Finite Group Databasebeta

← #117726

Subgroups — #117726

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#17787C211yesmaximal, characteristic
3#17486C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#10853C211yesmaximal, characteristic
6#8079C211yesmaximal, characteristic
7C2×#1301C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1998—22no—
16#1997C412yes—
17#1997C412yes—
18#1742—21no—
19#1742—21no—
20C2×#341—21no—
21C2×#341—21no—
22#1565—22no—
23#1565—22no—
24#1564—22no—
25#1493—22no—
26#1487—22no—
27#1481—22no—
28#1481—22no—
29#1481—22no—
30C2×Q8⋊C8—22no—
31#1301—22no—
32#1301—22no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C23×C8—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8Q812yes—
51C2×C4×Q8—21no—
52C2×C4×Q8D412yes—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431D412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423D412yes—
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⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8D412yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16C2×C412yes—
88C4×M16D412yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#343—22no—
94#343—22no—
95#343—22no—
96#343—22no—
97#341—42no—
98#341—42no—
99#335—24no—
100#334—24no—
101Q8⋊C8—28no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8C4⋊C412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q811yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16C22⋊C412yes—
129C2×M16C22⋊C412yes—
130C2×M16SD1612yes—
131C2×M16Q1612yes—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153#125—41no—
154#125—41no—
155#125—41no—
156#125—41no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163C22⋊Q8—41no—
164C22⋊Q8—41no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173#117—41no—
174#117—41no—
175#117—41no—
176#117—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C4⋊C412yes—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#97—22no—
218#97—22no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—41no—
225C24C22⋊Q811yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4C2×SD1611yescharacteristic
255C22×C4C2×Q1611yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259C22×C4#13711yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262M16—22no—
263M16—22no—
264M16—22no—
265M16—22no—
266M16—22no—
267M16—41no—
268M16—41no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16—42no—
273M16—81no—
274M16—81no—
275M16—81no—
276M16—81no—
277M16#10314yes—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—82no—
292C2×C8—82no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—81no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—42no—
316C42—42no—
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12011yescharacteristic
322C42#12011yescharacteristic
323C42#12011yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12411yescharacteristic
330C42#12411yescharacteristic
331C42#12411yescharacteristic
332C42#12711yescharacteristic
333C23—41no—
334C23—41no—
335C23#38011yescharacteristic
336C23#39611yescharacteristic
337C23#39611yescharacteristic
338C23#47511yescharacteristic
339C23#47711yescharacteristic
340C23#47911yescharacteristic
341C23#47911yescharacteristic
342Q8—44no—
343Q8—44no—
344Q8—82no—
345Q8—82no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#39011yescharacteristic
368C2×C4#39211yescharacteristic
369C2×C4#39311yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39711yescharacteristic
373C2×C4#39911yescharacteristic
374C2×C4C2×#10311yescharacteristic
375C2×C4#41611yescharacteristic
376C2×C4#41711yescharacteristic
377C2×C4#41911yescharacteristic
378C2×C4#46011yescharacteristic
379C2×C4#46111yescharacteristic
380C2×C4#46111yescharacteristic
381C2×C4#46211yescharacteristic
382C2×C4#46211yescharacteristic
383C2×C4#46211yescharacteristic
384C2×C4#46311yescharacteristic
385C2×C4#46311yescharacteristic
386C2×C4#46411yescharacteristic
387C2×C4#46411yescharacteristic
388C2×C4#46411yescharacteristic
389C2×C4#46411yescharacteristic
390C2×C4#46511yescharacteristic
391C2×C4#46511yescharacteristic
392C2×C4#46711yescharacteristic
393C2×C4#46711yescharacteristic
394C2×C4#46811yescharacteristic
395C2×C4#46811yescharacteristic
396C2×C4#47011yescharacteristic
397C2×C4#47011yescharacteristic
398C8—24no—
399C8—42no—
400C8—82no—
401C8—82no—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22#143911yescharacteristic
416C22#180311yescharacteristic
417C22#180511yescharacteristic
418C22#206611yescharacteristic
419C22#206611yescharacteristic
420C22#206811yescharacteristic
421C22#206811yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—82no—
431C4#185812yes—
432C4#189211yescharacteristic
433C4#189411yescharacteristic
434C4#198111yescharacteristic
435C4#198311yescharacteristic
436C4#205511yescharacteristic
437C4#205511yescharacteristic
438C4#205611yescharacteristic
439C4#205611yescharacteristic
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#1107711yescharacteristic
447C2#1113711yescharacteristic
448C2#1114111yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213372
#177872
#174862
C2×#14812
#108532
#80792
C2×#13012