Finite Group Databasebeta

← #97538

Subgroups — #97538

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic
2#11207C212yesmaximal
3#11050C211yesmaximal, characteristic
4C2×#1272C211yesmaximal, characteristic
5C2×#1271C212yesmaximal
6C22×C4⋊Q8—21no—
7C22×#103—21no—
8#2734—22no—
9#2734—24no—
10#2734C414yescomplemented
11C2×#398—21no—
12C2×#395—22no—
13C2×#391C2211yescharacteristic
14C2×#389C2212yes—
15C2×#387—22no—
16C2×#384C2211yescharacteristic
17C2×#340—21no—
18C2×#340—22no—
19C2×#340—22no—
20C2×#336C2211yescharacteristic
21C2×#336C2212yes—
22#1272—22no—
23#1271—24no—
24C23×Q8—21no—
25C23×Q8D411yescharacteristic
26C23×C8—21no—
27C23×C8—22no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—24no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—42no—
32C2×C4⋊Q8—42no—
33C2×C4⋊Q8—42no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D411yescharacteristic
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44C2×#103—42no—
45C2×#103—42no—
46C2×#103—42no—
47#398—24no—
48#398—24no—
49#395—28no—
50#395—28no—
51#391—22no—
52#391C2×C412yes—
53#391D412yes—
54#389—22no—
55#389—22no—
56#389C2×C414yes—
57#389D414yes—
58#387—28no—
59#387—28no—
60#384—42no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65#340—44no—
66#340—42no—
67#340—44no—
68#336—22no—
69#336—22no—
70#336—22no—
71C22×Q8—24no—
72C22×Q8—21no—
73C22×Q8—21no—
74C22×Q8—21no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—81no—
82C22×Q8—81no—
83C22×Q8—81no—
84C22×Q8—81no—
85C22×Q8C22⋊C412yes—
86C22×Q8D812yes—
87C22×Q8SD1612yes—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—41no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C22×C8—24no—
96C22×C8—22no—
97C22×C8—42no—
98C22×C8—41no—
99C22×C8—42no—
100C22×C8—41no—
101C22×C8—42no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—42no—
107C22×C8—41no—
108C22×C8—42no—
109C4⋊Q8—48no—
110C4⋊Q8—48no—
111C4⋊Q8—48no—
112C4⋊Q8—48no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4SD1614yes—
150C2×C4⋊C4Q1614yes—
151C2×C42—24no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D412yes—
159#103—48no—
160#95—24no—
161#95—28no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166C24#12011yescharacteristic
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—48no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×Q8—81no—
180C2×Q8—81no—
181C2×Q8—81no—
182C2×Q8—81no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—84no—
189C2×Q8#10214yes—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—82no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4#12011yescharacteristic
232C22×C4#12012yes—
233C22×C4C2×D811yescharacteristic
234C22×C4C2×SD1611yescharacteristic
235C22×C4C2×SD1612yes—
236C22×C4C2×Q1612yes—
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4AGL(1,ℤ/8)11yescharacteristic
239C22×C4#13712yes—
240C22×C4#13712yes—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8—81no—
249C2×C8—81no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—81no—
253C2×C8—81no—
254C2×C8—82no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—416no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—81no—
270C4⋊C4—82no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4#10318yescomplemented
277C42—41no—
278C42—42no—
279C42—41no—
280C42—48no—
281C23#37911yescharacteristic
282C23#44711yescharacteristic
283C23#44812yes—
284C23#45011yescharacteristic
285C23#45112yes—
286C23#45311yescharacteristic
287C23#45412yes—
288C23#45511yescharacteristic
289C23#45612yes—
290C23B(4,𝔽2)11yescharacteristic
291C23#45811yescharacteristic
292Q8—44no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296Q8—816no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—161no—
350C2×C4—162no—
351C2×C4—161no—
352C2×C4C2×#10211yescharacteristic
353C2×C4C2×#10312yes—
354C2×C4#41811yescharacteristic
355C2×C4#41912yes—
356C2×C4#44711yescharacteristic
357C2×C4#44812yes—
358C2×C4#45011yescharacteristic
359C2×C4#45112yes—
360C8—82no—
361C8—84no—
362C22#159511yescharacteristic
363C22#159912yes—
364C22#160111yescharacteristic
365C22#160512yes—
366C22#160711yescharacteristic
367C22#161112yes—
368C22#161311yescharacteristic
369C22#161712yes—
370C22#162311yescharacteristic
371C22#162712yes—
372C22#163311yescharacteristic
373C22#163712yes—
374C22#164512yes—
375C22#164912yes—
376C22#165112yes—
377C22#165711yescharacteristic
378C22#166011yescharacteristic
379C22#166412yes—
380C22#178611yescharacteristic
381C22#178712yes—
382C22#179011yescharacteristic
383C22#179211yescharacteristic
384C22#179312yes—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—44no—
394C4—48no—
395C4—82no—
396C4—84no—
397C4—162no—
398C2#774111yescharacteristic
399C2#775711yescharacteristic
400C2#776612yes—
401C2#779412yes—
402C2#781711yescharacteristic
403C2#782911yescharacteristic
404C2#840311yescharacteristic
405C2#840612yes—
406C2#844712yes—
407C2#845111yescharacteristic
408C2#850711yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2734C4semidirect
276. C4⋊C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
#112072
#110502
C2×#12722
C2×#12712