Finite Group Databasebeta

← #194189

Subgroups — #194189

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8974C218yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#477—24no—
12C2×#477—22no—
13C2×#477C418yescomplemented
14C2×#477C2214yes—
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C22×#107—21no—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20#2356—22no—
21#2356—22no—
22#2355—22no—
23#2355C2214yes—
24#2313—22no—
25#2310—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2283—22no—
29#2274—21no—
30#1990—22no—
31#1987C2214yes—
32#1981—22no—
33#1956C2214yes—
34#1824—22no—
35#1787—22no—
36#1479—24no—
37C2×Q8⋊C8C2214yes—
38#1296C2214yes—
39#1279C2214yes—
40C23×Q8—21no—
41C23×Q8D411yescharacteristic
42C23×C8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C414yes—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C414yes—
56C2×C4×Q8D414yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8C2312yes—
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#477—216no—
68#477—28no—
69#477C2×C4116yes—
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74C2×#107—28no—
75C2×#107—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C414yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—28no—
83C2×#103—24no—
84C2×#103—28no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—24no—
90#391—24no—
91#387—28no—
92#387—28no—
93#384D412yes—
94#384C2312yes—
95#383—24no—
96#382—24no—
97#382—21no—
98#382—21no—
99#381—24no—
100#380—21no—
101#380—21no—
102#380D412yes—
103#380D412yes—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C22⋊C4—24no—
112#376—24no—
113#340—28no—
114Q8⋊C8—28no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—84no—
126C22×Q8C22⋊C414yes—
127C22×Q8C2×D412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—24no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—42no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8C22×C414yes—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8C22⋊C418yes—
158C4×Q8C22×C414yes—
159C4×Q8C2×D414yes—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203#107—216no—
204C4⋊C8—24no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—42no—
209C4⋊C8C22×C414yes—
210#103—216no—
211#103—88no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224C24#12011yescharacteristic
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—28no—
233C2×Q8—28no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—84no—
242C2×Q8—88no—
243C2×Q8—88no—
244C2×Q8C2×C22⋊C414yes—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—82no—
258C22×C4—84no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C4×D412yes—
261C22×C4C4×D412yes—
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4#12312yes—
268C22×C4#13512yes—
269C22×C4#13512yes—
270C22×C4#13712yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—48no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—82no—
301C4⋊C4C2×C22⋊C414yes—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C42—24no—
305C42—24no—
306C42—28no—
307C42—42no—
308C42—42no—
309C42—44no—
310C42—88no—
311C42C2×C22⋊C412yes—
312C42C4×D412yes—
313C42C4×D412yes—
314C42#12112yes—
315C42#12112yes—
316C42#12312yes—
317C42C23×C411yescharacteristic
318C42C22×D411yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C23—22no—
322C23—22no—
323C23#38612yes—
324C23#45212yes—
325C23#45212yes—
326C23C2×#12011yescharacteristic
327Q8—24no—
328Q8—24no—
329Q8—48no—
330Q8—42no—
331Q8—42no—
332Q8—48no—
333Q8—84no—
334Q8—88no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—48no—
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—44no—
354C2×C4—44no—
355C2×C4—84no—
356C2×C4—88no—
357C2×C4—84no—
358C2×C4#38612yes—
359C2×C4#38612yes—
360C2×C4#38612yes—
361C2×C4#41614yes—
362C2×C4#41914yes—
363C2×C4#45212yes—
364C2×C4#45212yes—
365C2×C4#48514yes—
366C2×C4#48514yes—
367C2×C4C22×C22⋊C411yescharacteristic
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×#12111yescharacteristic
371C2×C4C2×#12111yescharacteristic
372C2×C4C2×#12311yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C2×C4C2×#13511yescharacteristic
375C2×C4C2×#13711yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4#57814yes—
378C8—44no—
379C8—82no—
380C8—82no—
381C22—22no—
382C22—24no—
383C22—22no—
384C22—24no—
385C22#187314yes—
386C22C2×#38611yescharacteristic
387C22C2×#45211yescharacteristic
388C22C2×#45211yescharacteristic
389C22#301112yes—
390C22#302112yes—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—44no—
396C4—44no—
397C4—84no—
398C4—84no—
399C4—84no—
400C4#187314yes—
401C4C2×#41611yescharacteristic
402C4C2×#41911yescharacteristic
403C4#295012yes—
404C4#304412yes—
405C4C2×#48511yescharacteristic
406C4C2×#48511yescharacteristic
407C2—24no—
408C2#1889712yes—
409C2#1889912yes—
410C2C2×#187311yescharacteristic
411C2#2093511yescharacteristic
412C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8974C2semidirect
13. C2×#477C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
#341872
#218342
#218042
C2×#14792
#89742
#80392
#78932