Finite Group Databasebeta

← #119181

Subgroups — #119181

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21885C211yesmaximal, characteristic
2C2×#1552C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#10833C211yesmaximal, characteristic
5#8044C211yesmaximal, characteristic
6#8022C211yesmaximal, characteristic
7C2×#1290C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1987—21no—
16#1981—21no—
17#1892—22no—
18#1803—21no—
19#1803—21no—
20C2×#336—22no—
21#1552—22no—
22#1552C414yescomplemented
23C2×#332—21no—
24C2×#332—21no—
25#1506—22no—
26#1506—22no—
27C2×#328—22no—
28#1296—21no—
29#1296—21no—
30#1294—22no—
31#1290—22no—
32C23×C8—21no—
33C23×C8D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533D412yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432D412yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2×C412yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×#97—21no—
80C2×#97—21no—
81#384D411yescharacteristic
82#384D411yescharacteristic
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#376—21no—
89#376—21no—
90#376—21no—
91#376—21no—
92C2×#95—21no—
93C2×#95D412yes—
94#340—22no—
95#340—22no—
96#336—44no—
97#332—24no—
98#332—24no—
99#328—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C22⋊C412yes—
125C4⋊Q8C22⋊C412yes—
126#125—22no—
127#125—22no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150#117—82no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
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—22no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197#103—82no—
198#103—82no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203#97—22no—
204#97—22no—
205#97—41no—
206#97—41no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213C24#12111yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—81no—
223C2×Q8—81no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4#11711yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—82no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—82no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—82no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—82no—
307C42C2×C22⋊C411yescharacteristic
308C42#11711yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12111yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42#12411yescharacteristic
320C42#12611yescharacteristic
321C42#12611yescharacteristic
322C42#12611yescharacteristic
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#38811yescharacteristic
328C23#39411yescharacteristic
329C23#39411yescharacteristic
330C23#47112yes—
331C23#47312yes—
332Q8—42no—
333Q8—42no—
334Q8—82no—
335Q8—82no—
336Q8—81no—
337Q8—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#38111yescharacteristic
384C2×C4#38611yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#39411yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39711yescharacteristic
392C2×C4#39711yescharacteristic
393C2×C4#39711yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#45311yescharacteristic
396C2×C4#45411yescharacteristic
397C2×C4#45511yescharacteristic
398C2×C4#45611yescharacteristic
399C8—82no—
400C8—82no—
401C8—82no—
402C8—82no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22#144011yescharacteristic
412C22#193012yes—
413C22#203911yescharacteristic
414C22#204011yescharacteristic
415C22#204611yescharacteristic
416C22#204711yescharacteristic
417C22#205212yes—
418C22#205311yescharacteristic
419C22#205311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—84no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4#188211yescharacteristic
436C4#188311yescharacteristic
437C4#203711yescharacteristic
438C4#203811yescharacteristic
439C2—22no—
440C2—22no—
441C2#1051411yescharacteristic
442C2#1051511yescharacteristic
443C2#1051611yescharacteristic
444C2#1051711yescharacteristic
445C2#1116411yescharacteristic
446C2#1117611yescharacteristic
447C2#1119511yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
22. #1552C4semidirect

Maximal subgroups

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

subgroupindex
#218852
C2×#15522
C2×#15062
#108332
#80442
#80222
C2×#12902