Finite Group Databasebeta

← #127402

Subgroups — #127402

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1801C211yesmaximal, characteristic
4#7238C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic
8C2×#431C411yescharacteristic
9C2×#423—21no—
10C22×#101—21no—
11#1985—21no—
12#1985—21no—
13#1984C412yes—
14#1984C2212yes—
15#1821C412yes—
16#1817—21no—
17#1817—21no—
18#1801—21no—
19#1801—21no—
20#1801—21no—
21#1801C412yes—
22#1801C2212yes—
23#1801C2212yes—
24#1784—21no—
25C4×#101—21no—
26C4×#101C412yes—
27C2×#343C412yes—
28C23×Q8D411yescharacteristic
29C22×M16—22no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×Q8Q811yescharacteristic
41C2×C4×Q8C2311yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4Q811yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—41no—
49#432—41no—
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54#431—41no—
55#431—41no—
56#423—22no—
57#423—22no—
58#423—22no—
59#423—41no—
60#423—41no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—41no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8C2×C412yes—
65C2×#101—22no—
66C2×#101—22no—
67C2×#101—22no—
68C2×#101—22no—
69C2×#101—44no—
70#407—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C4×M16—22no—
74C2×#97—21no—
75C2×C4×C8C2×C412yes—
76#343—22no—
77#343—22no—
78#343C2×C414yes—
79#343C2×C414yes—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—42no—
89C22×Q8C22⋊C411yescharacteristic
90C22×Q8C22⋊C411yescharacteristic
91C22×Q8C4⋊C411yescharacteristic
92C22×Q8C2×D411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—44no—
104C2×M16—44no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—42no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C4211yescharacteristic
131C2×C4⋊C4C4211yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C22×C411yescharacteristic
140C2×C4⋊C4C22×C411yescharacteristic
141C2×C4⋊C4C2×Q811yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42C4211yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C4⋊C8—42no—
167C4⋊C8—42no—
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174#101—44no—
175#101—44no—
176#101—42no—
177#101—44no—
178#101—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—42no—
186#97—42no—
187C4×C8—42no—
188C4×C8—42no—
189C24#9511yescharacteristic
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8AGL(1,ℤ/8)12yes—
207C2×Q8#13712yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4#9511yescharacteristic
232C22×C4#9511yescharacteristic
233C22×C4#9511yescharacteristic
234C22×C4#9511yescharacteristic
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4#9911yescharacteristic
241C22×C4#9911yescharacteristic
242C22×C4C2×C4211yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4C2×C4⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249M16—44no—
250M16—44no—
251M16—44no—
252M16—44no—
253M16—84no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—84no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C42—21no—
274C42—21no—
275C42—21no—
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—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C23—22no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23#34211yescharacteristic
302C23#34211yescharacteristic
303C23C2×#9511yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—44no—
307Q8—44no—
308Q8—42no—
309Q8—42no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
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—21no—
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—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4H(3,ℤ/4)12yes—
357C2×C4#34211yescharacteristic
358C2×C4#34211yescharacteristic
359C2×C4C2×#9911yescharacteristic
360C2×C4C2×#9911yescharacteristic
361C2×C4#41712yes—
362C2×C4#41811yescharacteristic
363C2×C4#41911yescharacteristic
364C2×C4#42812yes—
365C8—82no—
366C8—84no—
367C8—84no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—42no—
377C22—42no—
378C22#139212yes—
379C22C2×H(3,ℤ/4)11yescharacteristic
380C22#173611yescharacteristic
381C22C2×#34211yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—21no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4#213312yes—
399C2—22no—
400C2—21no—
401C2—21no—
402C2—41no—
403C2#798311yescharacteristic
404C2C2×#139211yescharacteristic
405C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7238C2semidirect

Maximal subgroups

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

subgroupindex
#218202
#207272
C2×#18012
#72382