Finite Group Databasebeta

← #10690757

Subgroups — #10690757

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#431C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3#71534C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal
5#67675C212yesmaximal, complemented
6#9114—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C3×#431—24no—
11C2×C3×C4⋊C8—21no—
12C2×C3×C22⋊C8C2212yes—
13C2×C3×C4×C8—21no—
14#5213—21no—
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2212yes—
19#4940—21no—
20#4940—21no—
21#4939C411yescharacteristic, complemented
22#4939C411yescharacteristic, complemented
23#4928C411yescharacteristic, complemented
24#4928C411yescharacteristic, complemented
25#4470—24no—
26C2×#431D311yescharacteristic, complemented
27#2854—31no—
28C2×#327—32no—
29#1434—31no—
30#1294—32no—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C22×#67—21no—
35C23×Dic3—22no—
36C22×D12C2×C411yescharacteristic
37C22×C3×C8—22no—
38C2×C3×C42—22no—
39C2×C3×C42—22no—
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45#930—22no—
46#930—22no—
47#929—42no—
48#929—42no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C3×C4⋊C8—24no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8D414yes—
62C3×C4×C8—24no—
63C3⋊#95—24no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—42no—
70C2×#123—61no—
71C2×#121—31no—
72C22×C42C2×D311yescharacteristic
73#431—24no—
74C2×C4⋊C8—21no—
75C2×C22⋊C8C2×D312yes—
76C2×C4×C8—21no—
77#390—31no—
78#390—31no—
79#389—31no—
80#389—31no—
81#388—61no—
82#388—61no—
83#386—61no—
84#380—31no—
85C2×#95—32no—
86C2×#95—61no—
87C2×#95—61no—
88#327—64no—
89C24×C3C2×D411yescharacteristic
90C23×D3—21no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4D812yes—
103C22×C3×C4SD1612yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4#4112yes—
108C2×#67—42no—
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—42no—
112C22×Dic3—24no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C2×D12—22no—
123C2×D12—41no—
124C2×D12—41no—
125C2×C3×C8—24no—
126C2×C3×C8—42no—
127C2×C3×C8—42no—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—21no—
131C3×C42—21no—
132C3×C42—21no—
133C3×C42—21no—
134#217—41no—
135#217—41no—
136(C2×D3)⋊C4—42no—
137(C2×D3)⋊C4—42no—
138(C2×D3)⋊C4—42no—
139(C2×D3)⋊C4—42no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145Dic3⋊C4—42no—
146Dic3⋊C4—42no—
147C22×D4—31no—
148C22×D4—61no—
149C23×C4—62no—
150C23×C4D1211yescharacteristic
151C23×C4D1211yescharacteristic
152C23×C4C22×D311yescharacteristic
153C22×C8—22no—
154#123—122no—
155#123—122no—
156#121—62no—
157#121—62no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42C4×D311yescharacteristic
172C2×C42C4×D311yescharacteristic
173C2×C42#6711yescharacteristic
174C2×C42#6711yescharacteristic
175C4⋊C8—24no—
176C22⋊C8—22no—
177C22⋊C8#6714yes—
178C4×C8—24no—
179#95—64no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—62no—
184#95—62no—
185#95—122no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3#12112yes—
192C23×C3C2×D811yescharacteristic
193C23×C3C2×SD1611yescharacteristic
194C23×C3#13512yes—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—44no—
220C2×C3×C4#10212yes—
221C2×C3×C4#10212yes—
222C2×C3×C4#10412yes—
223C2×C3×C4C4×D412yes—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12311yescharacteristic
226#67—82no—
227#67—82no—
228C2×Dic3—44no—
229C2×Dic3—44no—
230C2×Dic3—44no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—82no—
241C2×Dic3—82no—
242C2×Dic3—82no—
243C2×Dic3—82no—
244D12—82no—
245C3×C8—44no—
246C24—61no—
247C24C2×D1211yescharacteristic
248C2×D4—62no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—122no—
254C2×D4—122no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—64no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—122no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—122no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4#20412yes—
273C22×C4D2412yes—
274C22×C4(C2×D3)⋊C411yescharacteristic
275C22×C4(C2×D3)⋊C411yescharacteristic
276C22×C4(C2×D3)⋊C411yescharacteristic
277C22×C4(C2×D3)⋊C411yescharacteristic
278C22×C4C2×C4×D311yescharacteristic
279C22×C4#23512yes—
280C22×C4C2×#6711yescharacteristic
281C22×C4C2×#6712yes—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—62no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—122no—
289C4⋊C4—122no—
290C22⋊C4—122no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—122no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C42—22no—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3#38611yescharacteristic
313C22×C3#41011yescharacteristic
314C22×C3#41011yescharacteristic
315C22×C3C2×#10211yescharacteristic
316C22×C3#41611yescharacteristic
317C22×C3C2×#10411yescharacteristic
318C22×C3#42111yescharacteristic
319C22×C3#46512yes—
320C22×C3#46612yes—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—21no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—41no—
335Dic3—84no—
336Dic3—81no—
337Dic3—81no—
338Dic3—81no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23C2×#20411yescharacteristic
351C23C2×D2411yescharacteristic
352C23#90412yes—
353C23C2×(C2×D3)⋊C411yescharacteristic
354C23#93012yes—
355D4—242no—
356D4—242no—
357D4—242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—124no—
377C2×C4—124no—
378C2×C4—124no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—242no—
389C2×C4—242no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4#82112yes—
393C2×C4#82112yes—
394C2×C4#82512yes—
395C2×C4C4×#6712yes—
396C2×C4#92911yescharacteristic
397C2×C4#95311yescharacteristic
398C8—44no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3#150011yescharacteristic
402C2×C3#150211yescharacteristic
403C2×C3#151711yescharacteristic
404C2×C3#152111yescharacteristic
405C2×C3#187111yescharacteristic
406C2×C3#189311yescharacteristic
407C2×C3#189911yescharacteristic
408D3—161no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22#500611yescharacteristic
425C22#500611yescharacteristic
426C22C2×#82111yescharacteristic
427C22#511311yescharacteristic
428C22#511412yes—
429C22#511512yes—
430C22C2×#82511yescharacteristic
431C22#513911yescharacteristic
432C22#521311yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—41no—
440C4—244no—
441C4—241no—
442C4—241no—
443C4—241no—
444C3#911411yescharacteristic, complemented, Sylow
445C2—22no—
446C2—22no—
447C2—481no—
448C2#7078811yescharacteristic
449C2#7078911yescharacteristic
450C2#7079011yescharacteristic
451C2#7079111yescharacteristic
452C2#7159511yescharacteristic
453C2#7197311yescharacteristic
454C2#7198411yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#431C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#431D3
hypocenterC3#9114
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
1. C2×C3×#431C2semidirect
3. #71534C2semidirect
5. #67675C2semidirect
21. #4939C4semidirect
22. #4939C4semidirect
23. #4928C4semidirect
24. #4928C4semidirect
26. C2×#431D3semidirect
444. C3#9114semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4312
#820802
#715342
C2×#44702
#676752
#91143