Finite Group Databasebeta

← #127229

Subgroups — #127229

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20717C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#7216C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6C2×#432—21no—
7C2×#432C411yescharacteristic
8C2×#431C411yescharacteristic
9C2×#407—21no—
10#1822—21no—
11#1822C2212yes—
12#1821—21no—
13#1821C412yes—
14#1817—21no—
15#1798—21no—
16#1798—21no—
17#1779—21no—
18#1750C412yes—
19#1750C2212yes—
20#1742—21no—
21#1742C412yes—
22#1464—21no—
23#1464—21no—
24#1464C412yes—
25#1464C2212yes—
26C2×#324—21no—
27C2×#323C412yes—
28C24×C4D411yescharacteristic
29C22×M16—22no—
30C2×#117—41no—
31C2×#117—41no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C2×#117D411yescharacteristic
38C2×#117D411yescharacteristic
39C2×#117D411yescharacteristic
40C2×#117Q811yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—22no—
50#432—41no—
51#432—41no—
52#432—41no—
53#432—41no—
54#431—41no—
55#431—41no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—41no—
59#407—22no—
60#407—22no—
61#407—41no—
62#407—41no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70C2×#97—41no—
71C2×#97C2×C412yes—
72C2×C4×C8C2×C412yes—
73#343—22no—
74#343—22no—
75#343—22no—
76#343C2×C414yes—
77#336—22no—
78#336—22no—
79#324—22no—
80#324—22no—
81#323—22no—
82#323C2×C414yes—
83C25C4⋊C411yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4C4211yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—42no—
102C2×M16—44no—
103C2×M16—44no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—42no—
107C22×C8—42no—
108#117—41no—
109#117—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C22×C411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4C4211yescharacteristic
141C2×C22⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C22⋊C4C4⋊C411yescharacteristic
146C2×C22⋊C4C22×C411yescharacteristic
147C2×C22⋊C4C2×Q811yescharacteristic
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42C4211yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168C22⋊C8—24no—
169C22⋊C8—44no—
170C22⋊C8—44no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#97—42no—
181#97—42no—
182#97—42no—
183C4×C8—42no—
184C24—21no—
185C24—21no—
186C24—22no—
187C24—41no—
188C24—42no—
189C24—81no—
190C24#9511yescharacteristic
191C24#9511yescharacteristic
192C24C2×C4⋊C411yescharacteristic
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
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—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4#9511yescharacteristic
232C22×C4#9511yescharacteristic
233C22×C4#9511yescharacteristic
234C22×C4#9511yescharacteristic
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9911yescharacteristic
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4#9911yescharacteristic
241C22×C4C2×C4211yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247M16—44no—
248M16—44no—
249M16—84no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—84no—
259C2×C8—84no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—81no—
310C23—81no—
311C23—84no—
312C23—84no—
313C23H(3,ℤ/4)12yes—
314C23#34211yescharacteristic
315C23#34211yescharacteristic
316C23C2×#9511yescharacteristic
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4#34211yescharacteristic
365C2×C4#34211yescharacteristic
366C2×C4C2×#9911yescharacteristic
367C2×C4C2×#9911yescharacteristic
368C2×C4#42112yes—
369C8—82no—
370C8—82no—
371C8—84no—
372C8—84no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—84no—
392C22—81no—
393C22—84no—
394C22#139512yes—
395C22C2×H(3,ℤ/4)11yescharacteristic
396C22#173311yescharacteristic
397C22C2×#34211yescharacteristic
398C4—22no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—41no—
416C2—42no—
417C2—81no—
418C2#794311yescharacteristic
419C2#1257911yescharacteristic
420C2C2×#139511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
#207172
#206942
C2×#14642
#72162