Finite Group Databasebeta

← #370280

Subgroups — #370280

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1951C211yesmaximal, characteristic
2C2×#1951C211yesmaximal, characteristic
3#21159C212yesmaximal
4C2×#1941C211yesmaximal, characteristic
5C2×C4×#108C211yesmaximal, characteristic
6#20435C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11216C212yesmaximal
9#10913C212yesmaximal
10#10869C211yesmaximal, characteristic
11#10869C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×C4×M16C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2296C2211yescharacteristic
24C2×#389C2212yes—
25#1951—24no—
26#1951C418yescomplemented
27#1944—22no—
28#1943—22no—
29#1942—22no—
30#1941—22no—
31#1941—22no—
32#1941—22no—
33C4×#108—24no—
34#1771—22no—
35C8×C4⋊C4—22no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#340C2212yes—
44#1729C2211yescharacteristic
45#1728C2211yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C2×#108—22no—
59C2×#108—22no—
60C2×#108C2×C414yes—
61C2×#108D414yes—
62C2×#108Q814yes—
63C2×#108Q814yes—
64#428—24no—
65#428—24no—
66#428—42no—
67#428C2×C418yes—
68C2×#107—22no—
69C2×#107—22no—
70C2×#106—21no—
71C2×#106—21no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—21no—
75C2×#106—21no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106C2×C414yes—
80#424—22no—
81#424—22no—
82C2×C4⋊C8—22no—
83#405—22no—
84#405—22no—
85#405—22no—
86#405C2×C414yes—
87C4×M16—24no—
88C4×M16—42no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—22no—
97C2×C4×C8D412yes—
98C2×C4×C8D412yes—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389D412yes—
104#389D412yes—
105#385—22no—
106C4×C4⋊C4—22no—
107C2×#95—22no—
108#343—24no—
109#343—24no—
110#343—42no—
111#343C2×C418yes—
112#341—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#336—22no—
119#336—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4#4112yes—
123C23×C4C2411yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16C22×C412yes—
143C2×M16C22×C414yes—
144C2×M16C2×D414yes—
145C2×M16C2×Q814yes—
146C2×M16C2×Q814yes—
147C2×M16#4112yes—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C22×C412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8#4112yes—
166C22×C8#4112yes—
167#117—21no—
168#117—21no—
169#117—21no—
170#117—21no—
171#117—21no—
172#117—21no—
173#117C22×C412yes—
174#117#4112yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205#108—28no—
206#108C4⋊C4116yescomplemented
207#107—28no—
208#107—44no—
209#106—24no—
210#106—24no—
211#106—28no—
212#106—44no—
213#106—42no—
214#106—42no—
215C4⋊C8—28no—
216#97—24no—
217#97—44no—
218#97—42no—
219C4×C8—24no—
220C4×C8—24no—
221C4×C8—24no—
222C4×C8—44no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226#95—42no—
227#95—42no—
228C24C2×#4111yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4#12112yes—
246C22×C4#12112yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C42−1+411yescharacteristic
259C22×C42−1+411yescharacteristic
260C22×C42−1+411yescharacteristic
261M16—24no—
262M16—24no—
263M16—24no—
264M16—22no—
265M16—22no—
266M16—24no—
267M16—22no—
268M16—22no—
269M16—24no—
270M16—41no—
271M16—41no—
272M16—42no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16C2×C4⋊C418yes—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8C2×C4⋊C414yes—
297C2×C8C4×D412yes—
298C2×C8C4×D412yes—
299C2×C8C4×Q814yes—
300C2×C8#12112yes—
301C2×C8#12112yes—
302C2×C8C22⋊Q814yes—
303C2×C8#12514yes—
304C2×C8C4⋊Q812yes—
305C2×C8C4⋊Q812yes—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4C4×D414yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—41no—
335C42—41no—
336C23—22no—
337C23—22no—
338C23C4×#4111yescharacteristic
339C23C2×#12111yescharacteristic
340C23#52911yescharacteristic
341C23#54811yescharacteristic
342C23#55311yescharacteristic
343C23#55711yescharacteristic
344C23#55711yescharacteristic
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—162no—
367C2×C4—162no—
368C2×C4#47114yes—
369C2×C4C22×C4⋊C411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×C4×Q811yescharacteristic
373C2×C4#51911yescharacteristic
374C2×C4#52011yescharacteristic
375C2×C4C2×#12111yescharacteristic
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4#52511yescharacteristic
378C2×C4C2×#12511yescharacteristic
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4#53211yescharacteristic
381C2×C4#53611yescharacteristic
382C2×C4#53711yescharacteristic
383C2×C4#54111yescharacteristic
384C2×C4#54211yescharacteristic
385C2×C4D4211yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4#54711yescharacteristic
388C2×C4#54811yescharacteristic
389C2×C4D4×Q811yescharacteristic
390C2×C4D4×Q811yescharacteristic
391C2×C4#55211yescharacteristic
392C2×C4#55411yescharacteristic
393C2×C4#55411yescharacteristic
394C2×C4#55611yescharacteristic
395C2×C4#55711yescharacteristic
396C2×C4#55711yescharacteristic
397C8—28no—
398C8—42no—
399C8—42no—
400C8—42no—
401C8—44no—
402C8#38914yes—
403C8#38914yes—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22C4×#12111yescharacteristic
413C22#235111yescharacteristic
414C22#237011yescharacteristic
415C22#241611yescharacteristic
416C22#245311yescharacteristic
417C22#250911yescharacteristic
418C22#254611yescharacteristic
419C22C2×#47112yes—
420C22#306712yes—
421C4—24no—
422C4—44no—
423C4—42no—
424C4—84no—
425C4—82no—
426C4—162no—
427C4C2×#38911yescharacteristic
428C4#232311yescharacteristic
429C4D4×C4⋊C411yescharacteristic
430C4#236511yescharacteristic
431C4#241511yescharacteristic
432C4#245211yescharacteristic
433C4#250611yescharacteristic
434C4#258411yescharacteristic
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2#1404611yescharacteristic
440C2C4×#47112yes—
441C2#2480212yes—
442C2#2870011yescharacteristic
443C2#2870111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1951C4semidirect
206. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19512
#211592
C2×#19412
C2×C4×#1082
#204352
#112902
#112162
#109132
#108692