Finite Group Databasebeta

← #414279

Subgroups — #414279

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#20435C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11270C211yesmaximal, characteristic
6#11239C211yesmaximal, characteristic
7#11239C211yesmaximal, characteristic
8#11204C212yesmaximal
9#10896C212yesmaximal
10#10844C211yesmaximal, characteristic
11#10827C212yesmaximal
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#423C2212yes—
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2296C2211yescharacteristic
21C2×#389C2212yes—
22#2088—22no—
23#2088—22no—
24#2087—22no—
25#2086—22no—
26#1942—21no—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1771—21no—
31#1771—21no—
32C8×C4⋊C4—21no—
33C8×C4⋊C4—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2211yescharacteristic
46#1728C2211yescharacteristic
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2311yescharacteristic
59#428—24no—
60#428—42no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—24no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—42no—
68C2×#106—22no—
69#424—22no—
70#424Q814yes—
71#423—24no—
72#423—42no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77#405—22no—
78#405—42no—
79#405Q814yes—
80C2×#97—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—42no—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389D411yescharacteristic
94#389D411yescharacteristic
95#389D411yescharacteristic
96#389D411yescharacteristic
97#385—21no—
98#385—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C2×#95—22no—
102#343—24no—
103#343—24no—
104#343—24no—
105#343—42no—
106#341—24no—
107#341—22no—
108#341—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4C2411yescharacteristic
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16C2×Q812yes—
144C2×M16C2×Q812yes—
145C2×M16#4114yes—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×Q812yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—22no—
166#117C2×Q812yes—
167#117C2×Q812yes—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4#4114yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201#107—24no—
202#107—24no—
203#107—44no—
204#107—42no—
205#107—42no—
206#106—44no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—44no—
210C4⋊C8—44no—
211#97—24no—
212#97—42no—
213C4×C8—24no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—42no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—42no—
224#95—42no—
225C24C2×#4111yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4H(5,𝔽2)12yes—
259C22×C42−1+412yes—
260M16—24no—
261M16—24no—
262M16—24no—
263M16—24no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16—42no—
268M16—42no—
269M16—42no—
270M16—42no—
271M16—42no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8C22⋊Q812yes—
297C2×C8C22⋊Q812yes—
298C2×C8C22⋊Q812yes—
299C2×C8C22⋊Q812yes—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4C4⋊Q812yes—
319C4⋊C4C4⋊Q812yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—41no—
332C42—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23C2×#12111yescharacteristic
338C23#53311yescharacteristic
339C23#53311yescharacteristic
340C23#55212yes—
341C23#55312yes—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—162no—
375C2×C4—162no—
376C2×C4#47212yes—
377C2×C4#47312yes—
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4#52511yescharacteristic
381C2×C4#52912yes—
382C2×C4C2×C4⋊Q811yescharacteristic
383C2×C4#53211yescharacteristic
384C2×C4#53211yescharacteristic
385C2×C4#53312yes—
386C2×C4#53411yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4#53912yes—
389C2×C4#54412yes—
390C2×C4#54612yes—
391C2×C4#54812yes—
392C2×C4#55212yes—
393C2×C4#55312yes—
394C2×C4#55412yes—
395C2×C4#55712yes—
396C8—22no—
397C8—22no—
398C8—22no—
399C8—22no—
400C8—42no—
401C8—42no—
402C8—44no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22#253611yescharacteristic
413C22#254311yescharacteristic
414C22#259012yes—
415C22#260612yes—
416C22#285711yescharacteristic
417C22C2×#47211yescharacteristic
418C22C2×#47311yescharacteristic
419C22#306711yescharacteristic
420C22#306711yescharacteristic
421C4—22no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—84no—
428C4—81no—
429C4—81no—
430C4—162no—
431C4#239611yescharacteristic
432C4#255012yes—
433C4#255812yes—
434C4#256511yescharacteristic
435C4#257911yescharacteristic
436C4#262711yescharacteristic
437C2—22no—
438C2—22no—
439C2#1516611yescharacteristic
440C2#2821711yescharacteristic
441C2#2821911yescharacteristic
442C2#2859111yescharacteristic
443C2#2859211yescharacteristic
444C2#3322111yescharacteristic
445C2#3322211yescharacteristic
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
#236702
C2×#19422
#204352
#113312
#112702
#112392
#112042
#108962
#108442
#108272