Finite Group Databasebeta

← #119231

Subgroups — #119231

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2C2×#1549C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic
4#10840C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1990C414yescomplemented
16#1987—22no—
17#1824—22no—
18C2×#339—21no—
19C2×#339—21no—
20#1549—22no—
21#1549—22no—
22C2×#332—22no—
23#1446—22no—
24#1445—22no—
25C4⋊#95—21no—
26C4⋊#95—21no—
27#1294—22no—
28#1286—22no—
29#1274—22no—
30C23×Q8—21no—
31C22×M16—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—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⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×#117—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D412yes—
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42C2311yescharacteristic
51#431—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431D412yes—
59#423—21no—
60#423—21no—
61#423—21no—
62#423D412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2×C412yes—
71#384C2×C412yes—
72#382—21no—
73#382—21no—
74#382—21no—
75#382—21no—
76#382—21no—
77#380—21no—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81#376—21no—
82#376—21no—
83#376—21no—
84#376—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95D412yes—
91#340—28no—
92#339—42no—
93#339—42no—
94#332—28no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—84no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C22⋊C414yes—
159C2×C22⋊C4—41no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—41no—
171C2×C42SD1612yes—
172C2×C42Q1612yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
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—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—41no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—84no—
212#95—84no—
213C24#12011yescharacteristic
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—84no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—84no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12411yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4C2×SD1611yescharacteristic
266C22×C4C2×Q1611yescharacteristic
267C22×C4#13511yescharacteristic
268C22×C4#13511yescharacteristic
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4#13711yescharacteristic
271C22×C4#13711yescharacteristic
272C22×C4#13711yescharacteristic
273M16—42no—
274M16—42no—
275M16—81no—
276M16—81no—
277C2×C8—44no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—81no—
283C2×C8—81no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—82no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—81no—
311C42—81no—
312C42#11711yescharacteristic
313C42#11711yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42#12011yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12311yescharacteristic
322C42#12411yescharacteristic
323C42#12411yescharacteristic
324C42#12611yescharacteristic
325C42#12611yescharacteristic
326C42#12611yescharacteristic
327C42#12611yescharacteristic
328C23—22no—
329C23—22no—
330C23#38611yescharacteristic
331C23#39411yescharacteristic
332C23#39411yescharacteristic
333C23#44811yescharacteristic
334C23#44911yescharacteristic
335C23#45111yescharacteristic
336C23#45211yescharacteristic
337Q8—84no—
338Q8—84no—
339Q8—84no—
340Q8—84no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4#38111yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39411yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39711yescharacteristic
386C2×C4#39711yescharacteristic
387C2×C4#39711yescharacteristic
388C2×C4C4×SD1611yescharacteristic
389C2×C4C4×Q1611yescharacteristic
390C2×C4#44011yescharacteristic
391C2×C4#44111yescharacteristic
392C2×C4#46011yescharacteristic
393C2×C4#46111yescharacteristic
394C2×C4#46211yescharacteristic
395C2×C4#46211yescharacteristic
396C2×C4#46311yescharacteristic
397C2×C4#46411yescharacteristic
398C2×C4#46411yescharacteristic
399C2×C4#46411yescharacteristic
400C2×C4#46511yescharacteristic
401C2×C4#46711yescharacteristic
402C2×C4#46811yescharacteristic
403C2×C4#47011yescharacteristic
404C2×C4#48311yescharacteristic
405C2×C4#48411yescharacteristic
406C2×C4#48511yescharacteristic
407C2×C4#48511yescharacteristic
408C2×C4#48711yescharacteristic
409C2×C4#48811yescharacteristic
410C2×C4#48911yescharacteristic
411C2×C4#49111yescharacteristic
412C8—82no—
413C8—82no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22#144011yescharacteristic
424C22#187811yescharacteristic
425C22#188011yescharacteristic
426C22#190412yes—
427C22#191212yes—
428C22#203511yescharacteristic
429C22#203611yescharacteristic
430C22#204211yescharacteristic
431C22#204311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4—162no—
444C4—162no—
445C4#192711yescharacteristic
446C4#192811yescharacteristic
447C4#193311yescharacteristic
448C4#193511yescharacteristic
449C4#205511yescharacteristic
450C4#205611yescharacteristic
451C4#207211yescharacteristic
452C4#207411yescharacteristic
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2#929812yes—
458C2#930212yes—
459C2#1115611yescharacteristic
460C2#1118311yescharacteristic
461C2#1119011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1990C4semidirect

Maximal subgroups

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

subgroupindex
#218342
C2×#15492
#142822
#108402
#80262
C2×#12862
C2×#12742