Finite Group Databasebeta

← #10819907

Subgroups — #10819907

Order 768.

884 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2957C211yesmaximal, characteristic, complemented
2#83903C211yesmaximal, characteristic
3#83812C212yesmaximal, complemented
4#79249C211yesmaximal, characteristic, complemented
5#72086C211yesmaximal, characteristic, complemented
6#72021C212yesmaximal, complemented
7#70702C211yesmaximal, characteristic, complemented
8#70417C212yesmaximal, complemented
9#70397C212yesmaximal, complemented
10#69897C211yesmaximal, characteristic, complemented
11#69889C211yesmaximal, characteristic, complemented
12#23196—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C3×#568C2212yes—
15#5832C2212yes—
16C2×#950C2212yes—
17#5745C2212yes—
18C2×#904C2211yescharacteristic
19C3×#444—22no—
20C3×#443—22no—
21C2×C3×#108C2211yescharacteristic
22C3×#421C2212yes—
23C3×#405C2211yescharacteristic
24#5240C2212yes—
25#5236C2212yes—
26#5191C411yescharacteristic, complemented
27#5183C411yescharacteristic, complemented
28#5175C412yescomplemented
29#5168C412yescomplemented
30#5155C411yescharacteristic, complemented
31#5151C411yescharacteristic, complemented
32#5145—24no—
33#5144—24no—
34#5139C2212yes—
35#5139C2212yes—
36C2×#822C2211yescharacteristic
37#5122C2212yes—
38#5106C2211yescharacteristic
39#5101C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#5006C2212yes—
42#4998C2211yescharacteristic
43#4895—22no—
44#4894—22no—
45#4555C2211yescharacteristic
46#4555C2212yes—
47#4532C2211yescharacteristic
48#3147—31no—
49#2957—31no—
50#2957—32no—
51#2957D311yescharacteristic, complemented
52#1976—32no—
53#1976—31no—
54#1976—31no—
55#1884—32no—
56#1884—32no—
57#1772—31no—
58#1772—31no—
59C2×C3×#41C2312yes—
60C2×#235C2312yes—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135—21no—
64C3×#135—21no—
65C2×C3×Q16C2×C411yescharacteristic
66C2×C3×SD16C2×C412yes—
67C2×C3×D8C2×C411yescharacteristic
68C3×#131—24no—
69C3×#131—24no—
70C2×C3×M16—42no—
71C2×C3×M16C2312yes—
72C2×C3×M16C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#117C2311yescharacteristic
75#950—22no—
76#950—22no—
77#950—22no—
78#950—22no—
79#947C2×C412yes—
80#944C2×C412yes—
81C2×#216C2×C412yes—
82C2×#215C2×C412yes—
83#940C2×C412yes—
84#939—42no—
85#936C2×C412yes—
86C2×#214C2×C412yes—
87C2×#213C2×C412yes—
88C4×#67—42no—
89#926C2312yes—
90#926C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C2×#208C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94#907—24no—
95D3×M16—42no—
96C2×Dic12C2×C411yescharacteristic
97#904—21no—
98#904—21no—
99#904—21no—
100#904—21no—
101C2×D24C2×C411yescharacteristic
102C2×#204C2×C412yes—
103#901—24no—
104C3×#108D412yescomplemented
105C3×#108D412yescomplemented
106C3×#104—24no—
107C3×#97—21no—
108C3×C4×C8—21no—
109#837—24no—
110#834—42no—
111#825—24no—
112#825—24no—
113#822D412yescomplemented
114#822D412yescomplemented
115#818—21no—
116#817—21no—
117#815—21no—
118#815—22no—
119C8×Dic3—21no—
120C8×Dic3—22no—
121C8×Dic3C2×C411yescharacteristic
122C8×Dic3C2×C411yescharacteristic
123C8×Dic3C2×C411yescharacteristic
124C8×Dic3C2×C411yescharacteristic
125#808—21no—
126#807—21no—
127#806—42no—
128C2×#135—32no—
129C2×#135—31no—
130C2×#135C2×D311yescharacteristic
131#568—32no—
132#568C2×D312yes—
133#532—32no—
134#495—31no—
135#495—32no—
136#495—31no—
137#494—31no—
138#493—31no—
139#492—32no—
140#444—22no—
141#444—64no—
142#444—62no—
143#443—22no—
144#443—64no—
145#443—62no—
146C2×#108—31no—
147C2×#108C2×D311yescharacteristic
148#427—31no—
149#427—31no—
150#421—32no—
151#421—32no—
152#421—32no—
153#421C2×D312yes—
154#410—32no—
155#410—32no—
156#405—32no—
157#405—31no—
158#405C2×D311yescharacteristic
159C2×C4×C8—31no—
160#339—31no—
161#339—32no—
162#339—31no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—22no—
166C3×#41—22no—
167C2×C3×Q8C22×C412yes—
168C2×C3×D4—22no—
169C2×C3×D4C22×C412yes—
170C22×C3×C4—22no—
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C2×#67—42no—
174C22×Dic3—21no—
175#235—22no—
176#235—22no—
177#235—22no—
178#235—22no—
179C2×D12C22×C412yes—
180C2×C4×D3—22no—
181C2×Dic6C22×C412yes—
182C3×Q16—21no—
183C3×Q16—21no—
184C3×SD16—22no—
185C3×SD16—22no—
186C3×D8—21no—
187C3×D8—21no—
188C3×M16—22no—
189C3×M16—21no—
190C3×M16—44no—
191C3×M16C2×D412yes—
192C3×M16C2×D414yes—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C2×C3×C8—44no—
196C2×C3×C8C22×C411yescharacteristic
197C2×C3×C8C22×C411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C2×C3×C8#4111yescharacteristic
202C3×C4⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C42—21no—
205#217—22no—
206#217—21no—
207#217—42no—
208#216—22no—
209#216—22no—
210#215—22no—
211#215—22no—
212#214—22no—
213#214—22no—
214#213—22no—
215#213—22no—
216(C2×D3)⋊C4—42no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219Dic3⋊C4—42no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—22no—
223C4×Dic3C22×C411yescharacteristic
224C4×Dic3C22×C411yescharacteristic
225#208—21no—
226#208C2×D412yes—
227C2×C3⋊C8—21no—
228C2×C3⋊C8C22×C411yescharacteristic
229C2×C3⋊C8C22×C411yescharacteristic
230C2×C3⋊C8C2×D411yescharacteristic
231C2×C3⋊C8C2×D411yescharacteristic
232C2×C3⋊C8#4111yescharacteristic
233C2×C3⋊C8#4111yescharacteristic
234Dic12—21no—
235Dic12—21no—
236D24—21no—
237D24—21no—
238#204—22no—
239#204—22no—
240ΓL(1,𝔽25)—44no—
241D3×C8—44no—
242C2×#41—32no—
243C2×#41C22×D312yes—
244#135—21no—
245#135—21no—
246#135—21no—
247#135—21no—
248#135—62no—
249#135—62no—
250#135—62no—
251#135—62no—
252#135—61no—
253#135—61no—
254#135—61no—
255#135—61no—
256C2×Q16—32no—
257C2×Q16—31no—
258C2×Q16C4×D311yescharacteristic
259C2×SD16—32no—
260C2×SD16—32no—
261C2×SD16—32no—
262C2×SD16C4×D312yes—
263C2×D8—32no—
264C2×D8—31no—
265C2×D8C4×D311yescharacteristic
266#131—24no—
267#131—24no—
268#131—64no—
269#131—64no—
270C2×M16—31no—
271C2×M16—42no—
272C2×M16—122no—
273C2×M16C22×D311yescharacteristic
274C2×M16C22×D312yes—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C4⋊Q8—32no—
278#127—32no—
279#124—32no—
280#124—32no—
281#121—122no—
282C4×D4—122no—
283#117—32no—
284#117—31no—
285#117C22×D311yescharacteristic
286C2×C42—31no—
287#108—32no—
288#108—32no—
289#108#6712yescomplemented
290#108#6712yescomplemented
291#107—61no—
292#107—61no—
293#106—61no—
294#106—61no—
295#104—24no—
296#104—64no—
297#104—64no—
298#104—64no—
299#99—122no—
300#99—122no—
301#97—21no—
302#97—62no—
303#97—61no—
304C4×C8—21no—
305C4×C8—31no—
306C4×C8—31no—
307C4×C8—31no—
308C4×C8—31no—
309C4×C8—62no—
310C4×C8—61no—
311C23×C3—22no—
312C23×C3H(5,𝔽2)11yescharacteristic
313C22×D3—22no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—42no—
319C3×D4—42no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4C23×C411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330#67—42no—
331#67—42no—
332#67—84no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339D12—22no—
340D12—22no—
341C4×D3—42no—
342C4×D3—42no—
343Dic6—22no—
344Dic6—22no—
345C3×C8—24no—
346C3×C8—22no—
347C3×C8C4×D411yescharacteristic
348C3×C8C4×D411yescharacteristic
349C3×C8C4×D411yescharacteristic
350C3×C8C4×D411yescharacteristic
351C3⋊C8—22no—
352C3⋊C8C4×D411yescharacteristic
353C3⋊C8C4×D411yescharacteristic
354C3⋊C8C4×D411yescharacteristic
355C3⋊C8C4×D411yescharacteristic
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364C2×Q8—32no—
365C2×Q8C2×C4×D312yes—
366C2×D4—22no—
367C2×D4—32no—
368C2×D4—62no—
369C2×D4—122no—
370C2×D4C2×C4×D312yes—
371C22×C4—22no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4C23×D311yescharacteristic
375Q16—21no—
376Q16—21no—
377Q16—62no—
378Q16—62no—
379Q16—61no—
380Q16—61no—
381SD16—22no—
382SD16—22no—
383SD16—62no—
384SD16—62no—
385SD16—62no—
386SD16—62no—
387SD16—62no—
388SD16—62no—
389D8—21no—
390D8—21no—
391D8—62no—
392D8—62no—
393D8—61no—
394D8—61no—
395M16—21no—
396M16—22no—
397M16—32no—
398M16—44no—
399M16—61no—
400M16—124no—
401M16D3×D412yes—
402M16C2×#6714yes—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—31no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—44no—
412C2×C8—61no—
413C2×C8—124no—
414C2×C8C2×C4×D311yescharacteristic
415C2×C8C2×C4×D311yescharacteristic
416C2×C8#23511yescharacteristic
417C2×C8#23511yescharacteristic
418C2×C8C2×#6711yescharacteristic
419C2×C8C2×#6711yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—122no—
424C22⋊C4—21no—
425C22⋊C4—62no—
426C22⋊C4—61no—
427C22⋊C4—122no—
428C22⋊C4—122no—
429C42—21no—
430C42—31no—
431C42—31no—
432C42—61no—
433C42—62no—
434C42—61no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3#51811yescharacteristic
439C22×C3#55011yescharacteristic
440C22×C3#55011yescharacteristic
441C2×D3—42no—
442C2×D3—82no—
443C3×C4—42no—
444C3×C4—81no—
445C3×C4C2×C4×D411yescharacteristic
446C3×C4C2×C4×D411yescharacteristic
447C3×C4D4211yescharacteristic
448C3×C4#55911yescharacteristic
449Dic3—22no—
450Dic3—41no—
451Dic3—42no—
452Dic3—82no—
453Dic3—81no—
454C23—22no—
455C23—62no—
456C23#100911yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—62no—
460Q8—62no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—62no—
466D4—62no—
467D4—122no—
468D4—122no—
469D4—244no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—62no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4C22×C4×D311yescharacteristic
483C2×C4C2×#23511yescharacteristic
484C2×C4C2×D3×D411yescharacteristic
485C2×C4#100411yescharacteristic
486C2×C4D3×#4111yescharacteristic
487C2×C4C22×#6711yescharacteristic
488C8—22no—
489C8—24no—
490C8—31no—
491C8—31no—
492C8—31no—
493C8—31no—
494C8—62no—
495C8C4×#6711yescharacteristic
496C8C4×#6711yescharacteristic
497C8C4×#6711yescharacteristic
498C8C4×#6711yescharacteristic
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—82no—
503C2×C3#236611yescharacteristic
504D3—82no—
505C22—41no—
506C22—42no—
507C22—82no—
508C22—122no—
509C22—242no—
510C22#554511yescharacteristic
511C22#555311yescharacteristic
512C22#583411yescharacteristic
513C4—42no—
514C4—62no—
515C4—81no—
516C4—121no—
517C4—122no—
518C4—241no—
519C4—242no—
520C4C4×D3×D411yescharacteristic
521C4#555911yescharacteristic
522C4C2×C4×#6711yescharacteristic
523C4D4×#6711yescharacteristic
524C3#2319611yescharacteristic, complemented, Sylow
525C2—21no—
526C2—21no—
527C2—21no—
528C2—82no—
529C2—242no—
530C2#8234311yescharacteristic
subgroupG/H
centerC4D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2957C2
socleC2×C3#2366
prime radical#83903C2
cosocleC22×C3×C4C24
hypercenter#2957D3
hypocenterC3#23196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. C3×#2957C2semidirect
3. #83812C2semidirect
4. #79249C2semidirect
5. #72086C2semidirect
6. #72021C2semidirect
7. #70702C2semidirect
8. #70417C2semidirect
9. #70397C2semidirect
10. #69897C2semidirect
11. #69889C2semidirect
26. #5191C4semidirect
27. #5183C4semidirect
28. #5175C4semidirect
29. #5168C4semidirect
30. #5155C4semidirect
31. #5151C4semidirect
51. #2957D3semidirect
104. C3×#108D4semidirect
105. C3×#108D4semidirect
113. #822D4semidirect
114. #822D4semidirect
289. #108#67semidirect
290. #108#67semidirect
524. C3#23196semidirect

Maximal subgroups

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

subgroupindex
C3×#29572
#839032
#838122
#792492
#720862
#720212
#707022
#704172
#703972
#698972
#698892
#231963