Finite Group Databasebeta

← #116713

Subgroups — #116713

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2C2×#1574C211yesmaximal, characteristic
3C2×#1561C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5#10857C211yesmaximal, characteristic
6C2×#1299C211yesmaximal, characteristic
7#8043C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1990—21no—
16#1956—21no—
17#1872—22no—
18#1803—21no—
19#1803—21no—
20C2×#343—22no—
21#1574—22no—
22#1574—22no—
23C2×#335—21no—
24C2×#334—21no—
25#1561—22no—
26#1561—22no—
27C2×#332—21no—
28C2×#332—21no—
29#1485—22no—
30#1485C414yescomplemented
31C2×Q8⋊C8—22no—
32#1299—22no—
33#1296—22no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533Q812yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423D412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D412yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16C2×C412yes—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#343—44no—
97#340—22no—
98#340—22no—
99#335—24no—
100#334—24no—
101#332—24no—
102#332—24no—
103Q8⋊C8—28no—
104C22×Q8—21no—
105C22×Q8—21no—
106C23×C4—22no—
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—81no—
123C2×M16—81no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C4⋊C412yes—
137C4⋊Q8C4⋊C412yes—
138#125—22no—
139#125—22no—
140#125—41no—
141#125—41no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—22no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—81no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×Q811yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8C22⋊C414yes—
207#103—82no—
208#103—82no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#97—22no—
214#97—41no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221C24#12311yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4#11711yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12711yescharacteristic
268M16—24no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16—44no—
273M16—82no—
274M16—82no—
275M16—82no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—81no—
291C2×C8—81no—
292C2×C8—81no—
293C2×C8—81no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4—81no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—42no—
317C42—42no—
318C42C2×C22⋊C411yescharacteristic
319C42C2×C4⋊C411yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42#12011yescharacteristic
323C42#12011yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42#12311yescharacteristic
332C42#12411yescharacteristic
333C42#12611yescharacteristic
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#38811yescharacteristic
341C23#39411yescharacteristic
342C23#39911yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—42no—
346Q8—42no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
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—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—164no—
387C2×C4#38011yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38611yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#39011yescharacteristic
393C2×C4#39211yescharacteristic
394C2×C4#39311yescharacteristic
395C2×C4#39411yescharacteristic
396C2×C4#39611yescharacteristic
397C2×C4#39611yescharacteristic
398C2×C4#39711yescharacteristic
399C2×C4#45311yescharacteristic
400C2×C4#45411yescharacteristic
401C2×C4#45511yescharacteristic
402C2×C4#45611yescharacteristic
403C2×C4#47111yescharacteristic
404C2×C4#47111yescharacteristic
405C2×C4#47112yes—
406C2×C4#47211yescharacteristic
407C2×C4#47311yescharacteristic
408C2×C4#47312yes—
409C8—44no—
410C8—44no—
411C8—82no—
412C8—81no—
413C8—82no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#143911yescharacteristic
426C22#203711yescharacteristic
427C22#203811yescharacteristic
428C22#204611yescharacteristic
429C22#204711yescharacteristic
430C22#205111yescharacteristic
431C22#205211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—82no—
439C4—82no—
440C4—162no—
441C4#181511yescharacteristic
442C4#181611yescharacteristic
443C4#185911yescharacteristic
444C4#186011yescharacteristic
445C4#190812yes—
446C4#200611yescharacteristic
447C4#200711yescharacteristic
448C4#205111yescharacteristic
449C4#205211yescharacteristic
450C4#206412yes—
451C2—41no—
452C2—41no—
453C2#1049211yescharacteristic
454C2#1049311yescharacteristic
455C2#1049411yescharacteristic
456C2#1049511yescharacteristic
457C2#1106211yescharacteristic
458C2#1110511yescharacteristic
459C2#1111311yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
30. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218602
C2×#15742
C2×#15612
C2×#14852
#108572
C2×#12992
#80432