Finite Group Databasebeta

← #10696413

Subgroups — #10696413

Order 768.

790 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#82378C211yesmaximal, characteristic
2C2×#5007C211yesmaximal, characteristic, complemented
3C2×#4694C211yesmaximal, characteristic, complemented
4C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
5#70456C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10070—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940—21no—
11C2×#930—22no—
12C2×#874C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#832—21no—
17C2×#821—22no—
18C2×#816—22no—
19#5007—22no—
20#5007—22no—
21#4988—21no—
22C2×#807—21no—
23#4960—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4694—22no—
27#4694—22no—
28C3×D4⋊C8—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#474—31no—
32#2541—31no—
33#1981—31no—
34C2×#331—31no—
35C2×D4⋊C8D311yescharacteristic, complemented
36#1279—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C22×#67—24no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12D412yes—
43C22×Dic6—21no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53#940—42no—
54#940—42no—
55#930—44no—
56#930—44no—
57C2×(C2×D3)⋊C4—24no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3D412yes—
61C2×C4×Dic3—21no—
62C22×C3⋊C8—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D412yescomplemented
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8D412yescomplemented
75#832—24no—
76C3⋊#95—21no—
77#821—28no—
78#816—44no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yescomplemented
82C2×C4⋊Q8—31no—
83C2×#127—31no—
84C2×#127—61no—
85C2×#121—62no—
86C2×C4×D4C2×D311yescharacteristic
87#486—62no—
88#486—62no—
89#474—62no—
90#474—62no—
91C2×#107—61no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94C2×#103—62no—
95C2×#102—62no—
96C2×#102—61no—
97C2×C4×C8C2×D311yescharacteristic
98#392—62no—
99#390—61no—
100C4×C4⋊C4—31no—
101#331—62no—
102D4⋊C8—22no—
103C24×C3—21no—
104C23×D3—22no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4SD1612yes—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—48no—
120C2×#67—48no—
121C2×#67—84no—
122C22×Dic3—22no—
123C22×Dic3—41no—
124C2×D12—22no—
125C2×D12—22no—
126C2×D12—44no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—42no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—42no—
135C2×D12—41no—
136C2×D12—41no—
137C2×Dic6—22no—
138C2×Dic6—42no—
139C2×Dic6—42no—
140C2×Dic6—41no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C2×C3×C8—41no—
144C2×C3×C8—42no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153(C2×D3)⋊C4—48no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4⋊Dic3D814yes—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×Q8—61no—
166C22×D4—32no—
167C22×D4—64no—
168C22×D4—61no—
169C22×D4—61no—
170C22×D4D1211yescharacteristic
171C23×C4—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—61no—
175C4⋊Q8—32no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179#127—61no—
180#127—61no—
181#127—61no—
182#127—61no—
183#127—122no—
184#127—122no—
185#121—124no—
186#121—124no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—64no—
197C2×C42—61no—
198C2×C42C22×D311yescharacteristic
199#107—64no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—32no—
203C4⋊C8—61no—
204#103—124no—
205#102—68no—
206#102—64no—
207C4×C8—21no—
208C4×C8#6712yescomplemented
209#95—61no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3#12011yescharacteristic
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—44no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—41no—
224C3×D4—41no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—82no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237C2×C3×C4C2×D812yes—
238C2×C3×C4C2×SD1611yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)12yes—
242C2×C3×C4#13711yescharacteristic
243#67—816no—
244C2×Dic3—24no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—82no—
249D12—44no—
250D12—82no—
251D12—81no—
252D12—81no—
253D12—82no—
254D12—81no—
255D12—81no—
256D12—81no—
257D12—81no—
258Dic6—44no—
259Dic6—81no—
260Dic6—81no—
261C3×C8—42no—
262C3×C8—81no—
263C3⋊C8—42no—
264C24—21no—
265C24—62no—
266C2×Q8—62no—
267C2×Q8—121no—
268C2×Q8—122no—
269C2×Q8—122no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—124no—
278C2×D4—128no—
279C2×D4—128no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—122no—
287C2×D4—122no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—244no—
291C2×D4#20412yes—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—62no—
298C22×C4—121no—
299C22×C4C2×D1211yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—34no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—128no—
320C42—121no—
321C42—121no—
322C42#23511yescharacteristic
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C42C2×#6711yescharacteristic
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3#39211yescharacteristic
332C22×C3#44712yes—
333C22×C3#45011yescharacteristic
334C22×C3#45211yescharacteristic
335C22×C3#45512yes—
336C2×D3—84no—
337C2×D3—84no—
338C2×D3—82no—
339C2×D3—82no—
340C2×D3—82no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—81no—
344C3×C4#44811yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#45912yes—
347C3×C4#46511yescharacteristic
348C3×C4#46811yescharacteristic
349C3×C4#49311yescharacteristic
350C3×C4#49611yescharacteristic
351Dic3—44no—
352Dic3—82no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—122no—
359C23—122no—
360C23—122no—
361C23—124no—
362C23—124no—
363C23#88211yescharacteristic
364Q8—124no—
365Q8—241no—
366Q8—241no—
367D4—22no—
368D4—22no—
369D4—41no—
370D4—41no—
371D4—124no—
372D4—242no—
373D4—2416no—
374D4—241no—
375D4—241no—
376D4—241no—
377D4—241no—
378D4—241no—
379D4—242no—
380D4—241no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—64no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—242no—
392C2×C4#88211yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4#88411yescharacteristic
395C2×C4C2×#20411yescharacteristic
396C2×C4#90411yescharacteristic
397C2×C4#90811yescharacteristic
398C2×C4#90911yescharacteristic
399C2×C4D3×D812yes—
400C2×C4#91412yes—
401C2×C4#93011yescharacteristic
402C2×C4#93711yescharacteristic
403C2×C4#94011yescharacteristic
404C8—42no—
405C8—81no—
406C8—122no—
407C2×C3—82no—
408C2×C3#163912yes—
409C2×C3#164112yes—
410C2×C3#200011yescharacteristic
411C2×C3#200411yescharacteristic
412C2×C3#201111yescharacteristic
413D3—162no—
414C22—42no—
415C22—44no—
416C22—81no—
417C22—82no—
418C22—81no—
419C22—242no—
420C22—242no—
421C22—244no—
422C22—244no—
423C22—242no—
424C22#473111yescharacteristic
425C22#473311yescharacteristic
426C22#477312yes—
427C22#482512yes—
428C22#496011yescharacteristic
429C4—21no—
430C4—21no—
431C4—81no—
432C4—124no—
433C4—242no—
434C4#473411yescharacteristic
435C4#473811yescharacteristic
436C4#478612yes—
437C4#511411yescharacteristic
438C4#513411yescharacteristic
439C4#515111yescharacteristic
440C4#517611yescharacteristic
441C3#1007011yescharacteristic, complemented, Sylow
442C2—82no—
443C2—482no—
444C2#6818212yes—
445C2#6819012yes—
446C2#6996211yescharacteristic
447C2#7020911yescharacteristic
448C2#7038611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3#120
prime radical#82378C2
hypercenterC2×D4⋊C8D3
hypocenterC3#10070
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#5007C2semidirect
3. C2×#4694C2semidirect
4. C2×C3×D4⋊C8C2semidirect
5. #70456C2semidirect
6. C2×#4485C2semidirect
7. #67655C2semidirect
35. C2×D4⋊C8D3semidirect
70. #869D4semidirect
74. C3×C4×C8D4semidirect
81. C4⋊(C3⋊C8)D4semidirect
208. C4×C8#67semidirect
441. C3#10070semidirect

Maximal subgroups

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

subgroupindex
#823782
C2×#50072
C2×#46942
C2×C3×D4⋊C82
#704562
C2×#44852
#676552
#100703