Finite Group Databasebeta

← #416965

Subgroups — #416965

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1997C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5#20755C211yesmaximal, characteristic
6C4×#428C211yesmaximal, characteristic
7#11303C214yesmaximal
8#11249C214yesmaximal
9#10847C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#108C2212yes—
12C2×#428C2212yes—
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C4×#117C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1997—24no—
20#1988—22no—
21#1988—22no—
22#1988—21no—
23#1988—21no—
24#1987—22no—
25#1950—24no—
26#1832—22no—
27#1831—22no—
28#1776—22no—
29C4×#107—21no—
30C4×#106—21no—
31C2×#343C2212yes—
32C2×#343C2214yes—
33C2×#343C2212yes—
34C2×#341C2212yes—
35#1736C2212yes—
36#1736C2214yes—
37C2×#340C2214yes—
38#1729C2214yes—
39C23×Q8—21no—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C22×M16C2311yescharacteristic
43C22×M16C2311yescharacteristic
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8Q814yes—
53C2×#117C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C2×#108—24no—
60C2×#108—28no—
61#428—24no—
62#428—24no—
63C2×#107—22no—
64C2×#106—22no—
65#423—22no—
66#423—22no—
67#423—22no—
68#423—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C4×M16—22no—
78C4×M16D414yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4D411yescharacteristic
87C4×C4⋊C4D411yescharacteristic
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90#376—21no—
91C2×#95—21no—
92C43—21no—
93#343—24no—
94#343—24no—
95#343—28no—
96#341—24no—
97#340—28no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#336—28no—
103C22×Q8—24no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—42no—
107C22×Q8—82no—
108C22×Q8—82no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4C2411yescharacteristic
114C2×M16—21no—
115C2×M16—22no—
116C2×M16—21no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16C2×D412yes—
128C2×M16C2×D412yes—
129C2×M16#4114yes—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×D414yes—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×Q814yes—
168C2×C4⋊C4#4114yes—
169C2×C22⋊C4—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—82no—
177C2×C42C2×D412yes—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185#108—48no—
186#107—28no—
187#106—28no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C4⋊C8—42no—
194#97—24no—
195C4×C8—24no—
196#95—21no—
197#95—21no—
198#95—22no—
199#95—41no—
200#95—41no—
201C24C2×#4111yescharacteristic
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—88no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—88no—
208C2×Q8—82no—
209C2×Q8—82no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q812yes—
230C22×C4C22⋊Q812yes—
231C22×C4#12612yes—
232C22×C4#12612yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×Q811yescharacteristic
237C22×C4C2×#4111yescharacteristic
238C22×C4C2×#4111yescharacteristic
239C22×C4C2×#4112yes—
240C22×C4C2×#4112yes—
241C22×C42−1+412yes—
242C22×C42−1+412yes—
243M16—24no—
244M16—22no—
245M16—22no—
246M16—22no—
247M16—42no—
248M16—42no—
249M16—44no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—44no—
254M16#12414yes—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—28no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—44no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—44no—
267C2×C8—44no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—82no—
289C42—82no—
290C42—82no—
291C42#12112yes—
292C42#12112yes—
293C42#12112yes—
294C42#12112yes—
295C42C22⋊Q812yes—
296C42C22⋊Q812yes—
297C42C22⋊Q812yes—
298C42C22⋊Q812yes—
299C23—21no—
300C23—21no—
301C23—22no—
302C23C2×#12111yescharacteristic
303C23C2×C22⋊Q811yescharacteristic
304C23C2×#12611yescharacteristic
305C23#55212yes—
306C23#55612yes—
307Q8—88no—
308Q8—88no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4#39612yes—
334C2×C4#39612yes—
335C2×C4#39612yes—
336C2×C4#39612yes—
337C2×C4#47112yes—
338C2×C4#47212yes—
339C2×C4C2×#12111yescharacteristic
340C2×C4C2×#12111yescharacteristic
341C2×C4C2×C22⋊Q811yescharacteristic
342C2×C4C2×C22⋊Q811yescharacteristic
343C2×C4#52512yes—
344C2×C4C2×#12411yescharacteristic
345C2×C4#52912yes—
346C2×C4#53211yescharacteristic
347C2×C4#53612yes—
348C2×C4#53611yescharacteristic
349C2×C4#53611yescharacteristic
350C2×C4#54112yes—
351C2×C4#54712yes—
352C2×C4#54812yes—
353C2×C4#55612yes—
354C2×C4#55612yes—
355C2×C4#55712yes—
356C2×C4#55712yes—
357C8—24no—
358C8—42no—
359C8—44no—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22C2×#39611yescharacteristic
371C22#241512yes—
372C22#251412yes—
373C22#256511yescharacteristic
374C22#256811yescharacteristic
375C22C2×#47111yescharacteristic
376C22C2×#47211yescharacteristic
377C22#306912yes—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—84no—
382C4—82no—
383C4—84no—
384C4—82no—
385C4#206314yes—
386C4#245312yes—
387C4#254511yescharacteristic
388C4#254711yescharacteristic
389C4#258112yes—
390C4#262111yescharacteristic
391C4#285011yescharacteristic
392C2—21no—
393C2—21no—
394C2—22no—
395C2#1523711yescharacteristic
396C2C2×#206311yescharacteristic
397C2#2360311yescharacteristic
398C2#2472012yes—
399C2#2472312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19972
#213132
C2×#19882
C2×#19502
#207552
C4×#4282
#113032
#112492
#108472