Finite Group Databasebeta

← #10690183

Subgroups — #10690183

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5230C211yesmaximal, characteristic
2#73869C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5C2×#4469C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#9015—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C22×Dic3⋊C4C2211yescharacteristic
12C2×#877—21no—
13C2×C3×#103—22no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5230—22no—
17#5230C414yescomplemented
18C2×#835—22no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#4976—22no—
24#4942—21no—
25#4647—22no—
26#4469—22no—
27C2×#395—31no—
28#2316—31no—
29#1787—31no—
30#1787D311yescharacteristic
31C2×#324—31no—
32#1294—31no—
33#1294—31no—
34C22×C3×Q8—21no—
35C22×C3×Q8—21no—
36C22×C3×Q8—21no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×Q8—44no—
40C22×C3×Q8C2×C412yes—
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×C4⋊Dic3D412yes—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—44no—
59C2×Dic3⋊C4C2×C412yes—
60C2×Dic3⋊C4D412yes—
61C2×C4×Dic3—22no—
62C22×C3⋊C8—21no—
63#877—42no—
64#877—42no—
65C3×#103—44no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69#835—44no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—24no—
74C3⋊#95C2×C412yes—
75#830—21no—
76#830—21no—
77C23×Q8C2×D311yescharacteristic
78C2×#117—61no—
79C22×C4⋊C4—31no—
80C22×C4⋊C4—61no—
81C2×#103—22no—
82C2×#103—62no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#395—34no—
86#395—62no—
87#389—62no—
88#389—62no—
89#380—31no—
90#380—61no—
91#380C2×D311yescharacteristic
92C2×#95—31no—
93C2×#95—61no—
94#324—62no—
95C24×C3C2×D411yescharacteristic
96C2×C3×Q8—24no—
97C2×C3×Q8—24no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—44no—
101C2×C3×Q8—42no—
102C2×C3×Q8—42no—
103C2×C3×Q8—44no—
104C2×C3×Q8—44no—
105C2×C3×Q8—44no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C22×C3×C4—24no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—42no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3—42no—
134C22×Dic3C22⋊C412yes—
135C22×Dic3C22⋊C412yes—
136C22×Dic3SD1612yes—
137C22×Dic3Q1612yes—
138C2×C3×C8—42no—
139C2×C3×C8—41no—
140C3×C4⋊C4—42no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—41no—
144#217—41no—
145C4⋊Dic3—42no—
146Dic3⋊C4—22no—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149Dic3⋊C4—22no—
150Dic3⋊C4—44no—
151Dic3⋊C4—48no—
152Dic3⋊C4—82no—
153C4×Dic3—44no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—44no—
162C22×Q8C4×D312yes—
163C23×C4—21no—
164C23×C4—31no—
165C23×C4—31no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C22×C8—21no—
169C22×C8—61no—
170#117—122no—
171#117—122no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—124no—
183C2×C4⋊C4#6712yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—61no—
186C2×C42—62no—
187#103—44no—
188#103—124no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#95—21no—
194#95—32no—
195#95—64no—
196#95—62no—
197#95—61no—
198#95—61no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3C2×C22⋊C411yescharacteristic
202C23×C3#12011yescharacteristic
203C23×C3#12311yescharacteristic
204C23×C3C2×SD1611yescharacteristic
205C23×C3C2×Q1611yescharacteristic
206C23×C3AGL(1,ℤ/8)11yescharacteristic
207C23×C3#13711yescharacteristic
208C3×Q8—24no—
209C3×Q8—24no—
210C3×Q8—44no—
211C3×Q8—44no—
212C3×Q8—42no—
213C3×Q8—42no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—24no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—44no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—24no—
239C2×Dic3—22no—
240C2×Dic3—42no—
241C2×Dic3—44no—
242C2×Dic3—44no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—81no—
250C2×Dic3#10314yescomplemented
251C3×C8—81no—
252C3⋊C8—81no—
253C24D3×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—32no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—64no—
282C22×C4—62no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—122no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—122no—
290C22×C4C2×C4×D311yescharacteristic
291C22×C4#23511yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—122no—
299C2×C8—121no—
300C4⋊C4—42no—
301C4⋊C4—62no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—122no—
306C4⋊C4—128no—
307C4⋊C4—124no—
308C4⋊C4—242no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—124no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3#38611yescharacteristic
323C22×C3C2×#10311yescharacteristic
324C22×C3#41711yescharacteristic
325C22×C3#45011yescharacteristic
326C22×C3#45111yescharacteristic
327C22×C3#45612yes—
328C22×C3B(4,𝔽2)12yes—
329C22×C3#48311yescharacteristic
330C22×C3#48411yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—44no—
336C3×C4—82no—
337Dic3—24no—
338Dic3—44no—
339Dic3—42no—
340Dic3—82no—
341C23—22no—
342C23—22no—
343C23D3×C22⋊C411yescharacteristic
344C23#88311yescharacteristic
345C23D3×SD1611yescharacteristic
346C23#91411yescharacteristic
347C23D3×Q1611yescharacteristic
348C23#91811yescharacteristic
349C23#93711yescharacteristic
350Q8—24no—
351Q8—24no—
352Q8—44no—
353Q8—44no—
354Q8—42no—
355Q8—42no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—34no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—64no—
378C2×C4—62no—
379C2×C4—122no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—241no—
389C2×C4#67⋊C411yescharacteristic
390C2×C4#88411yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4#93811yescharacteristic
393C8—81no—
394C8—241no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#160412yes—
400C2×C3#160512yes—
401C2×C3#187211yescharacteristic
402C2×C3#188611yescharacteristic
403C2×C3#188911yescharacteristic
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22D3×#10311yescharacteristic
413C22#480311yescharacteristic
414C22#480511yescharacteristic
415C22#481111yescharacteristic
416C22#495611yescharacteristic
417C22#503212yes—
418C22#516911yescharacteristic
419C22#518611yescharacteristic
420C22#520512yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—44no—
426C4—64no—
427C4—82no—
428C4—124no—
429C4—122no—
430C4—242no—
431C3#901511yescharacteristic, complemented, Sylow
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2#6949911yescharacteristic
437C2#7026811yescharacteristic
438C2#7034511yescharacteristic
439C2#7097512yes—
440C2#7098312yes—
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1787D3
hypocenterC3#9015
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
17. #5230C4semidirect
250. C2×Dic3#103semidirect
431. C3#9015semidirect

Maximal subgroups

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

subgroupindex
C2×#52302
#738692
C3×#17872
#721322
C2×#44692
#676752
#676242
#90153