Finite Group Databasebeta

← #378796

Subgroups — #378796

Order 512.

834 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×#2089C211yesmaximal, characteristic
2#21033C212yesmaximal
3C2×C4×#107C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11270C211yesmaximal, characteristic
7#11239C211yesmaximal, characteristic
8#10896C212yesmaximal
9#10827C212yesmaximal
10#10822C211yesmaximal, characteristic
11#10802C212yesmaximal
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21#2089C418yescomplemented
22#1921—22no—
23#1920—22no—
24#1919—22no—
25#1918—22no—
26C4×#107—21no—
27C4×#107—21no—
28C4×#107—21no—
29C4×#107—21no—
30#1771—21no—
31#1771—21no—
32C8×C4⋊C4—21no—
33C8×C4⋊C4—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2211yescharacteristic
46#1728C2212yes—
47#1728C2211yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60#428—24no—
61#428C2×C418yes—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—24no—
69C2×#107C2×C414yes—
70C2×#106—22no—
71#424—22no—
72#424C2×C414yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75#405—22no—
76#405—22no—
77#405—24no—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—24no—
82C2×C4×C8—21no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87#389D411yescharacteristic
88#389D411yescharacteristic
89#389D411yescharacteristic
90#389D411yescharacteristic
91#385—21no—
92#385—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C2×#95—22no—
100#343—24no—
101#343—24no—
102#343—24no—
103#343C2×C418yes—
104#341—22no—
105#341—24no—
106#341C2×C414yes—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#336—21no—
116#336—21no—
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4C2411yescharacteristic
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16C22×C414yes—
144C2×M16C22×C412yes—
145C2×M16#4112yes—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8C22×C412yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8#4112yes—
161C22×C8#4114yes—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—22no—
166#117C22×C412yes—
167#117#4112yes—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—84no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201#107—24no—
202#107—24no—
203#107—24no—
204#107—24no—
205#107—28no—
206#106—28no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209#97—24no—
210#97—42no—
211C4×C8—24no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—42no—
222#95—42no—
223C24C2×#4111yescharacteristic
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4C2×#4112yes—
255C22×C4C2×#4111yescharacteristic
256C22×C42−1+412yes—
257C22×C42−1+412yes—
258M16—24no—
259M16—24no—
260M16—24no—
261M16—24no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16—42no—
268M16—42no—
269M16—42no—
270C2×C8—24no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—24no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8C4×D412yes—
295C2×C8C4×D412yes—
296C2×C8#12312yes—
297C2×C8#12312yes—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4C4×D412yes—
315C4⋊C4C4×D412yes—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C42—22no—
320C42—21no—
321C42—21no—
322C42—22no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—84no—
330C42—84no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23C4×#4111yescharacteristic
336C23C2×#12111yescharacteristic
337C23#52911yescharacteristic
338C23#55212yes—
339C23#55212yes—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4#47212yes—
375C2×C4#47312yes—
376C2×C4C2×C4×D411yescharacteristic
377C2×C4C2×C4×D411yescharacteristic
378C2×C4C4×#4112yes—
379C2×C4#51912yes—
380C2×C4C2×#12311yescharacteristic
381C2×C4#52511yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#52912yes—
384C2×C4#53211yescharacteristic
385C2×C4#53611yescharacteristic
386C2×C4#53611yescharacteristic
387C2×C4#54112yes—
388C2×C4#54712yes—
389C2×C4#54812yes—
390C2×C4#55212yes—
391C2×C4#55212yes—
392C2×C4#55212yes—
393C2×C4#55612yes—
394C8—22no—
395C8—22no—
396C8—22no—
397C8—22no—
398C8—42no—
399C8—44no—
400C8—42no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22C4×#12111yescharacteristic
411C22#237112yes—
412C22#248611yescharacteristic
413C22#252412yes—
414C22#253611yescharacteristic
415C22C2×#47211yescharacteristic
416C22C2×#47311yescharacteristic
417C22#306711yescharacteristic
418C22#306711yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—42no—
424C4—42no—
425C4—81no—
426C4—81no—
427C4—84no—
428C4—84no—
429C4C4×#12311yescharacteristic
430C4#232711yescharacteristic
431C4#236212yes—
432C4#248411yescharacteristic
433C4#251912yes—
434C4#259911yescharacteristic
435C2—22no—
436C2—22no—
437C2#1422311yescharacteristic
438C2C4×#47211yescharacteristic
439C2C4×#47311yescharacteristic
440C2#2767311yescharacteristic
441C2#2767511yescharacteristic
442C2#2821711yescharacteristic
443C2#2821911yescharacteristic
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
21. #2089C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20892
#210332
C2×C4×#1072
#204352
#113652
#112702
#112392
#108962
#108272
#108222
#108022