Finite Group Databasebeta

← #10690655

Subgroups — #10690655

Order 768.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79276C211yesmaximal, characteristic, complemented
2C4×#908C211yesmaximal, characteristic
3#70783C211yesmaximal, characteristic, complemented
4#70425C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9095—31nomaximal, Sylow
9C2×#908C411yescharacteristic
10C4×#235C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133C411yescharacteristic
16#5131C411yescharacteristic
17#5121C411yescharacteristic
18#4925C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20#4834—21no—
21#4830—21no—
22#67⋊C8—22no—
23#4711—21no—
24#4710—21no—
25C4×D24—21no—
26C4×#204—21no—
27#4685—21no—
28C4×D3×C8—21no—
29#4484—21no—
30#4483—21no—
31#4483—21no—
32#4483—21no—
33#4468—22no—
34C8×#41—31no—
35C4×AGL(1,ℤ/8)—31no—
36#1477—31no—
37C2×D4⋊C8—31no—
38#1460—31no—
39#1283D311yescharacteristic, complemented
40#1278—31no—
41C2×#235C2×C411yescharacteristic
42C22×D12C2×C411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16C2×C411yescharacteristic
45C2×C3×M16C2×C411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C42C2311yescharacteristic
48C4×#67—21no—
49C4×#67—21no—
50C2×(C2×D3)⋊C4—21no—
51#926C2×C411yescharacteristic
52C2×C4⋊Dic3C2×C411yescharacteristic
53C22×C3⋊C8—21no—
54#908—22no—
55#908C814yescomplemented
56C2×D24—21no—
57C2×#204—21no—
58C2×D3×C8—22no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12D411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67C4×Dic6D411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#830—21no—
77#821—21no—
78#821—21no—
79#821—21no—
80(C2×D3)⋊C8—22no—
81#818—21no—
82#817—21no—
83#816—21no—
84Dic3⋊C8—22no—
85C8×Dic3—22no—
86C4⋊(C3⋊C8)—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D411yescharacteristic
90C4×C3⋊C8D411yescharacteristic
91C2×AGL(1,ℤ/8)—31no—
92C4×#41—31no—
93C2×C4×D4—31no—
94C8×Q8—61no—
95#442—61no—
96#440—61no—
97C4×SD16—61no—
98C4×D8—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—31no—
102#431—61no—
103#431C2×D311yescharacteristic
104#428—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106#418—31no—
107#417—31no—
108C4×M16C2×D311yescharacteristic
109C2×C4×C8—31no—
110C2×C4×C8—61no—
111Q8⋊C8—61no—
112D4⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115#323—62no—
116C23×D3—21no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—21no—
121C22×Dic3—21no—
122#235—21no—
123#235—21no—
124#235C2×C812yes—
125#235M1612yes—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12C22⋊C411yescharacteristic
132C2×D12C2×C812yescomplemented
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—81no—
139C2×C4×D3—81no—
140C2×Dic6C22⋊C411yescharacteristic
141C3×M16—21no—
142C3×M16—21no—
143C3×M16—21no—
144C3×M16C2×C812yescomplemented
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C2×C3×C8—81no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—41no—
172D24—24no—
173#204—24no—
174D3×C8—44no—
175C2×#41—31no—
176C22×D4—31no—
177C23×C4—61no—
178AGL(1,ℤ/8)—34no—
179AGL(1,ℤ/8)—62no—
180C2×SD16—61no—
181C2×D8—61no—
182C2×M16C4×D311yescharacteristic
183C2×M16C4×D311yescharacteristic
184C22×C8—21no—
185C22×C8—62no—
186C22×C8—61no—
187C4×Q8—31no—
188C4×Q8—31no—
189C4×D4—31no—
190C4×D4—31no—
191C4×D4—61no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197#117—31no—
198#117—61no—
199C2×C4⋊C4—31no—
200C2×C22⋊C4—61no—
201C2×C42—61no—
202C2×C42C22×D311yescharacteristic
203#107—61no—
204#106—61no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—62no—
209C4⋊C8—61no—
210#103—61no—
211#102—61no—
212#102—61no—
213#102—61no—
214C22⋊C8—62no—
215C22⋊C8—61no—
216C22⋊C8D1212yes—
217#97—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—31no—
221C4×C8—31no—
222C4×C8—62no—
223C4×C8—61no—
224C4×C8—61no—
225C23×C3C4×D411yescharacteristic
226C22×D3—21no—
227C22×D3—22no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—81no—
231C22×D3—81no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×C3×C4C22×C811yescharacteristic
242C2×C3×C4C2×M1611yescharacteristic
243C2×C3×C4#13111yescharacteristic
244C2×C3×C4#13111yescharacteristic
245#67—22no—
246#67—22no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—41no—
251C2×Dic3—81no—
252D12—22no—
253D12—21no—
254D12—22no—
255D12—41no—
256D12—41no—
257D12C22⋊C812yes—
258C4×D3—22no—
259C4×D3—22no—
260C4×D3—42no—
261C4×D3—41no—
262C4×D3—41no—
263C4×D3—82no—
264C4×D3—81no—
265C4×D3—81no—
266Dic6—21no—
267Dic6C22⋊C812yes—
268C3×C8—22no—
269C3×C8—42no—
270C3×C8—41no—
271C3×C8—81no—
272C3⋊C8—22no—
273C3⋊C8—41no—
274C3⋊C8—81no—
275C24—61no—
276#41—32no—
277#41—32no—
278#41—61no—
279#41—61no—
280C2×Q8—31no—
281C2×D4—31no—
282C2×D4—32no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—241no—
296C22×C4—241no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4C2×D1211yescharacteristic
299C22×C4#23511yescharacteristic
300SD16—64no—
301D8—64no—
302M16—21no—
303M16—21no—
304M16—21no—
305M16D3×C812yescomplemented
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—81no—
313C2×C8—124no—
314C2×C8—122no—
315C2×C8—121no—
316C4⋊C4—31no—
317C4⋊C4—31no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—61no—
328C42—61no—
329C42—61no—
330C42D3×D411yescharacteristic
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42#23911yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3#38611yescharacteristic
337C22×C3C8×D411yescharacteristic
338C22×C3#43511yescharacteristic
339C2×D3—22no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—42no—
343C2×D3—82no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—82no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4C2×C22⋊C811yescharacteristic
351C3×C4#40811yescharacteristic
352C3×C4C8×D411yescharacteristic
353C3×C4#43511yescharacteristic
354C3×C4#47111yescharacteristic
355C3×C4#47211yescharacteristic
356Dic3—22no—
357Dic3—41no—
358Dic3—41no—
359Dic3—81no—
360C23—61no—
361C23—62no—
362C23—121no—
363C23—121no—
364C23—241no—
365C23—241no—
366C23C4×D1211yescharacteristic
367Q8—32no—
368Q8—61no—
369D4—32no—
370D4—62no—
371D4—62no—
372D4—62no—
373D4—61no—
374D4—62no—
375D4—121no—
376D4—121no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—242no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4D3×C22⋊C411yescharacteristic
394C2×C4#88211yescharacteristic
395C2×C4#88411yescharacteristic
396C2×C4#89311yescharacteristic
397C2×C4#89411yescharacteristic
398C2×C4#89511yescharacteristic
399C2×C4C2×D3×C811yescharacteristic
400C2×C4#90111yescharacteristic
401C2×C4D3×M1611yescharacteristic
402C2×C4#90711yescharacteristic
403C8—22no—
404C8—42no—
405C8—41no—
406C8—62no—
407C8—81no—
408C8—121no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3#186911yescharacteristic
413C2×C3#190611yescharacteristic
414C2×C3#190711yescharacteristic
415D3—42no—
416D3—82no—
417C22—21no—
418C22—21no—
419C22—62no—
420C22—121no—
421C22—121no—
422C22—122no—
423C22—242no—
424C22—242no—
425C22—241no—
426C22—241no—
427C22#466511yescharacteristic
428C22C8×D1211yescharacteristic
429C22#470611yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—62no—
434C4—121no—
435C4—121no—
436C4—241no—
437C4D3×C22⋊C811yescharacteristic
438C4#472711yescharacteristic
439C4#474911yescharacteristic
440C4#475411yescharacteristic
441C4#483411yescharacteristic
442C4#483611yescharacteristic
443C3#909511yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—122no—
447C2—242no—
448C2#6866811yescharacteristic
449C2#6900611yescharacteristic
450C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radicalC2×#908C4
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9095
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
1. #79276C2semidirect
3. #70783C2semidirect
4. #70425C2semidirect
5. C2×#4483C2semidirect
6. C3×#1283C2semidirect
7. #67667C2semidirect
39. #1283D3semidirect
55. #908C8semidirect
132. C2×D12C2×C8semidirect
144. C3×M16C2×C8semidirect
305. M16D3×C8semidirect
443. C3#9095semidirect

Maximal subgroups

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

subgroupindex
#792762
C4×#9082
#707832
#704252
C2×#44832
C3×#12832
#676672
#90953