Finite Group Databasebeta

← #6371910

Subgroups — #6371910

Order 512.

2598 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#430C211yesmaximal, characteristic
2C2×#2911C212yesmaximal
3C2×#2911C211yesmaximal, characteristic
4C2×#2873C211yesmaximal, characteristic
5#21182C2116yesmaximal, complemented
6#21181C218yesmaximal, complemented
7C2×#1742C212yesmaximal
8C23×M16C2212yes—
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2212yes—
13C2×#430C4116yescomplemented
14C2×#430C2218yes—
15C22×#108C2214yes—
16#2911—22no—
17#2911—22no—
18#2911—24no—
19#2911—24no—
20#2911—24no—
21#2911C2218yes—
22#2911C2218yes—
23C2×#428—24no—
24C2×#428C2218yes—
25C2×#428C2218yes—
26C2×#427—24no—
27C2×#427C2214yes—
28#2873—24no—
29#2873C2218yes—
30C2×#405—22no—
31C2×#405C2214yes—
32C2×C4×M16—22no—
33#1952C22132yescomplemented
34#1951C22132yescomplemented
35#1742C2218yes—
36#1742C2218yes—
37C2×#343C2218yes—
38C24×C4C2311yescharacteristic
39C22×M16—21no—
40C22×M16—22no—
41C22×M16—22no—
42C22×M16—22no—
43C22×M16—22no—
44C22×M16—28no—
45C22×M16—24no—
46C22×M16C2×C4116yes—
47C22×M16C2×C418yes—
48C22×M16D412yes—
49C22×M16D412yes—
50C22×M16D414yes—
51C22×M16Q818yes—
52C22×M16Q818yes—
53C22×M16C2314yes—
54C22×M16C2314yes—
55C22×M16C2314yes—
56C22×M16C2312yes—
57C22×M16C2314yes—
58C22×M16C2314yes—
59C23×C8—21no—
60C23×C8—22no—
61C23×C8C2312yes—
62C2×#117—21no—
63C2×#117—22no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117D414yes—
77C2×#117D412yes—
78C2×#117C2312yes—
79C2×#117C2314yes—
80C2×#117C2314yes—
81C2×#117C2314yes—
82C22×C4⋊C4D412yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42—22no—
88#430C2×C4164yescomplemented
89C2×#108C2×C4132yescomplemented
90C2×#108C23116yes—
91#428—216no—
92#428—216no—
93#428—216no—
94#428—216no—
95#428—216no—
96#428C23132yes—
97#427—28no—
98#427—28no—
99#427—216no—
100#427—216no—
101C2×#107—28no—
102C2×#107C2318yes—
103C2×#106—28no—
104C2×#106C2318yes—
105#405—216no—
106#405—28no—
107#405C23116yes—
108C4×M16—216no—
109C2×#97—24no—
110C2×C4×C8—24no—
111#343C23132yes—
112C25#4111yescharacteristic
113C23×C4—22no—
114C23×C4—24no—
115C23×C4C22×C414yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q814yes—
123C23×C4C2×Q814yes—
124C23×C4#4114yes—
125C23×C4#4111yescharacteristic
126C23×C4#4112yes—
127C23×C4C2412yes—
128C23×C4C2411yescharacteristic
129C2×M16—216no—
130C2×M16—216no—
131C2×M16—216no—
132C2×M16—28no—
133C2×M16—216no—
134C2×M16—24no—
135C2×M16—22no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—28no—
140C2×M16—28no—
141C2×M16—28no—
142C2×M16—24no—
143C2×M16—28no—
144C2×M16C4⋊C4116yes—
145C2×M16C4⋊C4116yes—
146C2×M16C4⋊C4132yes—
147C2×M16C22×C4116yes—
148C2×M16C22×C4116yes—
149C2×M16C22×C4116yes—
150C2×M16C22×C4116yes—
151C2×M16C2×D418yes—
152C2×M16C2×D418yes—
153C2×M16C2×Q818yes—
154C2×M16C2×Q8116yes—
155C2×M16C2×Q8116yes—
156C2×M16C2418yes—
157C2×M16C2414yes—
158C22×C8—216no—
159C22×C8—28no—
160C22×C8—22no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8C22×C418yes—
166C22×C8C2×D414yes—
167C22×C8C2×D414yes—
168C22×C8C2×Q818yes—
169C22×C8C2414yes—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—22no—
175#117—24no—
176#117—22no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—24no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193#117C2×D414yes—
194#117C2×D418yes—
195#117C2414yes—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4C2×D418yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2418yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226#108C22×C4164yescomplemented
227#107—216no—
228#107—232no—
229#106—216no—
230#106—232no—
231#97—216no—
232C4×C8—216no—
233C24—82no—
234C24C4×Q814yes—
235C24#12112yes—
236C24#12111yescharacteristic
237C24#12111yescharacteristic
238C24#12514yes—
239C24C2×#4111yescharacteristic
240C24C2×#4112yes—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—84no—
247C22×C4—88no—
248C22×C4C2×C4⋊C414yes—
249C22×C4C2×C4⋊C418yes—
250C22×C4C2×C4⋊C414yes—
251C22×C4C4×D418yes—
252C22×C4C4×D414yes—
253C22×C4C4×D414yes—
254C22×C4C4×Q814yes—
255C22×C4C4×Q818yes—
256C22×C4#12111yescharacteristic
257C22×C4#12112yes—
258C22×C4#12112yes—
259C22×C4#12111yescharacteristic
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4C22⋊Q814yes—
264C22×C4C22⋊Q818yes—
265C22×C4C22⋊Q814yes—
266C22×C4#12514yes—
267C22×C4#12518yes—
268C22×C4C4⋊Q814yes—
269C22×C4C4⋊Q818yes—
270C22×C4C4⋊Q814yes—
271C22×C4C23×C412yes—
272C22×C4C23×C414yes—
273C22×C4C22×D412yes—
274C22×C4C22×D412yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×D412yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×Q812yes—
282C22×C4C22×Q814yes—
283C22×C4C22×Q814yes—
284C22×C4C22×Q812yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4114yes—
287C22×C4C2×#4112yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4C2511yescharacteristic
290M16—216no—
291M16—216no—
292M16—216no—
293M16—216no—
294M16—28no—
295M16—28no—
296M16—24no—
297M16C2×C4⋊C4132yes—
298M16C2×C4⋊C4132yes—
299M16C23×C4116yes—
300M16C22×Q818yes—
301C2×C8—216no—
302C2×C8—216no—
303C2×C8—216no—
304C2×C8—28no—
305C2×C8—216no—
306C2×C8—24no—
307C2×C8—28no—
308C2×C8C2×C4⋊C4116yes—
309C2×C8C2×C4⋊C4116yes—
310C2×C8C23×C418yes—
311C2×C8C22×D418yes—
312C2×C8C22×Q818yes—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4C22×D418yes—
320C22⋊C4—24no—
321C22⋊C4—28no—
322C22⋊C4—24no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C42—28no—
328C42—24no—
329C42—24no—
330C42—44no—
331C42—44no—
332C42—44no—
333C42—44no—
334C23—42no—
335C23—44no—
336C23—42no—
337C23—44no—
338C23—42no—
339C23—84no—
340C23—84no—
341C23#38918yes—
342C23#38914yes—
343C23#38914yes—
344C23C2×C4×Q812yes—
345C23C2×C4×Q814yes—
346C23C2×#12111yescharacteristic
347C23C2×#12112yes—
348C23C2×#12111yescharacteristic
349C23C2×#12112yes—
350C23C2×#12512yes—
351C23C2×#12514yes—
352C23C22×#4111yescharacteristic
353C2×C4—28no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—88no—
357C2×C4—88no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4#38918yes—
361C2×C4#38914yes—
362C2×C4#38918yes—
363C2×C4#38918yes—
364C2×C4#38914yes—
365C2×C4#38918yes—
366C2×C4#38918yes—
367C2×C4C22×C4⋊C412yes—
368C2×C4C22×C4⋊C412yes—
369C2×C4C22×C4⋊C414yes—
370C2×C4C22×C4⋊C414yes—
371C2×C4C2×C4×D412yes—
372C2×C4C2×C4×D414yes—
373C2×C4C2×C4×D414yes—
374C2×C4C2×C4×D412yes—
375C2×C4C2×C4×Q814yes—
376C2×C4C2×C4×Q812yes—
377C2×C4C2×#12112yes—
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12112yes—
380C2×C4C2×#12111yescharacteristic
381C2×C4C2×C22⋊Q814yes—
382C2×C4C2×C22⋊Q812yes—
383C2×C4C2×C22⋊Q814yes—
384C2×C4C2×C22⋊Q812yes—
385C2×C4C2×#12514yes—
386C2×C4C2×#12512yes—
387C2×C4C2×C4⋊Q814yes—
388C2×C4C2×C4⋊Q814yes—
389C2×C4C2×C4⋊Q812yes—
390C2×C4C2×C4⋊Q812yes—
391C2×C4C24×C411yescharacteristic
392C2×C4C23×D411yescharacteristic
393C2×C4C23×D411yescharacteristic
394C2×C4C23×Q811yescharacteristic
395C2×C4C23×Q811yescharacteristic
396C2×C4C22×#4111yescharacteristic
397C8—216no—
398C8—28no—
399C8C22×C4⋊C4116yes—
400C22—22no—
401C22—24no—
402C22—22no—
403C22—24no—
404C22—22no—
405C22—44no—
406C22—42no—
407C22—44no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—48no—
412C22C2×#38912yes—
413C22C2×#38912yes—
414C22C2×#38914yes—
415C22C2×#38914yes—
416C22C22×C4×Q811yescharacteristic
417C22C22×#12111yescharacteristic
418C22C22×#12511yescharacteristic
419C4—48no—
420C4—88no—
421C4C2×#38912yes—
422C4C2×#38914yes—
423C4C2×#38914yes—
424C4C2×#38912yes—
425C4C23×C4⋊C411yescharacteristic
426C4C22×C4×D411yescharacteristic
427C4C22×C22⋊Q811yescharacteristic
428C4C22×C4⋊Q811yescharacteristic
429C2—21no—
430C2—21no—
431C2—22no—
432C2—24no—
433C2—24no—
434C2—22no—
435C2C22×#38911yescharacteristic
436C2#6046912yes—
subgroupG/H
centerC2×C4C2×#121
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×#121
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #21182C2semidirect
6. #21181C2semidirect
13. C2×#430C4semidirect
33. #1952C22semidirect
34. #1951C22semidirect
88. #430C2×C4semidirect
89. C2×#108C2×C4semidirect
226. #108C22×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4302
C2×#29112
C2×#28732
#211822
#211812
C2×#17422