Finite Group Databasebeta

← #411148

Subgroups — #411148

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1944C211yesmaximal, characteristic
2C2×#1771C211yesmaximal, characteristic
3#15194C212yesmaximal
4#11379C211yesmaximal, characteristic
5#11377C211yesmaximal, characteristic
6#11327C211yesmaximal, characteristic
7#11327C211yesmaximal, characteristic
8#11227C212yesmaximal
9#10881C212yesmaximal
10#10880C212yesmaximal
11#10821C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1944—24no—
23#1771—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1444—22no—
36#1441—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#106—22no—
60C2×#106—24no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×#97—42no—
68C2×#97D414yes—
69C2×C4×C8—22no—
70C2×C4×C8—42no—
71#389—24no—
72#389—24no—
73#389—24no—
74#389—24no—
75#389—42no—
76#389—42no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D412yes—
89#340—24no—
90#340—22no—
91#340—22no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340Q818yes—
97#336—22no—
98#336—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C22×C8C2×Q814yes—
124C22×C8#4112yes—
125C22×C8#4112yes—
126C22×C8#4112yes—
127C22×C8#4112yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C2×Q814yes—
162C2×C4⋊C4C2×Q814yes—
163C2×C4⋊C4#4114yes—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C2×D412yes—
177C2×C42#4112yes—
178#107—44no—
179#106—28no—
180C4⋊C8—28no—
181#97—28no—
182#97—44no—
183C4×C8—44no—
184#95—24no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194C24C22×D411yescharacteristic
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#12112yes—
245C22×C4#12112yes—
246C22×C4#13714yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×Q812yes—
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4C2×#4111yescharacteristic
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+412yes—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8#12314yes—
276C2×C8C4⋊Q814yes—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—48no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—42no—
306C42—42no—
307C23C2×#12111yescharacteristic
308C23#53211yescharacteristic
309C23#53211yescharacteristic
310C23#54612yes—
311C23D4×Q812yes—
312C23C2×#13712yes—
313C23#57512yes—
314C23#57612yes—
315C23#57712yes—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4C2×#12311yescharacteristic
369C2×C4#52511yescharacteristic
370C2×C4#52511yescharacteristic
371C2×C4#52911yescharacteristic
372C2×C4#52911yescharacteristic
373C2×C4C2×C4⋊Q811yescharacteristic
374C2×C4#53312yes—
375C2×C4#53312yes—
376C2×C4#53511yescharacteristic
377C2×C4#53611yescharacteristic
378C2×C4#53912yes—
379C2×C4#53912yes—
380C2×C4#55212yes—
381C2×C4#55212yes—
382C2×C4#55312yes—
383C2×C4#55412yes—
384C2×C4#55412yes—
385C2×C4#55712yes—
386C8—24no—
387C8—24no—
388C8—42no—
389C8—42no—
390C22#253511yescharacteristic
391C22#254611yescharacteristic
392C22#255612yes—
393C22#255612yes—
394C22#285411yescharacteristic
395C22#305612yes—
396C22#306212yes—
397C22#317212yes—
398C22#317311yescharacteristic
399C22#317611yescharacteristic
400C22#317912yes—
401C22#318311yescharacteristic
402C22#318511yescharacteristic
403C22#330912yes—
404C22#331112yes—
405C22#331412yes—
406C22#332012yes—
407C22#338512yes—
408C22#338712yes—
409C22#338812yes—
410C22#338912yes—
411C4—42no—
412C4—42no—
413C4—44no—
414C4—84no—
415C4—84no—
416C4—82no—
417C4—84no—
418C4—162no—
419C4#239911yescharacteristic
420C4#248411yescharacteristic
421C4#250211yescharacteristic
422C4#259012yes—
423C4#259012yes—
424C4#263311yescharacteristic
425C2#1510611yescharacteristic
426C2#2815512yes—
427C2#2868612yes—
428C2#2896012yes—
429C2#2896312yes—
430C2#2896712yes—
431C2#2897212yes—
432C2#3314011yescharacteristic
433C2#3314411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19442
C2×#17712
#151942
#113792
#113772
#113272
#112272
#108812
#108802
#108212