Finite Group Databasebeta

← #121432

Subgroups — #121432

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1950C211yesmaximal, characteristic
2#21146C211yesmaximal, characteristic
3C2×#1566C211yesmaximal, characteristic
4C2×#1301C211yesmaximal, characteristic
5#7901C211yesmaximal, characteristic
6C2×#1279C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C22×#108—21no—
10C2×#423—21no—
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1950—22no—
18#1950—22no—
19#1940—22no—
20#1939—22no—
21C2×#341—21no—
22#1728—21no—
23#1566—22no—
24#1566—22no—
25C2×#334—22no—
26#1301—22no—
27#1279—22no—
28#1279C411yescharacteristic
29#1279C411yescharacteristic
30#1279C411yescharacteristic
31#1279C411yescharacteristic
32#1278—22no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C23×C8—21no—
38C23×C8—21no—
39C22×C4⋊C4Q812yes—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#431—21no—
43#431—21no—
44#431—21no—
45#431Q812yes—
46C2×#108—24no—
47C2×#108—42no—
48C2×#108—42no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—42no—
57#423—42no—
58#423D412yes—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2×C411yescharacteristic
68C2×C4⋊C8C2×C411yescharacteristic
69C2×C4⋊C8D412yes—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8Q812yes—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8D412yes—
86C2×C4×C8Q812yes—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C411yescharacteristic
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4D412yes—
92C2×#95—21no—
93#341—24no—
94#336—24no—
95#334—28no—
96C23×C4—22no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16SD1612yes—
123C2×M16SD1612yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—44no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C4⋊C412yes—
146C2×C4⋊C4C4⋊C412yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42C2×Q811yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166#108—48no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—48no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C4⋊C412yes—
183C4⋊C8C4⋊C412yes—
184C22⋊C8—22no—
185C22⋊C8—41no—
186#97—41no—
187#97—41no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8C4⋊C412yes—
197C4×C8C4⋊C412yes—
198#95—21no—
199#95—21no—
200#95—22no—
201C24C22⋊Q811yescharacteristic
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12511yescharacteristic
224C22×C4#12511yescharacteristic
225C22×C4C4⋊Q811yescharacteristic
226C22×C4C4⋊Q811yescharacteristic
227C22×C4C4⋊Q811yescharacteristic
228C22×C4C4⋊Q811yescharacteristic
229C22×C4C2×SD1611yescharacteristic
230C22×C4C2×SD1611yescharacteristic
231C22×C4#13511yescharacteristic
232C22×C4#13511yescharacteristic
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4AGL(1,ℤ/8)11yescharacteristic
235C22×C4#13711yescharacteristic
236C22×C4#13711yescharacteristic
237M16—22no—
238M16—22no—
239M16—22no—
240M16—22no—
241M16—41no—
242M16—41no—
243M16—41no—
244M16—41no—
245M16—42no—
246M16—42no—
247M16—81no—
248M16—81no—
249M16—81no—
250M16—81no—
251M16—81no—
252M16—81no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—82no—
281C2×C8—81no—
282C2×C8—81no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4#10814yescomplemented
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—42no—
293C42—42no—
294C42C2×C4⋊C411yescharacteristic
295C42C2×C4⋊C411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×Q811yescharacteristic
298C42#12011yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42#12311yescharacteristic
306C42#12511yescharacteristic
307C42#12711yescharacteristic
308C42C4⋊Q811yescharacteristic
309C42C4⋊Q811yescharacteristic
310C23—22no—
311C23—22no—
312C23#38511yescharacteristic
313C23#39511yescharacteristic
314C23#39811yescharacteristic
315C23#47511yescharacteristic
316C23#47611yescharacteristic
317C23#47811yescharacteristic
318C23#47911yescharacteristic
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4#38411yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38911yescharacteristic
344C2×C4#38911yescharacteristic
345C2×C4#38911yescharacteristic
346C2×C4#39211yescharacteristic
347C2×C4#39511yescharacteristic
348C2×C4#39511yescharacteristic
349C2×C4#39811yescharacteristic
350C2×C4#39811yescharacteristic
351C2×C4#39911yescharacteristic
352C2×C4C2×#10812yes—
353C2×C4#43012yes—
354C2×C4C4×SD1611yescharacteristic
355C2×C4C4×SD1611yescharacteristic
356C2×C4#44111yescharacteristic
357C2×C4#44211yescharacteristic
358C2×C4#44811yescharacteristic
359C2×C4#44811yescharacteristic
360C2×C4#44911yescharacteristic
361C2×C4#44911yescharacteristic
362C2×C4#45011yescharacteristic
363C2×C4#45011yescharacteristic
364C2×C4#45211yescharacteristic
365C2×C4#45211yescharacteristic
366C2×C4#46511yescharacteristic
367C2×C4#46511yescharacteristic
368C2×C4#46811yescharacteristic
369C2×C4#46811yescharacteristic
370C2×C4#47511yescharacteristic
371C2×C4#47611yescharacteristic
372C2×C4#47811yescharacteristic
373C2×C4#47911yescharacteristic
374C2×C4#49811yescharacteristic
375C2×C4#49911yescharacteristic
376C2×C4#50111yescharacteristic
377C2×C4#50111yescharacteristic
378C8—24no—
379C8—42no—
380C8—42no—
381C8—42no—
382C8—42no—
383C8—42no—
384C8—82no—
385C8—82no—
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22#144411yescharacteristic
396C22#185512yes—
397C22#186112yes—
398C22#186411yescharacteristic
399C22#186511yescharacteristic
400C22#205811yescharacteristic
401C22#205911yescharacteristic
402C22#208811yescharacteristic
403C22#208811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—82no—
412C4#184811yescharacteristic
413C4#184911yescharacteristic
414C4#186411yescharacteristic
415C4#186511yescharacteristic
416C4#194111yescharacteristic
417C4#194311yescharacteristic
418C4#195012yes—
419C4#200011yescharacteristic
420C4#200011yescharacteristic
421C4#205811yescharacteristic
422C4#205911yescharacteristic
423C2—21no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2#890812yes—
428C2#890912yes—
429C2#1132611yescharacteristic
430C2#1132711yescharacteristic
431C2#1134111yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
287. C4⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
C2×#19502
#211462
C2×#15662
C2×#13012
#79012
C2×#12792
C2×#12782