Finite Group Databasebeta

← #127196

Subgroups — #127196

Order 512.

681 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20694C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7216C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6#2920—21no—
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C22×#97C411yescharacteristic
10C22×C4×C8C411yescharacteristic
11#1822—21no—
12#1822C2212yes—
13#1821—21no—
14#1821C412yes—
15#1750—21no—
16#1750—21no—
17#1750C412yes—
18#1750C2212yes—
19#1742—21no—
20#1742C412yes—
21C2×#336—21no—
22#1464—21no—
23#1464C2212yes—
24#1463—21no—
25#1462C412yes—
26#1458—21no—
27#1458C412yes—
28C24×C4D411yescharacteristic
29C22×M16—22no—
30C23×C8—21no—
31C23×C8—21no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117—41no—
35C2×#117—41no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117C2×C411yescharacteristic
39C2×#117D411yescharacteristic
40C2×#117D411yescharacteristic
41C2×#117D411yescharacteristic
42C2×#117Q811yescharacteristic
43C2×#117C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4Q811yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#407—22no—
56#407—22no—
57#407—22no—
58#407—41no—
59#407—41no—
60C2×C22⋊C8—28no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C4×M16—41no—
66C2×#97—21no—
67C2×#97—41no—
68C2×#97—41no—
69C2×#97C2×C412yes—
70C2×C4×C8—21no—
71C2×C4×C8—41no—
72C2×C4×C8—41no—
73C2×C4×C8C2×C412yes—
74#343—22no—
75#343—22no—
76#343—22no—
77#343C2×C414yes—
78#336—22no—
79#336—22no—
80#324—22no—
81#324—22no—
82#323—22no—
83#323C2×C414yes—
84C25C4⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—24no—
90C23×C4—42no—
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—44no—
105C2×M16—44no—
106C22×C8—24no—
107C22×C8—24no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—42no—
117#117—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C4211yescharacteristic
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C4⋊C411yescharacteristic
142C2×C4⋊C4C4⋊C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C2×D411yescharacteristic
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4C4211yescharacteristic
150C2×C22⋊C4C22⋊C411yescharacteristic
151C2×C22⋊C4C4⋊C411yescharacteristic
152C2×C22⋊C4C4⋊C411yescharacteristic
153C2×C22⋊C4C4⋊C411yescharacteristic
154C2×C22⋊C4C4⋊C411yescharacteristic
155C2×C22⋊C4C22×C411yescharacteristic
156C2×C22⋊C4C2×Q811yescharacteristic
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42C4211yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C4⋊C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C4⋊C8—42no—
174C4⋊C8—42no—
175C22⋊C8—28no—
176C22⋊C8—28no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—42no—
183#97—42no—
184#97—42no—
185#97—42no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8—42no—
189C24—22no—
190C24—21no—
191C24—21no—
192C24—41no—
193C24—42no—
194C24—42no—
195C24#9511yescharacteristic
196C24#9511yescharacteristic
197C24C2×C4⋊C411yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9911yescharacteristic
242C22×C4#9911yescharacteristic
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#10414yes—
246C22×C4C2×C4211yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252M16—44no—
253M16—44no—
254M16—84no—
255M16—84no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—84no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C23—22no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—48no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23#33912yes—
323C23#34211yescharacteristic
324C23#34211yescharacteristic
325C23C2×#9511yescharacteristic
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4H(3,ℤ/4)12yes—
372C2×C4#33912yes—
373C2×C4#34211yescharacteristic
374C2×C4#34211yescharacteristic
375C2×C4C2×#9911yescharacteristic
376C2×C4C2×#9911yescharacteristic
377C2×C4C2×#10412yes—
378C8—44no—
379C8—44no—
380C8—84no—
381C8—84no—
382C22—24no—
383C22—24no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22#139212yes—
403C22#139512yes—
404C22C2×H(3,ℤ/4)11yescharacteristic
405C22C2×#33911yescharacteristic
406C22C2×#34211yescharacteristic
407C4—21no—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4#128412yes—
420C4#128412yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—42no—
426C2—81no—
427C2C2×#128411yescharacteristic
428C2C2×#139211yescharacteristic
429C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7216C2semidirect

Maximal subgroups

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

subgroupindex
#206942
C2×#17502
#173572
#72162