C4×Q8SD1612yes—
107C4×Q8Q1612yes—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133#106—44no—
134C4⋊C8—22no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8—41no—
141C4⋊C8—44no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—81no—
155C4⋊C8—82no—
156C4⋊C8SD1612yes—
157#103—44no—
158#97—44no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163#95—21no—
164#95—21no—
165#95—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8#10312yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12311yescharacteristic
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4#10112yes—
228C4⋊C4#10312yes—
229C42—21no—
230C42—21no—
231C42—22no—
232C42—22no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42—42no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—81no—
246C42C2×SD1611yescharacteristic
247C42C2×SD1611yescharacteristic
248C42C2×Q1611yescharacteristic
249C42#13511yescharacteristic
250C42AGL(1,ℤ/8)11yescharacteristic
251C42AGL(1,ℤ/8)11yescharacteristic
252C42AGL(1,ℤ/8)11yescharacteristic
253C42AGL(1,ℤ/8)11yescharacteristic
254C42AGL(1,ℤ/8)11yescharacteristic
255C42#13711yescharacteristic
256C42#13711yescharacteristic
257C42#13711yescharacteristic
258C23#38611yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—41no—
262Q8—41no—
263Q8—161no—
264Q8—161no—
265Q8—161no—
266Q8—161no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4C2×#10111yescharacteristic
292C2×C4#41311yescharacteristic
293C2×C4C2×#10311yescharacteristic
294C2×C4#41711yescharacteristic
295C2×C4#41711yescharacteristic
296C2×C4#41811yescharacteristic
297C2×C4C4×SD1611yescharacteristic
298C2×C4#44011yescharacteristic
299C2×C4#44811yescharacteristic
300C2×C4#44911yescharacteristic
301C2×C4#45011yescharacteristic
302C2×C4#45111yescharacteristic
303C2×C4#45511yescharacteristic
304C2×C4#45611yescharacteristic
305C2×C4#46011yescharacteristic
306C2×C4#46411yescharacteristic
307C2×C4#46811yescharacteristic
308C2×C4#46811yescharacteristic
309C2×C4#46911yescharacteristic
310C2×C4#46911yescharacteristic
311C2×C4#47111yescharacteristic
312C2×C4#47211yescharacteristic
313C2×C4#48311yescharacteristic
314C2×C4#48411yescharacteristic
315C8—42no—
316C8—82no—
317C8—82no—
318C8—81no—
319C8—82no—
320C8—161no—
321C22#187211yescharacteristic
322C22#188011yescharacteristic
323C22#188211yescharacteristic
324C22#189511yescharacteristic
325C22#190111yescharacteristic
326C22#190211yescharacteristic
327C22#190711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—81no—
339C4—162no—
340C4#154911yescharacteristic
341C4#155011yescharacteristic
342C4#155711yescharacteristic
343C4#156111yescharacteristic
344C4#164211yescharacteristic
345C4#164511yescharacteristic
346C4#164811yescharacteristic
347C4#165111yescharacteristic
348C4#168611yescharacteristic
349C4#168811yescharacteristic
350C4#169811yescharacteristic
351C4#170211yescharacteristic
352C2#898311yescharacteristic
353C2#913711yescharacteristic
354C2#916711yescharacteristic
355C2#923011yescharacteristic
356C2#925211yescharacteristic
357C2#932011yescharacteristic
358C2#935511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#331962
#227222
#98982
#96922
#85642
#70502
#70312