Finite Group Databasebeta

← #119176

Subgroups — #119176

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21313C211yesmaximal, characteristic
2C2×#1549C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4#10856C211yesmaximal, characteristic
5#8079C211yesmaximal, characteristic
6#8079C211yesmaximal, characteristic
7#7854C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1988—22no—
16#1987—22no—
17C2×#344—21no—
18C2×#344—21no—
19#1736—21no—
20#1736—21no—
21#1549—22no—
22#1549C414yescomplemented
23#1548—22no—
24#1548—22no—
25C2×#333—22no—
26C2×#332—22no—
27#1301—22no—
28#1301—22no—
29#1274—22no—
30C23×Q8—21no—
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8D412yes—
46C2×#117D411yescharacteristic
47C2×#117D411yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42C2311yescharacteristic
51#431—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431C2×C412yes—
58#431D412yes—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D412yes—
78C2×C4⋊C8D412yes—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87#376—21no—
88C43—21no—
89#344—42no—
90#344—42no—
91#340—24no—
92#340—24no—
93#333—28no—
94#332—28no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—84no—
102C23×C4—41no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
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×M16—41no—
123C2×M16—41no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8C22⋊C414yes—
145#117—21no—
146#117—21no—
147#117—21no—
148#117—21no—
149#117—21no—
150#117—21no—
151#117SD1612yes—
152#117Q1612yes—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209#95—41no—
210#95—41no—
211C24#12311yescharacteristic
212C2×Q8—24no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—84no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—84no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—84no—
230C22×C4#11711yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12611yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4C2×SD1611yescharacteristic
249C22×C4C2×Q1611yescharacteristic
250C22×C4#13511yescharacteristic
251C22×C4#13511yescharacteristic
252C22×C4AGL(1,ℤ/8)11yescharacteristic
253C22×C4#13711yescharacteristic
254C22×C4#13711yescharacteristic
255C22×C4#13711yescharacteristic
256M16—42no—
257M16—42no—
258M16—81no—
259M16—81no—
260M16—81no—
261M16—81no—
262M16—81no—
263M16—81no—
264M16—81no—
265M16—81no—
266M16—81no—
267M16—81no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—82no—
279C2×C8—81no—
280C2×C8—81no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—22no—
298C42—22no—
299C42—41no—
300C42—41no—
301C42—82no—
302C42—82no—
303C42C2×C22⋊C411yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12011yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12111yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12411yescharacteristic
318C42#12611yescharacteristic
319C23—41no—
320C23—41no—
321C23#38811yescharacteristic
322C23#39411yescharacteristic
323C23#39711yescharacteristic
324C23#48311yescharacteristic
325C23#48411yescharacteristic
326C23#48511yescharacteristic
327C23#48511yescharacteristic
328Q8—84no—
329Q8—84no—
330Q8—84no—
331Q8—84no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4#38111yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38611yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#39411yescharacteristic
361C2×C4#39411yescharacteristic
362C2×C4#39411yescharacteristic
363C2×C4#39711yescharacteristic
364C2×C4#39711yescharacteristic
365C2×C4#39711yescharacteristic
366C2×C4C4×SD1611yescharacteristic
367C2×C4C4×Q1611yescharacteristic
368C2×C4#44011yescharacteristic
369C2×C4#44111yescharacteristic
370C2×C4#44811yescharacteristic
371C2×C4#44911yescharacteristic
372C2×C4#45111yescharacteristic
373C2×C4#45211yescharacteristic
374C2×C4#46011yescharacteristic
375C2×C4#46111yescharacteristic
376C2×C4#46211yescharacteristic
377C2×C4#46211yescharacteristic
378C2×C4#46311yescharacteristic
379C2×C4#46411yescharacteristic
380C2×C4#46411yescharacteristic
381C2×C4#46411yescharacteristic
382C2×C4#46511yescharacteristic
383C2×C4#46711yescharacteristic
384C2×C4#46811yescharacteristic
385C2×C4#47011yescharacteristic
386C2×C4#48711yescharacteristic
387C2×C4#48811yescharacteristic
388C2×C4#48911yescharacteristic
389C2×C4#49111yescharacteristic
390C8—82no—
391C8—82no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22#144011yescharacteristic
406C22#192711yescharacteristic
407C22#192811yescharacteristic
408C22#203511yescharacteristic
409C22#203611yescharacteristic
410C22#207211yescharacteristic
411C22#207411yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—82no—
419C4—82no—
420C4—84no—
421C4#187811yescharacteristic
422C4#188011yescharacteristic
423C4#193311yescharacteristic
424C4#193511yescharacteristic
425C4#204211yescharacteristic
426C4#204311yescharacteristic
427C4#205511yescharacteristic
428C4#205611yescharacteristic
429C4#208112yes—
430C4#208212yes—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2#1115811yescharacteristic
438C2#1118111yescharacteristic
439C2#1119011yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
22. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#213132
C2×#15492
C2×#15482
#108562
#80792
#78542