Finite Group Databasebeta

← #10694694

Subgroups — #10694694

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5114C211yesmaximal, characteristic
2C3×#1807C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#69529C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic
6#69510C211yesmaximal, characteristic, complemented
7#69509C211yesmaximal, characteristic
8#9781—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C22×#204C411yescharacteristic, complemented
13C2×C3×#106—21no—
14C3×#380C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5114—21no—
19#5114—21no—
20#5114—21no—
21#5114—21no—
22C2×#821C411yescharacteristic, complemented
23C2×#817C411yescharacteristic, complemented
24C2×#816C411yescharacteristic, complemented
25C2×C8×Dic3—21no—
26C2×#806—22no—
27C3×#340—21no—
28#4550—21no—
29#4527—22no—
30#4452—22no—
31#4451—22no—
32C2×#465—31no—
33#1996—31no—
34#1964—31no—
35#1807D311yescharacteristic, complemented
36C8×C22⋊C4—31no—
37#1520—31no—
38#1519—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12C2×C411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—42no—
44C22×C3×C8C2×C411yescharacteristic
45C22×C3×C8C2×C411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C22×C3×C8D411yescharacteristic
48C22×C3×C8D411yescharacteristic
49C22×C3×C8D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×#217D412yes—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4⋊Dic3C2×C411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C22×C3⋊C8—21no—
70C2×#204—21no—
71C2×#204—21no—
72C2×#204—42no—
73C3×#106—42no—
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95—21no—
77#830—21no—
78#830D412yes—
79#821—21no—
80#821—21no—
81#817—21no—
82#817—21no—
83#816—21no—
84#816—21no—
85C8×Dic3—42no—
86#806—44no—
87C22×SD16—31no—
88C23×C8C2×D311yescharacteristic
89C2×C22⋊Q8—31no—
90C2×#121—31no—
91#465—61no—
92#465—61no—
93#465—61no—
94#465—61no—
95C2×#106—21no—
96C2×#106—31no—
97C2×#103—31no—
98C2×#102—31no—
99C2×#99—62no—
100C2×C22⋊C8—31no—
101C2×C4×C8—61no—
102#380C2×D311yescharacteristic
103#380C2×D311yescharacteristic
104C4×C22⋊C4—31no—
105#342—62no—
106#340—21no—
107#336—61no—
108#328—62no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—42no—
125C2×#67—42no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—41no—
133C2×D12—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8C22⋊C411yescharacteristic
145C2×C3×C8C22⋊C411yescharacteristic
146C2×C3×C8C22⋊C411yescharacteristic
147C2×C3×C8C22⋊C411yescharacteristic
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—42no—
154#217—22no—
155#217—41no—
156#217—41no—
157#217SD1614yescomplemented
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171#204—42no—
172#204—42no—
173C22×Q8—31no—
174C22×D4—31no—
175C22×D4—61no—
176C23×C4C22×D311yescharacteristic
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—122no—
180C22×C8—42no—
181C22×C8—61no—
182C22×C8C4×D311yescharacteristic
183C22×C8C4×D311yescharacteristic
184C22×C8D1211yescharacteristic
185C22×C8D1211yescharacteristic
186C22×C8#6711yescharacteristic
187C22×C8#6711yescharacteristic
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192#121—61no—
193#121—61no—
194#121—61no—
195#121—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—61no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207#106—42no—
208#106—61no—
209#106—61no—
210#103—61no—
211#103—61no—
212#102—61no—
213#102—61no—
214#99—124no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C4×C8—122no—
218#95—21no—
219#95—21no—
220#95—61no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12711yescharacteristic
226C23×C3C2×SD1612yes—
227C23×C3#13512yes—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12411yescharacteristic
251#67—82no—
252#67—81no—
253#67—81no—
254C2×Dic3—24no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263D12—41no—
264D12—41no—
265D12—81no—
266Dic6—41no—
267Dic6—41no—
268Dic6—81no—
269C3×C8—21no—
270C3×C8—21no—
271C3×C8—21no—
272C3×C8#9912yescomplemented
273C3⋊C8—42no—
274C24—61no—
275C24D3×D411yescharacteristic
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—121no—
279C2×Q8—121no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—122no—
286C2×D4—122no—
287C2×D4—121no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4C2×D1211yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4#23911yescharacteristic
300C22×C4#23911yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303SD16—122no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—62no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8(C2×D3)⋊C411yescharacteristic
314C2×C8(C2×D3)⋊C411yescharacteristic
315C2×C8(C2×D3)⋊C411yescharacteristic
316C2×C8(C2×D3)⋊C411yescharacteristic
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C22⋊C4—34no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—62no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—121no—
336C42—121no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3#39011yescharacteristic
346C22×C3C4×SD1612yes—
347C22×C3#46512yes—
348C22×C3#49211yescharacteristic
349C22×C3#49511yescharacteristic
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—162no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—161no—
358C3×C4—161no—
359C3×C4C2×#9911yescharacteristic
360C3×C4#41011yescharacteristic
361Dic3—44no—
362Dic3—81no—
363Dic3—81no—
364Dic3—81no—
365C23—41no—
366C23—41no—
367C23—121no—
368C23—122no—
369C23—241no—
370C23—241no—
371C23#89311yescharacteristic
372C23#89511yescharacteristic
373C23D3×SD1612yes—
374C23#91612yes—
375C23#94011yescharacteristic
376Q8—121no—
377Q8—121no—
378Q8—241no—
379D4—121no—
380D4—121no—
381D4—241no—
382D4—242no—
383D4—241no—
384D4—241no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—64no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4#89311yescharacteristic
410C2×C4#89511yescharacteristic
411C2×C4C2×(C2×D3)⋊C411yescharacteristic
412C2×C4#94711yescharacteristic
413C8—21no—
414C8—21no—
415C8—21no—
416C8—122no—
417C8#80612yescomplemented
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3#196411yescharacteristic
423C2×C3#196811yescharacteristic
424C2×C3#197611yescharacteristic
425D3—161no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—241no—
435C22—241no—
436C22—241no—
437C22—482no—
438C22#485212yes—
439C22#486412yes—
440C22#498811yescharacteristic
441C22#516811yescharacteristic
442C22#517511yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—124no—
447C4—161no—
448C4—161no—
449C4—241no—
450C4—241no—
451C4—241no—
452C4C2×#80611yescharacteristic
453C4#500611yescharacteristic
454C3#978111yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—21no—
458C2—22no—
459C2—481no—
460C2#6952811yescharacteristic
461C2#7057411yescharacteristic
462C2#7065811yescharacteristic
subgroupG/H
centerC22#4988
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1807C2
socleC22×C3#390
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1807D3
hypocenterC3#9781
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C3×#1807C2semidirect
3. C8×#217C2semidirect
4. #69529C2semidirect
6. #69510C2semidirect
12. C22×#204C4semidirect
22. C2×#821C4semidirect
23. C2×#817C4semidirect
24. C2×#816C4semidirect
35. #1807D3semidirect
157. #217SD16semidirect
272. C3×C8#99semidirect
417. C8#806semidirect
454. C3#9781semidirect

Maximal subgroups

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

subgroupindex
C2×#51142
C3×#18072
C8×#2172
#695292
#695282
#695102
#695092
#97813