Finite Group Databasebeta

← #375108

Subgroups — #375108

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1946C211yesmaximal, characteristic
2C2×#1945C211yesmaximal, characteristic
3#21145C212yesmaximal
4#20799C211yesmaximal, characteristic
5C2×#1778C211yesmaximal, characteristic
6C2×#1771C211yesmaximal, characteristic
7#11221C212yesmaximal
8#11221C212yesmaximal
9#10860C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10851C212yesmaximal
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#427C2211yescharacteristic
16C2×#427C2212yes—
17C2×#424C2212yes—
18C2×#424C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2212yes—
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2285C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1946—24no—
27#1945—24no—
28#1940—24no—
29#1939—24no—
30#1847C414yescomplemented
31#1847C414yescomplemented
32#1778—24no—
33#1771—24no—
34C2×#341C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
40#1729C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2312yes—
50C2×#117C2312yes—
51C22×C4⋊C4—21no—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#108—22no—
56C2×#108—22no—
57C2×#108—22no—
58C2×#108—22no—
59C2×#108D414yes—
60C2×#108Q814yes—
61#427—22no—
62#427—22no—
63#427—22no—
64#427—22no—
65#427—42no—
66#427—42no—
67C2×#107C2×C412yes—
68C2×#107C2×C412yes—
69C2×#107C2×C414yes—
70C2×#106C2×C412yes—
71C2×#106C2×C412yes—
72C2×#106C2×C414yes—
73#424—24no—
74#424—24no—
75#424—42no—
76#424—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8D414yes—
82C4×M16—24no—
83C4×M16—24no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97C2×C412yes—
89C2×#97C2×C412yes—
90C2×#97D414yes—
91C2×#97Q814yes—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96#384C2×C414yes—
97#382C2×C414yes—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4Q814yes—
104C2×#95—21no—
105#341—24no—
106#339—24no—
107#339—24no—
108#339—42no—
109#339—42no—
110#336—24no—
111#336—24no—
112#336—22no—
113#336—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4C2411yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16C2×D414yes—
130C2×M16C2×Q814yes—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C22×C412yes—
150C22×C8C22×C411yescharacteristic
151C22×C8C22×C411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8C2×Q812yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C414yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×Q812yes—
194C2×C42C2×Q812yes—
195C2×C42C2×Q812yes—
196C2×C42C2×Q812yes—
197C2×C42#4112yes—
198#108—28no—
199#108—28no—
200#107—24no—
201#107—24no—
202#107—44no—
203#106—24no—
204#106—24no—
205#106—44no—
206C4⋊C8—28no—
207C4⋊C8—48no—
208C4⋊C8—48no—
209#97—44no—
210#97—44no—
211#97C4⋊C418yes—
212#97C4⋊C418yes—
213C4×C8—44no—
214C4×C8—44no—
215#95—24no—
216#95—24no—
217#95—24no—
218C24C2×#4111yescharacteristic
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4C4×D412yes—
234C22×C4C4×D412yes—
235C22×C4C22⋊Q812yes—
236C22×C4C22⋊Q812yes—
237C22×C4#12512yes—
238C22×C4#12512yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×Q811yescharacteristic
244C22×C4C22×Q811yescharacteristic
245C22×C4C22×Q811yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C4H(5,𝔽2)11yescharacteristic
251C22×C42−1+411yescharacteristic
252C22×C42−1+411yescharacteristic
253M16—24no—
254M16—24no—
255M16—24no—
256M16—24no—
257M16—42no—
258M16—42no—
259M16—42no—
260M16—42no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—24no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8C2×C4⋊C414yes—
288C2×C8C2×C4⋊C412yes—
289C2×C8C2×C4⋊C412yes—
290C2×C8C4×D412yes—
291C2×C8C4×D412yes—
292C2×C8C4×Q812yes—
293C2×C8C4×Q812yes—
294C2×C8#12114yes—
295C2×C8C22⋊Q814yes—
296C2×C8#12514yes—
297C2×C8C4⋊Q814yes—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42C2×C4⋊C414yes—
322C42C4×Q814yes—
323C42#12114yes—
324C42C4⋊Q814yes—
325C23—22no—
326C23—22no—
327C23C2×C4×D411yescharacteristic
328C23C2×C22⋊Q811yescharacteristic
329C23C2×#12511yescharacteristic
330C23#55011yescharacteristic
331C23#55611yescharacteristic
332C23#55711yescharacteristic
333C23#55911yescharacteristic
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#38914yes—
357C2×C4#38914yes—
358C2×C4C22×C4⋊C411yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C2×C4×Q811yescharacteristic
361C2×C4C2×C4×Q811yescharacteristic
362C2×C4#51811yescharacteristic
363C2×C4#51911yescharacteristic
364C2×C4C2×#12111yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×#12511yescharacteristic
368C2×C4C2×C4⋊Q811yescharacteristic
369C2×C4C2×C4⋊Q811yescharacteristic
370C2×C4#53711yescharacteristic
371C2×C4#53711yescharacteristic
372C2×C4#54411yescharacteristic
373C2×C4#54411yescharacteristic
374C2×C4D4211yescharacteristic
375C2×C4#54711yescharacteristic
376C2×C4#54711yescharacteristic
377C2×C4D4×Q811yescharacteristic
378C2×C4D4×Q811yescharacteristic
379C2×C4#55011yescharacteristic
380C2×C4#55411yescharacteristic
381C2×C4#55411yescharacteristic
382C2×C4#55511yescharacteristic
383C2×C4#55611yescharacteristic
384C2×C4#55711yescharacteristic
385C2×C4Q8211yescharacteristic
386C8—24no—
387C8—42no—
388C8—42no—
389C8—44no—
390C8—44no—
391C8#38914yes—
392C8#38914yes—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22C2×#38911yescharacteristic
402C22#236511yescharacteristic
403C22#236611yescharacteristic
404C22#253811yescharacteristic
405C22#255211yescharacteristic
406C22#257111yescharacteristic
407C22#258211yescharacteristic
408C22#295512yes—
409C22#295612yes—
410C4—24no—
411C4—24no—
412C4—44no—
413C4—82no—
414C4—84no—
415C4—84no—
416C4C2×#38911yescharacteristic
417C4#231611yescharacteristic
418C4#236511yescharacteristic
419C4#236911yescharacteristic
420C4#253711yescharacteristic
421C4#253811yescharacteristic
422C4#255111yescharacteristic
423C4#255211yescharacteristic
424C2—21no—
425C2—21no—
426C2—21no—
427C2—21no—
428C2#1415111yescharacteristic
429C2#2116411yescharacteristic
430C2#2116511yescharacteristic
431C2#2317212yes—
432C2#2318112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1847C4semidirect
31. #1847C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19462
C2×#19452
#211452
#207992
C2×#17782
C2×#17712
#112212
#108602
#108592
#108512