Finite Group Databasebeta

← #370607

Subgroups — #370607

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3162C211yesmaximal, characteristic
2#21304C212yesmaximal
3C2×#1946C211yesmaximal, characteristic
4C2×#1772C211yesmaximal, characteristic
5#11386C212yesmaximal
6#11221C212yesmaximal
7#10909C212yesmaximal
8#10865C211yesmaximal, characteristic
9#10860C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10696C211yesmaximal, characteristic
12C2×#533C2212yes—
13#3162—24no—
14#3162—24no—
15C2×#501C414yescomplemented
16C2×#501C414yescomplemented
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2212yes—
21C2×#427C2211yescharacteristic
22C2×#424C2212yes—
23C2×#405C2212yes—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#1985—24no—
28#1984—24no—
29#1946—24no—
30#1772—24no—
31C2×#341C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36#1729C2212yes—
37#1729C2212yes—
38#1728C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C4×#95C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49#533—24no—
50#533—24no—
51#533—42no—
52#533—42no—
53C2×C4⋊Q8C2×C414yes—
54C2×#125C2×C414yes—
55C2×C22⋊Q8—24no—
56C2×C4×Q8—24no—
57C2×#117C2312yes—
58C2×#117C2312yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#501—216no—
63#501—216no—
64C2×#108—22no—
65C2×#108—22no—
66C2×#108—22no—
67C2×#108—22no—
68C2×#108—22no—
69C2×#108Q814yescomplemented
70#427—22no—
71#427—22no—
72#427—24no—
73#427—24no—
74#427D414yes—
75#427D414yes—
76C2×#107C2×C412yes—
77C2×#107C2×C414yes—
78C2×#107C2×C412yes—
79C2×#106C2×C412yes—
80C2×#106C2×C414yes—
81C2×#106C2×C412yes—
82#424—24no—
83#424—42no—
84C2×C4⋊C8—24no—
85#405—24no—
86#405—42no—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97C2×C412yes—
94C2×#97C2×C412yes—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8Q814yescomplemented
99C2×#95—22no—
100#341—24no—
101#339—24no—
102#339—24no—
103#339—42no—
104#339—42no—
105#336—22no—
106#336—24no—
107#336—22no—
108#336—22no—
109#336—24no—
110#336D412yes—
111#336D412yes—
112#336Q814yescomplemented
113C22×Q8—24no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4#4112yes—
117C23×C4C2411yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16C2×Q814yes—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8C22×C412yes—
147C22×C8C22×C411yescharacteristic
148C22×C8C22×C411yescharacteristic
149C22×C8C2×D412yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×Q812yes—
155C22×C8C2×Q812yes—
156C22×C8C2×Q812yes—
157C22×C8C2×Q812yes—
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C4⋊Q8—28no—
163C4⋊Q8—44no—
164#125—28no—
165#125—44no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C4×Q8—48no—
169C4×Q8—48no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117C2×D414yes—
177#117C2×D414yes—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C22×C414yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C22×C412yes—
201C2×C42C2×D412yes—
202C2×C42C2×Q812yes—
203C2×C42C2×Q812yes—
204#108—28no—
205#108—44no—
206#107—24no—
207#107—28no—
208#107C22⋊C418yes—
209#106—24no—
210#106—28no—
211#106C22⋊C418yes—
212C4⋊C8—48no—
213#97—24no—
214#97—24no—
215#97—44no—
216#97—44no—
217C4×C8—28no—
218C4×C8—44no—
219#95—44no—
220C24C2×#4111yescharacteristic
221C2×Q8—48no—
222C2×Q8—48no—
223C2×Q8—84no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4C4×D414yes—
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×Q811yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C22×Q811yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4H(5,𝔽2)11yescharacteristic
252C22×C4H(5,𝔽2)11yescharacteristic
253C22×C42−1+411yescharacteristic
254C22×C42−1+411yescharacteristic
255M16—24no—
256M16—24no—
257M16—42no—
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—42no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8C2×C22⋊C412yes—
291C2×C8C2×C22⋊C412yes—
292C2×C8C4×Q814yes—
293C2×C8C4×Q812yes—
294C2×C8C4×Q812yes—
295C2×C8C4×Q812yes—
296C2×C8C4×Q812yes—
297C2×C8C22⋊Q814yes—
298C2×C8C22⋊Q814yes—
299C2×C8#12414yes—
300C2×C8C4⋊Q812yes—
301C2×C8C4⋊Q812yes—
302C2×C8C4⋊Q814yes—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—416no—
307C4⋊C4—48no—
308C4⋊C4C2×C22⋊C418yes—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—22no—
314C42—22no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—41no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—41no—
329C23—22no—
330C23—22no—
331C23C2×C4×D411yescharacteristic
332C23#53311yescharacteristic
333C23#53311yescharacteristic
334C23#54611yescharacteristic
335C23#54611yescharacteristic
336C23#55611yescharacteristic
337C23#55711yescharacteristic
338Q8—816no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4C22×C22⋊C411yescharacteristic
361C2×C4C2×C4×Q811yescharacteristic
362C2×C4C2×C4×Q811yescharacteristic
363C2×C4C2×C4×Q811yescharacteristic
364C2×C4#51811yescharacteristic
365C2×C4#51911yescharacteristic
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×#12411yescharacteristic
369C2×C4C2×C4⋊Q811yescharacteristic
370C2×C4C2×C4⋊Q811yescharacteristic
371C2×C4#53211yescharacteristic
372C2×C4#54211yescharacteristic
373C2×C4#54211yescharacteristic
374C2×C4#54411yescharacteristic
375C2×C4#54411yescharacteristic
376C2×C4D4211yescharacteristic
377C2×C4D4211yescharacteristic
378C2×C4#54711yescharacteristic
379C2×C4#54811yescharacteristic
380C2×C4#54811yescharacteristic
381C2×C4D4×Q811yescharacteristic
382C2×C4D4×Q811yescharacteristic
383C2×C4D4×Q811yescharacteristic
384C2×C4#55411yescharacteristic
385C2×C4#55411yescharacteristic
386C2×C4#55511yescharacteristic
387C2×C4Q8211yescharacteristic
388C8—24no—
389C8—22no—
390C8—22no—
391C8—44no—
392C8—42no—
393C8—82no—
394C8#39114yes—
395C8#39114yes—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22C4×C4⋊Q811yescharacteristic
405C22D4×C22⋊C411yescharacteristic
406C22#236511yescharacteristic
407C22#243911yescharacteristic
408C22#245111yescharacteristic
409C22#249211yescharacteristic
410C22#255011yescharacteristic
411C22#295512yes—
412C22#295612yes—
413C4—44no—
414C4—44no—
415C4—48no—
416C4—82no—
417C4—84no—
418C4C2×#39111yescharacteristic
419C4#233111yescharacteristic
420C4Q8×C22⋊C411yescharacteristic
421C4#236911yescharacteristic
422C4#244011yescharacteristic
423C4#245211yescharacteristic
424C4#249111yescharacteristic
425C4#255111yescharacteristic
426C2—21no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2#1405311yescharacteristic
431C2#2173511yescharacteristic
432C2#2173611yescharacteristic
433C2#2245312yes—
434C2#2317212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#501C4semidirect
16. C2×#501C4semidirect
69. C2×#108Q8semidirect
98. C2×C4×C8Q8semidirect
112. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31622
#213042
C2×#19462
C2×#17722
#113862
#112212
#109092
#108652
#108602
#108592
#106962