Finite Group Databasebeta

← #120874

Subgroups — #120874

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20801C211yesmaximal, characteristic
2#10855C211yesmaximal, characteristic
3#8026C211yesmaximal, characteristic
4#7981C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—21no—
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1847—22no—
18#1846—21no—
19#1846—21no—
20C2×#344—21no—
21C2×#344—21no—
22#1294—22no—
23#1289—22no—
24#1286—22no—
25#1286—21no—
26#1286—21no—
27#1278C414yescomplemented
28C22×M16—21no—
29C22×M16D411yescharacteristic
30C22×M16D411yescharacteristic
31C2×#117—21no—
32C2×#117—21no—
33C22×C4⋊C4—21no—
34C22×C42C2311yescharacteristic
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431C2×C412yes—
43#428—42no—
44#428—42no—
45#428—42no—
46#428—42no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#106—22no—
50C2×#106—22no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8Q812yes—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16C2×C412yes—
64C4×M16Q812yes—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2×C412yes—
69#384—21no—
70#384—21no—
71#382—21no—
72#382—21no—
73#382—21no—
74#382—21no—
75#382D411yescharacteristic
76#382D411yescharacteristic
77#382D412yes—
78#380—21no—
79#380—21no—
80#380—21no—
81#380D412yes—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C2×#95—21no—
87C2×#95—21no—
88#344—42no—
89#344—42no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16C4⋊C412yes—
107C2×M16C4⋊C412yes—
108C2×M16D812yes—
109C2×M16SD1612yes—
110C2×M16SD1612yes—
111C2×M16Q1612yes—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117#117—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—41no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×Q811yescharacteristic
146C2×C42C2×Q811yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154#107—44no—
155#107—44no—
156#106—44no—
157#106—44no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—41no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170#97—22no—
171#97—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—41no—
178#95—22no—
179#95—21no—
180#95—21no—
181#95—22no—
182#95—21no—
183#95—21no—
184#95—22no—
185#95—41no—
186#95—41no—
187#95—41no—
188C24#12111yescharacteristic
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×Q811yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22⋊Q811yescharacteristic
206C22×C4#12311yescharacteristic
207C22×C4#12311yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4#12311yescharacteristic
210C22×C4#12311yescharacteristic
211C22×C4#12411yescharacteristic
212C22×C4#12411yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12611yescharacteristic
215C22×C4#12611yescharacteristic
216C22×C4C4⋊Q811yescharacteristic
217C22×C4C2×D811yescharacteristic
218C22×C4C2×SD1611yescharacteristic
219C22×C4C2×SD1611yescharacteristic
220C22×C4C2×Q1611yescharacteristic
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4#13711yescharacteristic
224C22×C4#13711yescharacteristic
225M16—22no—
226M16—22no—
227M16—22no—
228M16—22no—
229M16—22no—
230M16—22no—
231M16—81no—
232M16—81no—
233M16—81no—
234M16—81no—
235M16#10614yes—
236M16#10714yes—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C42—81no—
269C42—81no—
270C42—81no—
271C42—81no—
272C42#11711yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42#12411yescharacteristic
281C42#12411yescharacteristic
282C42#12511yescharacteristic
283C42#12511yescharacteristic
284C42#12511yescharacteristic
285C42#12511yescharacteristic
286C42#12611yescharacteristic
287C42#12611yescharacteristic
288C23—41no—
289C23—41no—
290C23#38911yescharacteristic
291C23#39411yescharacteristic
292C23#39711yescharacteristic
293C23#45911yescharacteristic
294C23#46011yescharacteristic
295C23#46111yescharacteristic
296C23#46211yescharacteristic
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—162no—
322C2×C4—162no—
323C2×C4#38211yescharacteristic
324C2×C4#38511yescharacteristic
325C2×C4#38711yescharacteristic
326C2×C4#38711yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#39311yescharacteristic
330C2×C4#39811yescharacteristic
331C2×C4#39811yescharacteristic
332C2×C4#40011yescharacteristic
333C2×C4#40011yescharacteristic
334C2×C4#40011yescharacteristic
335C2×C4C2×#10611yescharacteristic
336C2×C4C2×#10711yescharacteristic
337C2×C4#42811yescharacteristic
338C2×C4#42811yescharacteristic
339C2×C4#44711yescharacteristic
340C2×C4#44811yescharacteristic
341C2×C4#45011yescharacteristic
342C2×C4#45111yescharacteristic
343C2×C4#47411yescharacteristic
344C2×C4#47511yescharacteristic
345C2×C4#47611yescharacteristic
346C2×C4#47711yescharacteristic
347C2×C4#48011yescharacteristic
348C2×C4#48111yescharacteristic
349C2×C4#48311yescharacteristic
350C2×C4#48411yescharacteristic
351C2×C4#48611yescharacteristic
352C2×C4#48611yescharacteristic
353C2×C4#48711yescharacteristic
354C2×C4#48711yescharacteristic
355C2×C4#48911yescharacteristic
356C2×C4#48911yescharacteristic
357C2×C4#49011yescharacteristic
358C2×C4#49111yescharacteristic
359C8—24no—
360C8—24no—
361C8—82no—
362C8—82no—
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22#144211yescharacteristic
375C22#194711yescharacteristic
376C22#194811yescharacteristic
377C22#204111yescharacteristic
378C22#204311yescharacteristic
379C22#207111yescharacteristic
380C22#207211yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—161no—
390C4—161no—
391C4—162no—
392C4—162no—
393C4#183411yescharacteristic
394C4#183611yescharacteristic
395C4#185211yescharacteristic
396C4#185311yescharacteristic
397C4#202311yescharacteristic
398C4#202411yescharacteristic
399C4#209911yescharacteristic
400C4#210011yescharacteristic
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2#1126511yescharacteristic
408C2#1126711yescharacteristic
409C2#1127711yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1278C4semidirect

Maximal subgroups

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

subgroupindex
#208012
#108552
#80262
#79812
#79572
C2×#12862
#78822