Finite Group Databasebeta

← #414278

Subgroups — #414278

Order 512.

784 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
1#23670C212yesmaximal
2C2×#1941C211yesmaximal, characteristic
3#20435C211yesmaximal, characteristic
4#11330C211yesmaximal, characteristic
5#11271C211yesmaximal, characteristic
6#11238C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#11204C212yesmaximal
9#10896C212yesmaximal
10#10844C211yesmaximal, characteristic
11#10827C212yesmaximal
12C2×#428C2212yes—
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
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#1941—21no—
27#1941—21no—
28#1941—21no—
29#1941—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—22no—
62C2×#106—21no—
63C2×#106—21no—
64C2×#106—24no—
65C2×#106—21no—
66C2×#106—21no—
67C2×#106—22no—
68C2×#106—42no—
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#389D412yes—
94#389D412yes—
95#385—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C2×#95—22no—
99#343—24no—
100#343—24no—
101#343—24no—
102#343—42no—
103#341—24no—
104#341—22no—
105#341—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4#4112yes—
121C23×C4C2411yescharacteristic
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
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×M16C2×Q812yes—
140C2×M16C2×Q812yes—
141C2×M16#4114yes—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8C2×Q812yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158#117—21no—
159#117—21no—
160#117—22no—
161#117—22no—
162#117C2×Q812yes—
163#117C2×Q812yes—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4#4114yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193#107—44no—
194#106—24no—
195#106—24no—
196#106—44no—
197#106—42no—
198#106—42no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—44no—
202C4⋊C8—44no—
203#97—24no—
204#97—42no—
205C4×C8—24no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—42no—
211#95—22no—
212#95—22no—
213#95—42no—
214#95—42no—
215C24C2×#4111yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×Q811yescharacteristic
238C22×C4C22×Q811yescharacteristic
239C22×C4C2×#4111yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4112yes—
242C22×C4C2×#4112yes—
243C22×C4H(5,𝔽2)12yes—
244C22×C42−1+412yes—
245M16—24no—
246M16—24no—
247M16—24no—
248M16—24no—
249M16—42no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—42no—
254M16—42no—
255M16—42no—
256M16—42no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8C22⋊Q812yes—
282C2×C8C22⋊Q812yes—
283C2×C8C22⋊Q812yes—
284C2×C8C22⋊Q812yes—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4C4⋊Q812yes—
302C4⋊C4C4⋊Q812yes—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C42—22no—
307C42—22no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C23—22no—
317C23—22no—
318C23C2×#12111yescharacteristic
319C23#53311yescharacteristic
320C23#53311yescharacteristic
321C23#55212yes—
322C23#55312yes—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—162no—
346C2×C4—162no—
347C2×C4#47114yes—
348C2×C4C2×C22⋊Q811yescharacteristic
349C2×C4C2×C22⋊Q811yescharacteristic
350C2×C4#52511yescharacteristic
351C2×C4#52912yes—
352C2×C4C2×C4⋊Q811yescharacteristic
353C2×C4#53211yescharacteristic
354C2×C4#53211yescharacteristic
355C2×C4#53312yes—
356C2×C4#53411yescharacteristic
357C2×C4#53611yescharacteristic
358C2×C4#53912yes—
359C2×C4#54412yes—
360C2×C4#54612yes—
361C2×C4#54812yes—
362C2×C4#55212yes—
363C2×C4#55312yes—
364C2×C4#55412yes—
365C2×C4#55712yes—
366C8—22no—
367C8—22no—
368C8—22no—
369C8—22no—
370C8—42no—
371C8—42no—
372C8—44no—
373C8—82no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#253611yescharacteristic
383C22#254311yescharacteristic
384C22#259012yes—
385C22#260612yes—
386C22#285711yescharacteristic
387C22C2×#47112yes—
388C22#306712yes—
389C4—24no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—84no—
394C4—82no—
395C4—162no—
396C4#239611yescharacteristic
397C4#255012yes—
398C4#255812yes—
399C4#256511yescharacteristic
400C4#257911yescharacteristic
401C4#262711yescharacteristic
402C2—22no—
403C2—22no—
404C2#1516611yescharacteristic
405C2#2821812yes—
406C2#2859312yes—
407C2#3322011yescharacteristic
408C2#3322111yescharacteristic
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×#19412
#204352
#113302
#112712
#112382
#112042
#108962
#108442
#108272