Finite Group Databasebeta

← #10883586

Subgroups — #10883586

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3059C211yesmaximal, characteristic, complemented
2C2×#5192C211yesmaximal, characteristic, complemented
3#83874C211yesmaximal, characteristic, complemented
4#83792C211yesmaximal, characteristic
5#72013C211yesmaximal, characteristic, complemented
6#71869C212yesmaximal
7#71864C212yesmaximal, complemented
8#70664C211yesmaximal, characteristic, complemented
9#70657C211yesmaximal, characteristic, complemented
10#70342C212yesmaximal, complemented
11#70334C212yesmaximal, complemented
12#28184—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C3×#536C2212yes—
16#5822C2211yescharacteristic
17#5812C2212yes—
18C2×#942C2212yes—
19C2×#901C2211yescharacteristic
20C3×#470—22no—
21C3×#467—22no—
22C3×#427C2211yescharacteristic
23C3×#419C2212yes—
24#5236C2211yescharacteristic
25#5236C2212yes—
26C2×#836C2212yes—
27#5225C2212yes—
28#5192—22no—
29#5192—22no—
30#5184—22no—
31Dic3×Q16—22no—
32#5145—22no—
33#5144—22no—
34#5139C2211yescharacteristic
35#5124C2211yescharacteristic
36C2×#819C2211yescharacteristic
37#5106C2211yescharacteristic
38#5101C2211yescharacteristic
39#5006C2212yes—
40#4559C2212yes—
41#4533C2212yes—
42#4531C2212yes—
43#4531C2212yes—
44C2×#473—31no—
45#3059D311yescharacteristic, complemented
46#2955—31no—
47#2950—31no—
48#2082—32no—
49#2009—32no—
50#1912—32no—
51#1905—32no—
52#1833—31no—
53#1814—31no—
54#1812—31no—
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C2×#235C2311yescharacteristic
58C2×C3×Q16—21no—
59C2×C3×Q16—21no—
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C2×C3×Q16D412yes—
63C2×C3×Q16D412yes—
64C3×#131—21no—
65C3×#131—21no—
66C3×#131—21no—
67C3×#131—21no—
68C2×C3×M16—21no—
69C2×C3×M16C2311yescharacteristic
70C22×C3×C8—21no—
71C22×C3×C8C2311yescharacteristic
72C3×C4⋊Q8—22no—
73C3×#124—22no—
74C3×#123—42no—
75C3×C22⋊Q8—24no—
76C3×#117C2312yes—
77#951—21no—
78#950—22no—
79#949—21no—
80Q8×Dic3—24no—
81Q8×Dic3—42no—
82C2×#216—22no—
83#942—21no—
84#942—24no—
85#942—42no—
86C2×#215—22no—
87#932—21no—
88#926—42no—
89#926C2312yes—
90#926C2311yescharacteristic
91C2×#208C2311yescharacteristic
92C2×#208C2312yes—
93C2×#208C2311yescharacteristic
94#901—21no—
95#901—21no—
96#901—21no—
97#901—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C3×#107—21no—
101C3×#106—21no—
102C3×#103—24no—
103#837—22no—
104#837—42no—
105#836—24no—
106#835—24no—
107#825—22no—
108#822—22no—
109#819—21no—
110#819—21no—
111#818—21no—
112#817—21no—
113#815D412yes—
114C8×Dic3D412yes—
115#806—42no—
116#575—31no—
117C2×#137—32no—
118C22×Q16C2×D311yescharacteristic
119C2×#131—31no—
120C2×#131C2×D311yescharacteristic
121#536C2×D312yes—
122#519—32no—
123#473—62no—
124#473—62no—
125#470—22no—
126#467—22no—
127#444—62no—
128#443—62no—
129#441—62no—
130C4×Q16—62no—
131#430—31no—
132C2×#108—31no—
133#427—31no—
134#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421—32no—
138#419—32no—
139#419C2×D312yes—
140C2×#101—32no—
141#410—32no—
142#405—31no—
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C3×#41—21no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C2×C3×Q8#4111yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4#4111yescharacteristic
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4C2411yescharacteristic
163C2×#67—21no—
164#235—21no—
165#235—21no—
166#235—21no—
167#235—21no—
168C2×D12#4111yescharacteristic
169C2×C4×D3—21no—
170C2×Dic6#4111yescharacteristic
171C3×Q16—22no—
172C3×Q16—22no—
173C3×Q16—22no—
174C3×Q16—22no—
175C3×M16—21no—
176C3×M16—21no—
177C3×M16—41no—
178C3×M16—41no—
179C2×C3×C8—21no—
180C2×C3×C8—41no—
181C2×C3×C8—41no—
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8#4111yescharacteristic
187C2×C3×C8#4111yescharacteristic
188C3×C4⋊C4—22no—
189C3×C4⋊C4—44no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—44no—
192C3×C42—22no—
193#217—21no—
194#217—22no—
195#216—22no—
196#216—44no—
197#215—22no—
198#215—44no—
199#214—22no—
200#213—22no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3—44no—
204C4×Dic3—22no—
205C4×Dic3—44no—
206C4×Dic3C2×D412yes—
207#208—21no—
208#208—21no—
209#208—22no—
210#208—21no—
211#208—21no—
212#208—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8C2×D412yes—
216ΓL(1,𝔽25)—41no—
217ΓL(1,𝔽25)—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41C22×D311yescharacteristic
222C22×Q8C22×D312yes—
223#137—61no—
224#137—61no—
225#137—64no—
226#137—122no—
227AGL(1,ℤ/8)—61no—
228AGL(1,ℤ/8)—61no—
229#135—62no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—62no—
235C2×Q16#6712yes—
236C2×Q16#6712yes—
237C2×SD16—62no—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—21no—
247C2×M16—31no—
248C2×M16—31no—
249C2×M16—32no—
250C2×M16—61no—
251C2×M16C22×D311yescharacteristic
252C22×C8—21no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255C4⋊Q8—22no—
256#124—22no—
257#123—42no—
258C22⋊Q8—24no—
259C4×Q8—64no—
260C4×Q8—122no—
261#117—31no—
262#117—32no—
263#117—122no—
264#117C22×D312yes—
265#108—62no—
266#108—61no—
267#108—61no—
268#107—21no—
269#107—61no—
270#106—21no—
271#106—61no—
272#104—62no—
273#104—62no—
274#104—122no—
275#103—24no—
276#103—64no—
277#101—64no—
278#99—122no—
279#97—32no—
280C4×C8—32no—
281C23×C3—21no—
282C23×C3H(5,𝔽2)11yescharacteristic
283C22×D3—21no—
284C3×Q8—24no—
285C3×Q8—21no—
286C3×Q8—24no—
287C3×Q8—21no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×D4—21no—
291C3×D4—21no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307#67—41no—
308#67—41no—
309C2×Dic3—22no—
310C2×Dic3—21no—
311C2×Dic3—42no—
312D12—21no—
313D12—21no—
314C4×D3—41no—
315C4×D3—41no—
316Dic6—21no—
317Dic6—21no—
318C3×C8—41no—
319C3×C8#12111yescharacteristic
320C3×C8#12111yescharacteristic
321C3×C8#12111yescharacteristic
322C3×C8#12111yescharacteristic
323C3⋊C8—22no—
324C3⋊C8—41no—
325C3⋊C8—42no—
326#41—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—61no—
331#41—61no—
332#41—61no—
333#41—61no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—31no—
340C2×Q8#23711yescharacteristic
341C2×Q8C2×#6714yes—
342C2×D4—21no—
343C2×D4—31no—
344C2×D4—61no—
345C2×D4#23711yescharacteristic
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—61no—
349C22×C4C23×D311yescharacteristic
350Q16—22no—
351Q16—22no—
352Q16—22no—
353Q16—22no—
354Q16—62no—
355Q16—124no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—62no—
360M16—21no—
361M16—21no—
362M16—41no—
363M16—41no—
364M16—61no—
365M16—61no—
366M16—61no—
367M16—62no—
368M16—61no—
369M16—62no—
370M16—121no—
371M16—121no—
372C2×C8—21no—
373C2×C8—32no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—61no—
377C2×C8—62no—
378C2×C8—121no—
379C2×C8—121no—
380C2×C8D3×D411yescharacteristic
381C2×C8D3×D411yescharacteristic
382C2×C8#23711yescharacteristic
383C2×C8#23711yescharacteristic
384C2×C8C2×#6711yescharacteristic
385C2×C8C2×#6711yescharacteristic
386C4⋊C4—22no—
387C4⋊C4—44no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—124no—
391C22⋊C4—22no—
392C22⋊C4—44no—
393C22⋊C4—61no—
394C22⋊C4—62no—
395C42—22no—
396C42—32no—
397C42—62no—
398C42—124no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3#53511yescharacteristic
403C22×C3#54611yescharacteristic
404C22×C3#55311yescharacteristic
405C2×D3—41no—
406C2×D3—81no—
407C3×C4—41no—
408C3×C4—44no—
409C3×C4—82no—
410C3×C4C2×#12111yescharacteristic
411C3×C4#52511yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#55311yescharacteristic
414Dic3—42no—
415Dic3—41no—
416Dic3—82no—
417C23—21no—
418C23—61no—
419C23#100411yescharacteristic
420Q8—24no—
421Q8—24no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—42no—
426Q8—61no—
427Q8—61no—
428D4—21no—
429D4—21no—
430D4—41no—
431D4—41no—
432D4—61no—
433D4—61no—
434D4—121no—
435D4—121no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4C2×D3×D411yescharacteristic
449C2×C4C2×#23711yescharacteristic
450C2×C4C2×#23711yescharacteristic
451C2×C4D3×#4111yescharacteristic
452C2×C4#100911yescharacteristic
453C2×C4C22×#6711yescharacteristic
454C8—41no—
455C8—62no—
456C8—121no—
457C8—122no—
458C8#93911yescharacteristic
459C8#93911yescharacteristic
460C8#93911yescharacteristic
461C8#93911yescharacteristic
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—81no—
466C2×C3#253511yescharacteristic
467D3—81no—
468C22—41no—
469C22—42no—
470C22—41no—
471C22—121no—
472C22—241no—
473C22#560711yescharacteristic
474C22#561411yescharacteristic
475C22#580611yescharacteristic
476C4—41no—
477C4—44no—
478C4—82no—
479C4—122no—
480C4—121no—
481C4—242no—
482C4#560611yescharacteristic
483C4#561811yescharacteristic
484C4C2×#93911yescharacteristic
485C4#582911yescharacteristic
486C3#2818411yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—81no—
491C2—241no—
492C2#8266611yescharacteristic
subgroupG/H
centerC2#82666
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3059C2
socleC2×C3#2535
prime radical#5822C22
hypercenter#3059D3
hypocenterC3#28184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #3059

Complements

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

NH ≅ G/Ntype
1. C3×#3059C2semidirect
2. C2×#5192C2semidirect
3. #83874C2semidirect
5. #72013C2semidirect
7. #71864C2semidirect
8. #70664C2semidirect
9. #70657C2semidirect
10. #70342C2semidirect
11. #70334C2semidirect
45. #3059D3semidirect
486. C3#28184semidirect

Maximal subgroups

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

subgroupindex
C3×#30592
C2×#51922
#838742
#837922
#720132
#718692
#718642
#706642
#706572
#703422
#703342
#281843