Finite Group Databasebeta

← #6392793

Subgroups — #6392793

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2C2×#2986C211yesmaximal, characteristic
3#60192C211yesmaximal, characteristic
4#22747C214yesmaximal, complemented
5#22746C214yesmaximal, complemented
6#22733C214yesmaximal, complemented
7Q8×#108C214yesmaximal, complemented
8#21995C214yesmaximal
9#21993C212yesmaximal
10#21372C214yesmaximal
11#21370C212yesmaximal
12C2×#533C2211yescharacteristic
13#2987—22no—
14#2986—22no—
15#2986C412yes—
16#2986C412yes—
17#2986C418yes—
18#2986C414yes—
19#2985—24no—
20Q8×M16C2214yes—
21Q8×M16C2214yes—
22Q8×M16C2214yes—
23Q8×M16C2214yes—
24#2962—24no—
25#2961—22no—
26#2960—22no—
27C2×#446C2214yes—
28C2×C8×Q8C2214yes—
29#2921C2214yes—
30#2920C2212yes—
31C2×#433C2214yes—
32C2×#431C2212yes—
33#2905C2212yes—
34C2×#423C2211yescharacteristic
35#2873C2212yes—
36C2×C4×M16C2211yescharacteristic
37#1950C2212yes—
38#1950C2212yes—
39#1950C2212yes—
40#1950C2212yes—
41#1950C2218yes—
42#1950C2218yes—
43#1950C2214yes—
44#1950C2218yes—
45#1950C2218yes—
46#1950C2214yes—
47#1855C2214yes—
48#1855C2214yes—
49#1855C2218yes—
50#1849C2214yes—
51#1848C2214yes—
52#1848C2214yes—
53#1838C2214yes—
54#1837C2214yes—
55#1837C2214yes—
56#1778C2218yes—
57C4×#108C2214yes—
58C4×#108C2214yes—
59C4×#108C2218yes—
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2312yes—
63#533C2×C411yescharacteristic
64#533C2×C411yescharacteristic
65#533C2×C414yes—
66#533C2×C412yes—
67#533D412yes—
68#533D412yes—
69#533D414yes—
70C2×C4⋊Q8Q812yes—
71C2×#125Q812yes—
72C2×C22⋊Q8Q814yes—
73C2×C4×Q8C2314yes—
74C2×#117C2312yes—
75C22×C42C2311yescharacteristic
76#446—24no—
77#446—24no—
78#446—28no—
79#446—28no—
80#446—24no—
81#446—24no—
82#446—28no—
83#446—28no—
84#446C2×C418yes—
85#446C2×C418yes—
86#446C2×C4116yes—
87C8×Q8—24no—
88C8×Q8—24no—
89C8×Q8—24no—
90C8×Q8—24no—
91C8×Q8C2×C418yes—
92C8×Q8C2×C418yes—
93C8×Q8C2×C4116yes—
94#433—28no—
95#433—28no—
96#433C2×C418yes—
97#433C2×C418yes—
98#433C2×C418yes—
99#433C2×C418yes—
100#432—22no—
101#432—22no—
102#431—22no—
103#431—22no—
104#431C2×C414yes—
105#431C2×C414yes—
106#431C2×C414yes—
107#431C2×C414yes—
108C2×#108C2314yes—
109C2×#108C2314yes—
110C2×#108C2314yes—
111C2×#108C2318yes—
112C2×#108C2318yes—
113C2×#108C2314yes—
114#424—24no—
115#424—24no—
116#423—21no—
117#423—21no—
118#423C2312yes—
119#423C2314yes—
120#423C2314yes—
121#423C2314yes—
122#423C2314yes—
123#423C2312yes—
124#423C2314yes—
125C2×C4⋊C8—22no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2318yes—
129#407C2314yes—
130C2×C22⋊C8C2314yes—
131#405—24no—
132#405—24no—
133C4×M16—22no—
134C4×M16C2312yes—
135C4×M16C2312yes—
136C4×M16C2318yes—
137C4×M16C2314yes—
138C4×M16C2318yes—
139C2×#97—21no—
140C2×#97C2314yes—
141C2×C4×C8—21no—
142C2×C4×C8C2314yes—
143C2×C4×C8C2314yes—
144#341C2314yes—
145#341C2318yes—
146#341C2318yes—
147#341C2314yes—
148#341C2314yes—
149#341C2314yes—
150C22×Q8C2×Q814yes—
151C23×C4C2×Q812yes—
152C23×C4C2411yescharacteristic
153C2×M16—21no—
154C2×M16—24no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—42no—
158C2×M16—44no—
159C2×M16C2411yescharacteristic
160C2×M16C2411yescharacteristic
161C2×M16C2414yes—
162C2×M16C2414yes—
163C2×M16C2414yes—
164C2×M16C2412yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8C2414yes—
169C22×C8C2414yes—
170C4⋊Q8C4⋊C412yes—
171C4⋊Q8C4⋊C412yes—
172C4⋊Q8C4⋊C414yes—
173C4⋊Q8C4⋊C414yes—
174C4⋊Q8C4⋊C414yes—
175#125C4⋊C414yes—
176#125C4⋊C412yes—
177#125C4⋊C412yes—
178#125C4⋊C418yes—
179C22⋊Q8C4⋊C414yes—
180C22⋊Q8C4⋊C414yes—
181C22⋊Q8C4⋊C414yes—
182C22⋊Q8C4⋊C414yes—
183C22⋊Q8C4⋊C418yes—
184C22⋊Q8C4⋊C418yes—
185C4×Q8C22×C414yes—
186C4×Q8C22×C414yes—
187C4×Q8C22×C418yes—
188C4×Q8C2×D414yes—
189C4×Q8C2×D414yes—
190C4×Q8C2×D418yes—
191#117C22×C414yes—
192#117C22×C412yes—
193#117C22×C412yes—
194#117C2×D414yes—
195#117C2×D414yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4C2×Q818yes—
198C2×C4⋊C4C2×Q814yes—
199C2×C22⋊C4C2×Q814yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×Q811yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42C2414yes—
209C2×C42C2412yes—
210#108—28no—
211#108—28no—
212#108—44no—
213#108—48no—
214#108—48no—
215#108—44no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—28no—
219C4⋊C8—28no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—28no—
225C4⋊C8—28no—
226C4⋊C8C22×C414yes—
227C4⋊C8C22×C414yes—
228C4⋊C8C22×C414yes—
229C4⋊C8C22×C414yes—
230C4⋊C8C22×C4116yes—
231C4⋊C8C22×C4116yes—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8C22×C418yes—
235C22⋊C8C22×C418yes—
236#97—22no—
237#97—22no—
238#97—28no—
239#97—24no—
240#97—28no—
241#97C22×C418yes—
242#97C22×C418yes—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—28no—
246C4×C8—24no—
247C4×C8—28no—
248C4×C8C22×C418yes—
249C4×C8C22×C418yes—
250C4×C8C22×C418yes—
251C4×C8C22×C418yes—
252C24C22×Q811yescharacteristic
253C2×Q8—48no—
254C2×Q8C2×C4⋊C414yes—
255C2×Q8C2×C4⋊C414yes—
256C2×Q8C2×C4⋊C418yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4C2×C4⋊C412yes—
261C22×C4C2×C4⋊C412yes—
262C22×C4C2×C4⋊C412yes—
263C22×C4C2×C4⋊C412yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C22×Q812yes—
270C22×C4C22×Q812yes—
271C22×C4C22×Q818yes—
272C22×C4C2511yescharacteristic
273M16—24no—
274M16—22no—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—22no—
280M16—28no—
281M16—22no—
282M16—42no—
283M16—44no—
284M162−1+414yes—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—24no—
296C2×C8—44no—
297C2×C8—42no—
298C2×C8C23×C418yes—
299C2×C8C23×C418yes—
300C4⋊C4C2×C4⋊C414yes—
301C4⋊C4C2×C4⋊C414yes—
302C4⋊C4C2×C4⋊C418yes—
303C4⋊C4C2×C4⋊C418yes—
304C4⋊C4C2×C4⋊C418yes—
305C4⋊C4C2×C4⋊C418yes—
306C4⋊C4C2×C4⋊C418yes—
307C4⋊C4C2×C4⋊C418yes—
308C4⋊C4C2×C4⋊C414yes—
309C4⋊C4C2×C4⋊C414yes—
310C22⋊C4C2×C4⋊C418yes—
311C22⋊C4C2×C4⋊C414yes—
312C22⋊C4C2×C4⋊C414yes—
313C42C2×C4⋊C411yescharacteristic
314C42C2×C4⋊C411yescharacteristic
315C42C2×C4⋊C412yes—
316C42C2×C4⋊C411yescharacteristic
317C42C2×C4⋊C412yes—
318C42C2×C4⋊C411yescharacteristic
319C42C23×C412yes—
320C42C23×C412yes—
321C42C23×C414yes—
322C42C23×C414yes—
323C42C22×D412yes—
324C42C22×D418yes—
325C42C22×D412yes—
326C23—22no—
327C23—42no—
328C23C22×C4⋊C411yescharacteristic
329C23C22×C4⋊C411yescharacteristic
330C23C23×Q811yescharacteristic
331Q8—28no—
332Q8—28no—
333Q8—48no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4C22×C4⋊C411yescharacteristic
349C2×C4C22×C4⋊C411yescharacteristic
350C2×C4C22×C4⋊C412yes—
351C2×C4C22×C4⋊C412yes—
352C2×C4C22×C4⋊C412yes—
353C2×C4C22×C4⋊C418yes—
354C2×C4C22×C4⋊C418yes—
355C2×C4C22×C4⋊C412yes—
356C2×C4C24×C411yescharacteristic
357C2×C4C23×D411yescharacteristic
358C2×C4C2×2−1+412yes—
359C2×C4#58512yes—
360C8—28no—
361C8—28no—
362C8—24no—
363C8—44no—
364C8—41no—
365C8—41no—
366C8—44no—
367C8—44no—
368C22—22no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—82no—
374C22C23×C4⋊C411yescharacteristic
375C22#350212yes—
376C4—21no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—28no—
382C4—28no—
383C4#291614yes—
384C4C4×2−1+412yes—
385C4#348611yescharacteristic
386C4#348711yescharacteristic
387C2—22no—
388C2—41no—
389C2—41no—
390C2#3424911yescharacteristic
391C2C2×#291612yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C4⋊C4
prime radicalC24C22×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22747C2semidirect
5. #22746C2semidirect
6. #22733C2semidirect
7. Q8×#108C2semidirect

Maximal subgroups

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

subgroupindex
#602492
C2×#29862
#601922
#227472
#227462
#227332
Q8×#1082
#219952
#219932
#213722
#213702