Finite Group Databasebeta

← #10697519

Subgroups — #10697519

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4849C211yesmaximal, characteristic, complemented
2C2×#4846C211yesmaximal, characteristic
3#74369C211yesmaximal, characteristic
4C2×#4484C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6#69582C211yesmaximal, characteristic, complemented
7C3×#1296C211yesmaximal, characteristic, complemented
8#10233—31nomaximal, Sylow
9C22×Dic12—21no—
10C2×#897—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17C2×#821—21no—
18C2×#818—21no—
19C2×#816—21no—
20#4994—22no—
21#4988—21no—
22#4987—21no—
23#4984—21no—
24C2×C4×C3⋊C8C2211yescharacteristic
25#4849—22no—
26#4849—22no—
27#4846—22no—
28#4846—22no—
29#4529—22no—
30#4484—22no—
31#4483—22no—
32C2×#474—31no—
33C2×#462—31no—
34#2538—31no—
35#1990—31no—
36C2×Q8⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60C2×Dic12—24no—
61C2×Dic12—42no—
62C2×Dic12—42no—
63#897—42no—
64#897—42no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69#869—21no—
70#869—21no—
71#869—21no—
72#869D412yes—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C4×Dic6—21no—
76C4×Dic6—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8D412yes—
80C3×C4⋊C8D412yes—
81C3⋊#95—22no—
82#821—24no—
83#818—24no—
84#816—24no—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C22×Q16—61no—
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—61no—
90C2×C4×Q8—31no—
91C2×C4×D4—31no—
92#474—62no—
93#474—62no—
94#462—62no—
95#462—62no—
96C2×#107—61no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#103—61no—
100C2×#102—61no—
101C2×C4×C8—31no—
102#396—62no—
103#390—61no—
104#389—61no—
105#384C2×D311yescharacteristic
106#380—61no—
107#340—62no—
108Q8⋊C8—62no—
109D4⋊C8—62no—
110C23×D3—21no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—41no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12Q1612yescomplemented
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—22no—
137C2×Dic6—41no—
138C2×Dic6—41no—
139C2×Dic6—41no—
140C2×Dic6—41no—
141C2×Dic6—41no—
142C2×Dic6D812yes—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—22no—
167C4⋊Dic3—21no—
168C4⋊Dic3—44no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173C4×Dic3—41no—
174C4×Dic3—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178Dic12—48no—
179C22×Q8—31no—
180C22×Q8—61no—
181C22×D4—31no—
182C23×C4—61no—
183C2×Q16—64no—
184C2×Q16—122no—
185C2×Q16—122no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—61no—
189C4⋊Q8—32no—
190C4⋊Q8—61no—
191C4⋊Q8—61no—
192C4⋊Q8—61no—
193C22⋊Q8—122no—
194C22⋊Q8—122no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×Q8—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4D1212yes—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—62no—
215C2×C42—61no—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8D1212yes—
221C4⋊C8D1212yes—
222#103—64no—
223#102—64no—
224C4×C8—61no—
225C4×C8—61no—
226#95—62no—
227C23×C3#12011yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4C2×Q1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4#13711yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266D12—22no—
267D12—22no—
268D12—41no—
269D12—41no—
270C4×D3—81no—
271C4×D3—81no—
272C4×D3—81no—
273C4×D3—81no—
274Dic6—22no—
275Dic6—22no—
276Dic6—41no—
277Dic6—41no—
278Dic6—42no—
279Dic6—42no—
280Dic6—81no—
281Dic6—81no—
282C3×C8—42no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—61no—
286C2×Q8—32no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×Q8—62no—
290C2×Q8—61no—
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—121no—
294C2×Q8—121no—
295C2×Q8—121no—
296C2×Q8—121no—
297C2×Q8—121no—
298C2×D4—32no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—61no—
304C22×C4—22no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4C2×D1211yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318Q16—128no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—62no—
326C2×C8—121no—
327C2×C8—121no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—62no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—62no—
339C4⋊C4—124no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C22⋊C4—124no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—242no—
348C42—121no—
349C42—121no—
350C42C2×D1211yescharacteristic
351C42C2×D1211yescharacteristic
352C42D3×D411yescharacteristic
353C42#23911yescharacteristic
354C22×C3#39211yescharacteristic
355C22×C3#44711yescharacteristic
356C22×C3#44911yescharacteristic
357C22×C3#45111yescharacteristic
358C22×C3#45211yescharacteristic
359C22×C3#45511yescharacteristic
360C22×C3#45611yescharacteristic
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—82no—
371C3×C4#44711yescharacteristic
372C3×C4#44911yescharacteristic
373C3×C4#45111yescharacteristic
374C3×C4#45211yescharacteristic
375C3×C4#46611yescharacteristic
376C3×C4#46711yescharacteristic
377C3×C4#46911yescharacteristic
378C3×C4#47011yescharacteristic
379Dic3—42no—
380Dic3—82no—
381Dic3—81no—
382Dic3—81no—
383C23—62no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23#88211yescharacteristic
391Q8—62no—
392Q8—62no—
393Q8—121no—
394Q8—121no—
395Q8—122no—
396Q8—122no—
397Q8—241no—
398Q8—241no—
399D4—62no—
400D4—62no—
401D4—121no—
402D4—121no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—62no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4#87411yescharacteristic
425C2×C4#88211yescharacteristic
426C2×C4#88211yescharacteristic
427C2×C4#89511yescharacteristic
428C2×C4#89511yescharacteristic
429C2×C4#89511yescharacteristic
430C2×C4D3×D811yescharacteristic
431C2×C4#91211yescharacteristic
432C2×C4#91411yescharacteristic
433C2×C4#91411yescharacteristic
434C2×C4#91511yescharacteristic
435C2×C4#91511yescharacteristic
436C2×C4D3×Q1611yescharacteristic
437C2×C4#91911yescharacteristic
438C8—42no—
439C8—42no—
440C8—122no—
441C2×C3#164111yescharacteristic
442C2×C3#164411yescharacteristic
443C2×C3#164911yescharacteristic
444C2×C3#165211yescharacteristic
445C2×C3#199911yescharacteristic
446C2×C3#200111yescharacteristic
447C2×C3#200711yescharacteristic
448D3—82no—
449C22—122no—
450C22—122no—
451C22—122no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—241no—
456C22#467111yescharacteristic
457C22#474611yescharacteristic
458C22#475211yescharacteristic
459C22#477311yescharacteristic
460C22#478311yescharacteristic
461C22#480911yescharacteristic
462C22#481011yescharacteristic
463C4—21no—
464C4—21no—
465C4—82no—
466C4—122no—
467C4—241no—
468C4—242no—
469C4—241no—
470C4#477311yescharacteristic
471C4#478311yescharacteristic
472C4#480911yescharacteristic
473C4#481011yescharacteristic
474C4#486511yescharacteristic
475C4#486711yescharacteristic
476C4#488311yescharacteristic
477C4#488711yescharacteristic
478C3#1023311yescharacteristic, complemented, Sylow
479C2—242no—
480C2#6840711yescharacteristic
481C2#6846111yescharacteristic
482C2#6846211yescharacteristic
483C2#6850611yescharacteristic
484C2#6888611yescharacteristic
485C2#6888811yescharacteristic
486C2#6900911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1296D3
hypocenterC3#10233
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. C2×#4849C2semidirect
4. C2×#4484C2semidirect
6. #69582C2semidirect
7. C3×#1296C2semidirect
38. #1296D3semidirect
124. C2×D12Q16semidirect
478. C3#10233semidirect

Maximal subgroups

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

subgroupindex
C2×#48492
C2×#48462
#743692
C2×#44842
C2×#44832
#695822
C3×#12962
#102333