Finite Group Databasebeta

← #116306

Subgroups — #116306

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#21145C211yesmaximal, characteristic
3C2×#1542C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7932C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#424—21no—
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2285C2211yescharacteristic
16#1998—22no—
17#1997—21no—
18#1997—21no—
19#1940—22no—
20#1939—22no—
21#1742—21no—
22#1742—21no—
23#1729—21no—
24#1542—22no—
25#1542—22no—
26C2×#333—21no—
27C2×#332—21no—
28#1520C414yes—
29#1518—22no—
30#1503—22no—
31#1301—22no—
32#1286—22no—
33#1283—22no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C23×C8—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8—21no—
52C2×C4×Q8Q812yes—
53C2×#117—21no—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431D412yes—
64#424—42no—
65#424—42no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8C2×C412yes—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#384—21no—
92#382D412yes—
93#380—21no—
94#380C2×C412yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D412yes—
97#343—22no—
98#343—22no—
99#343—22no—
100#343—22no—
101#336—24no—
102#333—24no—
103#332—24no—
104#328—28no—
105C22×Q8—21no—
106C22×Q8C4⋊C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16SD1612yes—
127C2×M16Q1612yes—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148#125—41no—
149#125—41no—
150#125—41no—
151#125—41no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174#117—41no—
175#117—41no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C4⋊C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—48no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218#97—41no—
219#97—41no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8—41no—
225C4×C8—41no—
226#95—22no—
227#95C22⋊C414yes—
228C24C4×D411yescharacteristic
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—82no—
234C2×Q8—82no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12611yescharacteristic
258C22×C4C2×SD1611yescharacteristic
259C22×C4C2×Q1611yescharacteristic
260C22×C4#13511yescharacteristic
261C22×C4#13511yescharacteristic
262C22×C4AGL(1,ℤ/8)11yescharacteristic
263C22×C4#13711yescharacteristic
264C22×C4#13711yescharacteristic
265C22×C4#13711yescharacteristic
266M16—22no—
267M16—22no—
268M16—22no—
269M16—22no—
270M16—41no—
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—42no—
276M16—81no—
277M16—81no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—82no—
291C2×C8—81no—
292C2×C8—81no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—82no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42#12011yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12411yescharacteristic
335C42#12411yescharacteristic
336C42#12611yescharacteristic
337C42#12711yescharacteristic
338C23—41no—
339C23—41no—
340C23#38511yescharacteristic
341C23#38611yescharacteristic
342C23#38811yescharacteristic
343C23C4×SD1611yescharacteristic
344C23C4×Q1611yescharacteristic
345C23#44011yescharacteristic
346C23#44111yescharacteristic
347Q8—82no—
348Q8—82no—
349Q8—82no—
350Q8—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#38011yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#39011yescharacteristic
374C2×C4#39211yescharacteristic
375C2×C4#39311yescharacteristic
376C2×C4#39411yescharacteristic
377C2×C4#39411yescharacteristic
378C2×C4#39611yescharacteristic
379C2×C4#39611yescharacteristic
380C2×C4#39711yescharacteristic
381C2×C4#39911yescharacteristic
382C2×C4#44811yescharacteristic
383C2×C4#44911yescharacteristic
384C2×C4#45011yescharacteristic
385C2×C4#45111yescharacteristic
386C2×C4#45111yescharacteristic
387C2×C4#45211yescharacteristic
388C2×C4#45211yescharacteristic
389C2×C4#45211yescharacteristic
390C2×C4#46111yescharacteristic
391C2×C4#46211yescharacteristic
392C2×C4#46311yescharacteristic
393C2×C4#46411yescharacteristic
394C2×C4#46511yescharacteristic
395C2×C4#46711yescharacteristic
396C2×C4#46811yescharacteristic
397C2×C4#47011yescharacteristic
398C2×C4#47511yescharacteristic
399C2×C4#47711yescharacteristic
400C2×C4#47911yescharacteristic
401C2×C4#47911yescharacteristic
402C2×C4#48711yescharacteristic
403C2×C4#48811yescharacteristic
404C2×C4#48911yescharacteristic
405C2×C4#49111yescharacteristic
406C8—42no—
407C8—42no—
408C8—82no—
409C8—82no—
410C8—82no—
411C8—82no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22#143911yescharacteristic
423C22#186311yescharacteristic
424C22#186511yescharacteristic
425C22#187711yescharacteristic
426C22#187811yescharacteristic
427C22#193311yescharacteristic
428C22#193511yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4#200011yescharacteristic
440C4#200111yescharacteristic
441C4#202411yescharacteristic
442C4#202611yescharacteristic
443C4#204211yescharacteristic
444C4#204311yescharacteristic
445C4#206611yescharacteristic
446C4#206811yescharacteristic
447C4#208012yes—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2#1106711yescharacteristic
455C2#1108611yescharacteristic
456C2#1109011yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213372
#211452
C2×#15422
#175752
C2×#13012
#79572
#79322