Finite Group Databasebeta

← #103374

Subgroups — #103374

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2057C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3#17070C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5C2×#1271C211yesmaximal, characteristic
6C2×#1271C211yesmaximal, characteristic
7#7025C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×#107—21no—
10C22×#103—21no—
11C22×#103—21no—
12C2×#398—21no—
13C2×#398—21no—
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16#2057—22no—
17#2057—22no—
18#2043—22no—
19#2043—22no—
20C2×#340—21no—
21C2×#340—22no—
22C2×#336C2211yescharacteristic
23C2×#336C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1441—21no—
26#1441—21no—
27#1441—22no—
28C4⋊#95C2211yescharacteristic
29#1271—22no—
30#1271—22no—
31C23×Q8—21no—
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8D412yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C42C2311yescharacteristic
51C2×#107—24no—
52C2×#107—44no—
53C2×#103—24no—
54C2×#103—24no—
55C2×#103—42no—
56C2×#103—44no—
57C2×#103—42no—
58#398—42no—
59#398—42no—
60#398—42no—
61#398—42no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77#340—24no—
78#340—44no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336D412yes—
84#336D412yes—
85C22×Q8—24no—
86C22×Q8—22no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C22×C8—24no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—42no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—42no—
112C22×C8—41no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—42no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4D812yes—
166C2×C4⋊C4SD1612yes—
167C2×C4⋊C4Q1614yes—
168C2×C42—42no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173#107—48no—
174#103—48no—
175#103—48no—
176#95—44no—
177#95—44no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—82no—
187C24#12011yescharacteristic
188C2×Q8—24no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—81no—
205C2×Q8—81no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12711yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×SD1611yescharacteristic
257C22×C4C2×Q1612yes—
258C22×C4#13511yescharacteristic
259C22×C4#13511yescharacteristic
260C22×C4AGL(1,ℤ/8)11yescharacteristic
261C22×C4AGL(1,ℤ/8)11yescharacteristic
262C22×C4AGL(1,ℤ/8)11yescharacteristic
263C22×C4#13711yescharacteristic
264C22×C4#13712yes—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—81no—
279C2×C8—81no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—88no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C23#39211yescharacteristic
314C23#44711yescharacteristic
315C23#44911yescharacteristic
316C23#45011yescharacteristic
317C23#45112yes—
318C23#45211yescharacteristic
319C23#45412yes—
320C23#45511yescharacteristic
321C23#45612yes—
322C23#45611yescharacteristic
323C23B(4,𝔽2)12yes—
324Q8—44no—
325Q8—44no—
326Q8—82no—
327Q8—82no—
328Q8—84no—
329Q8—84no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
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—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4#44711yescharacteristic
391C2×C4#44811yescharacteristic
392C2×C4#44911yescharacteristic
393C2×C4#44911yescharacteristic
394C2×C4#46212yes—
395C2×C4#46511yescharacteristic
396C2×C4#46611yescharacteristic
397C2×C4#46811yescharacteristic
398C2×C4#46911yescharacteristic
399C2×C4#49411yescharacteristic
400C2×C4#49711yescharacteristic
401C8—44no—
402C8—82no—
403C8—82no—
404C22#159712yes—
405C22#159812yes—
406C22#160212yes—
407C22#160312yes—
408C22#160412yes—
409C22#160512yes—
410C22#163612yes—
411C22#163712yes—
412C22#163911yescharacteristic
413C22#164611yescharacteristic
414C22#164711yescharacteristic
415C22#164812yes—
416C22#164912yes—
417C22#165411yescharacteristic
418C22#166312yes—
419C22#166412yes—
420C22#199911yescharacteristic
421C22#200011yescharacteristic
422C22#200311yescharacteristic
423C22#200511yescharacteristic
424C22#200711yescharacteristic
425C22#201611yescharacteristic
426C22#202111yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—21no—
438C4—84no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—161no—
444C2#776312yes—
445C2#776412yes—
446C2#777112yes—
447C2#777212yes—
448C2#991511yescharacteristic
449C2#991711yescharacteristic
450C2#996711yescharacteristic
451C2#1005911yescharacteristic
452C2#1006711yescharacteristic
453C2#1008411yescharacteristic
454C2#1023111yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC22×C42C23
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20572
C2×#20432
#170702
#113842
C2×#12712
#70252