Finite Group Databasebeta

← #10690791

Subgroups — #10690791

Order 768.

626 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
1C2×D3×#97C211yesmaximal, characteristic
2#74257C211yesmaximal, characteristic
3#69807C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic, complemented
5C2×#4465C211yesmaximal, characteristic, complemented
6C3×#1295C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#9121—31nomaximal, Sylow
9C22×D3×C8—21no—
10C2×C42×D3C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15C2×(C2×D3)⋊C8—21no—
16C2×#815—21no—
17#4987C411yescharacteristic
18#4987C411yescharacteristic
19#4984C411yescharacteristic
20#4984C411yescharacteristic
21C2×#803C2211yescharacteristic
22C2×#803C2211yescharacteristic
23D3×#97—22no—
24D3×#97—22no—
25#4668—21no—
26#4668—21no—
27#4653—21no—
28#4653—21no—
29#4550—21no—
30#4465—22no—
31#4465—22no—
32C22×#97—31no—
33#2851—31no—
34#1749—31no—
35C2×#330—31no—
36C2×#330—31no—
37#1295—31no—
38#1295D311yescharacteristic, complemented
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C2×D3×C8—24no—
61C2×D3×C8—24no—
62C2×D3×C8—24no—
63C42×D3—21no—
64C42×D3—21no—
65C42×D3—21no—
66C42×D3—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71C3×#97—21no—
72C3×#97—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3×#95—21no—
76C3×#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81(C2×D3)⋊C8—24no—
82#815—24no—
83#803—21no—
84#803—21no—
85#803—21no—
86#803D412yes—
87C23×C8—61no—
88C2×#125—31no—
89C22×C42—31no—
90C2×C22⋊C8—61no—
91C2×#97—31no—
92C2×#97—31no—
93C2×#97—62no—
94C2×#97—61no—
95C2×#97—62no—
96C2×#97C2×D311yescharacteristic
97C2×#97C2×D311yescharacteristic
98#389—31no—
99#389—31no—
100#387—61no—
101#387—61no—
102#385—61no—
103#385—61no—
104#382C2×D311yescharacteristic
105#380—31no—
106#380—31no—
107#336—61no—
108#330—62no—
109#330—62no—
110C23×D3C22⋊C411yescharacteristic
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×C4×D3—22no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—22no—
125C2×C4×D3—22no—
126C2×C4×D3—22no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—42no—
146(C2×D3)⋊C4—42no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—42no—
152C4⋊Dic3—42no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169D3×C8—216no—
170C23×C4—31no—
171C23×C4—31no—
172C23×C4—31no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—64no—
176C22×C8—64no—
177C22×C8—64no—
178C22×C8—61no—
179C22×C8—61no—
180#125—61no—
181#125—61no—
182#125—61no—
183#125—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4C4×D311yescharacteristic
191C2×C4⋊C4C4×D311yescharacteristic
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—31no—
195C2×C42—31no—
196C2×C42—31no—
197C2×C42—61no—
198C2×C42—61no—
199C2×C42—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202C22⋊C8—64no—
203#97—21no—
204#97—21no—
205#97—21no—
206#97—32no—
207#97—64no—
208#97—61no—
209#97—61no—
210#97—61no—
211#97D1212yes—
212#95—21no—
213#95—21no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218C23×C3C2×C22⋊C411yescharacteristic
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—21no—
222C22×D3—21no—
223C22×D3#10112yes—
224C22×D3#10412yes—
225C22×D3#10412yes—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3#10412yes—
259C4×D3—24no—
260C4×D3—24no—
261C4×D3—42no—
262C4×D3—42no—
263C4×D3—42no—
264C4×D3—42no—
265C3×C8—24no—
266C3×C8—42no—
267C3⋊C8—24no—
268C3⋊C8—42no—
269C24—31no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4C2×C4×D311yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—616no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—121no—
304C2×C8—121no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—122no—
312C4⋊C4—122no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C22⋊C4—122no—
318C22⋊C4—122no—
319C42—61no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—61no—
325C42C2×D1211yescharacteristic
326C42#23511yescharacteristic
327C42D3×D411yescharacteristic
328C42#23911yescharacteristic
329C22×C3#38611yescharacteristic
330C22×C3C2×#10111yescharacteristic
331C22×C3#41311yescharacteristic
332C22×C3C2×#10411yescharacteristic
333C22×C3C2×#10411yescharacteristic
334C22×C3C2×#10411yescharacteristic
335C22×C3#42111yescharacteristic
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—22no—
342C2×D3#33014yescomplemented
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—81no—
350C3×C4—81no—
351Dic3—22no—
352Dic3—41no—
353Dic3—41no—
354Dic3—81no—
355Dic3—81no—
356C23—32no—
357C23—32no—
358C23—32no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23D3×C22⋊C411yescharacteristic
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—32no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—64no—
378C2×C4—64no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4C4×D1211yescharacteristic
398C2×C4#88211yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#89311yescharacteristic
401C2×C4#89411yescharacteristic
402C2×C4#89511yescharacteristic
403C8—24no—
404C8—42no—
405C8—64no—
406C8—122no—
407C2×C3C2×#33011yescharacteristic
408C2×C3#152411yescharacteristic
409C2×C3#153111yescharacteristic
410C2×C3#153211yescharacteristic
411C2×C3#187111yescharacteristic
412C2×C3#189611yescharacteristic
413C2×C3#190111yescharacteristic
414D3—24no—
415C22—34no—
416C22—62no—
417C22—62no—
418C22—62no—
419C22—62no—
420C22—62no—
421C22—62no—
422C22#466511yescharacteristic
423C22#474511yescharacteristic
424C22D3×#10111yescharacteristic
425C22D3×#10411yescharacteristic
426C22D3×#10411yescharacteristic
427C22D3×#10411yescharacteristic
428C22#482111yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—62no—
436C4—81no—
437C4—81no—
438C4—121no—
439C4—121no—
440C4—241no—
441C4—241no—
442C3#912111yescharacteristic, complemented, Sylow
443C2—64no—
444C2#6834111yescharacteristic
445C2#6834611yescharacteristic
446C2D3×#33011yescharacteristic
447C2#6836511yescharacteristic
448C2#6875511yescharacteristic
449C2#6875611yescharacteristic
450C2#6899311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1295D3
hypocenterC3#9121
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
4. C2×#4465C2semidirect
5. C2×#4465C2semidirect
6. C3×#1295C2semidirect
7. #67630C2semidirect
38. #1295D3semidirect
342. C2×D3#330semidirect
442. C3#9121semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#972
#742572
#698072
C2×#44652
C3×#12952
#676302
#91213