Finite Group Databasebeta

← #118060

Subgroups — #118060

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2#21358C211yesmaximal, characteristic
3#21321C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#10835C211yesmaximal, characteristic
6#7893C211yesmaximal, characteristic
7C2×#1275C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993—21no—
16#1993—21no—
17#1992—22no—
18#1987—21no—
19#1981—21no—
20#1947—22no—
21#1872—22no—
22#1852—22no—
23C4×#107—22no—
24C4×#103—22no—
25#1735—22no—
26C2×#336—22no—
27#1481—22no—
28#1481C414yescomplemented
29C2×Q8⋊C8—22no—
30#1279—22no—
31#1275—22no—
32C23×C8—21no—
33C23×C8D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431D412yes—
62C2×#107—24no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×#103—24no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8D412yes—
81C2×C4×C8D412yes—
82#384D412yes—
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C2×#95—21no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#336—44no—
98Q8⋊C8—28no—
99C22×Q8—21no—
100C22×Q8—21no—
101C23×C4—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8Q1614yes—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4⋊C412yes—
126C4⋊Q8C4⋊C412yes—
127#125—22no—
128#125—22no—
129#125—41no—
130#125—41no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—22no—
148#117—22no—
149#117—41no—
150#117—41no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185#107—48no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194#103—48no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—41no—
200#97—41no—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8C22⋊C414yes—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211C24#12111yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12711yescharacteristic
246C22×C4C2×Q1612yes—
247C22×C4#13512yes—
248C22×C4AGL(1,ℤ/8)12yes—
249C22×C4#13712yes—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—82no—
264C2×C8—81no—
265C2×C8—81no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—82no—
288C42—42no—
289C42—42no—
290C42—82no—
291C42—81no—
292C42—81no—
293C42—81no—
294C42—81no—
295C42C2×C22⋊C411yescharacteristic
296C42C2×C4⋊C411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42#12311yescharacteristic
306C42#12311yescharacteristic
307C42#12311yescharacteristic
308C42#12311yescharacteristic
309C42#12411yescharacteristic
310C42#12411yescharacteristic
311C23—22no—
312C23—22no—
313C23#38811yescharacteristic
314C23#39211yescharacteristic
315C23#39611yescharacteristic
316C23#46212yes—
317C23#46312yes—
318Q8—44no—
319Q8—44no—
320Q8—82no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#38011yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38611yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#39011yescharacteristic
352C2×C4#39311yescharacteristic
353C2×C4#39411yescharacteristic
354C2×C4#39411yescharacteristic
355C2×C4#39611yescharacteristic
356C2×C4#39711yescharacteristic
357C2×C4#39911yescharacteristic
358C2×C4C4×Q1612yes—
359C2×C4#44012yes—
360C2×C4#44912yes—
361C2×C4#45112yes—
362C2×C4#46712yes—
363C2×C4#46812yes—
364C2×C4#47712yes—
365C2×C4#47912yes—
366C2×C4#48412yes—
367C2×C4#48512yes—
368C2×C4#49411yescharacteristic
369C2×C4#49511yescharacteristic
370C2×C4#49611yescharacteristic
371C2×C4#49711yescharacteristic
372C8—44no—
373C8—44no—
374C8—82no—
375C8—82no—
376C22—24no—
377C22—24no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#143911yescharacteristic
383C22#192812yes—
384C22#201012yes—
385C22#201211yescharacteristic
386C22#201611yescharacteristic
387C22#201712yes—
388C22#206812yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—84no—
398C4—82no—
399C4—82no—
400C4—162no—
401C4#186312yes—
402C4#196911yescharacteristic
403C4#197311yescharacteristic
404C4#203612yes—
405C4#209511yescharacteristic
406C4#209711yescharacteristic
407C2—22no—
408C2—22no—
409C2#1003312yes—
410C2#1003712yes—
411C2#1108311yescharacteristic
412C2#1111111yescharacteristic
413C2#1112411yescharacteristic
subgroupG/H
centerC22×C4#127
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#392
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#213852
#213582
#213212
C2×#14812
#108352
#78932
C2×#12752