Finite Group Databasebeta

← #118652

Subgroups — #118652

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1995C211yesmaximal, characteristic
2#17814C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#10840C211yesmaximal, characteristic
6#8095C211yesmaximal, characteristic
7#7906C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1995—22no—
16#1995—22no—
17C2×#339—22no—
18C2×#339—22no—
19#1571—22no—
20#1571—22no—
21#1567—22no—
22#1520—22no—
23#1518C414yescomplemented
24#1503—22no—
25#1493—22no—
26#1487—22no—
27#1481—22no—
28#1303—22no—
29#1280—22no—
30C22×M16—21no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533D412yes—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117—21no—
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51#431—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431C2×C412yes—
57#431D412yes—
58#431D412yes—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423D412yes—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#384—21no—
78#382C2×C412yes—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83#376—21no—
84#376—21no—
85#376—21no—
86#376—21no—
87C2×#95—21no—
88C2×#95—21no—
89#339—28no—
90#339—44no—
91#335—28no—
92#328—28no—
93Q8⋊C8—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4—22no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4⋊C412yes—
119C4⋊Q8C4⋊C412yes—
120#125—22no—
121#125—22no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—22no—
141#117—22no—
142#117—41no—
143#117—41no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42Q1614yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8C22⋊C414yes—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197#95—22no—
198#95—22no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24#12111yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C2×Q8—82no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12711yescharacteristic
238C22×C4C2×Q1612yes—
239C22×C4#13512yes—
240C22×C4AGL(1,ℤ/8)12yes—
241C22×C4#13712yes—
242M16—44no—
243M16—44no—
244M16—82no—
245M16—81no—
246M16—81no—
247C2×C8—24no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—81no—
255C2×C8—81no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—82no—
274C42—24no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—81no—
281C42—81no—
282C42C2×C4⋊C411yescharacteristic
283C42#11711yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12011yescharacteristic
287C42#12011yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42#12311yescharacteristic
294C42#12311yescharacteristic
295C42#12311yescharacteristic
296C42#12411yescharacteristic
297C42#12611yescharacteristic
298C23—41no—
299C23—41no—
300C23#39011yescharacteristic
301C23#39411yescharacteristic
302C23#39611yescharacteristic
303C23#46712yes—
304C23#46812yes—
305Q8—44no—
306Q8—82no—
307Q8—82no—
308Q8—82no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#38011yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#39211yescharacteristic
340C2×C4#39311yescharacteristic
341C2×C4#39411yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#39711yescharacteristic
344C2×C4#39911yescharacteristic
345C2×C4C4×Q1612yes—
346C2×C4#44012yes—
347C2×C4#44912yes—
348C2×C4#45112yes—
349C2×C4#46212yes—
350C2×C4#46312yes—
351C2×C4#47712yes—
352C2×C4#47912yes—
353C2×C4#48412yes—
354C2×C4#48512yes—
355C2×C4#49411yescharacteristic
356C2×C4#49511yescharacteristic
357C2×C4#49611yescharacteristic
358C2×C4#49711yescharacteristic
359C8—44no—
360C8—82no—
361C8—82no—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22#143911yescharacteristic
372C22#196911yescharacteristic
373C22#197311yescharacteristic
374C22#203612yes—
375C22#206812yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—84no—
385C4—82no—
386C4—82no—
387C4—162no—
388C4#186312yes—
389C4#192812yes—
390C4#201211yescharacteristic
391C4#201611yescharacteristic
392C4#209511yescharacteristic
393C4#209711yescharacteristic
394C2—22no—
395C2—21no—
396C2—22no—
397C2—21no—
398C2#1109411yescharacteristic
399C2#1111811yescharacteristic
400C2#1112411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1518C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19952
#178142
#175752
#174862
#108402
#80952
#79062